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v>
EASY MATHEMATICS
OR
ARITHMETIC AND ALGEBRA FOR GENERAL READERS
BEING AN ELEMENTARY TREATISE ADDRESSED TO
TEACHERS, PARENTS, SELF-TAUGHT
STUDENTS, AND ADULTS
BY
SIR OLIVER LODGE, F.R.S.
D.8C. LOND. ; HON. D.W. OX»X)BI>, VICTORIA, AND LIVKRPOOL.
LUD. ST. ANDREWS, OI.ASnoW, AND ABKRDRKN
PRINCIPAL or THE UNIVERSITY OF BIRMINOHAX
MACMILLAN AND CO., LIMITED
ST. MARTIN'S STREET. LONDON
1910
First Edition, 1905. Keprinted 1906, 1910.
GLASGOW : PRINTED AT THE UNIVERSITY PRESS BY ROBERT MACLEHOSE AND CO. LTD.
"The parent inherits a primal tendency to revert to the fixed and rooted form, while the child is ' free- swimming'; it is the natural explorer. And for ages we the parents through the teachers have been more and more successfully trying to train and educate our •free-swimmers' into fixed and rooted prisoners; thus atrophising or mutilating their discovering and inter- pretative powers just as our own were injured at the same age. ' Lady Welbt.
" There are several chapters in most arithmetic books that are wholly unnecessary .... but a writer of a 8chool-t>ook for elementary schools is not his own master ; he must comply with the often unwise demand of teachers and examiners." A. Sonnknschkin.
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PREFACR
Tuts book is written without the least regard to any demand but those of children and of life and mental activity generally. In places where the author is mistaken he cannot plead that ho has been hampered by artificial considerations. His object in writing it has been solely the earnest hope that the teaching of this subject may improve and may become lively and interesting. Dulness and bad teaching are synonymous terms. A few children are bom mentally deficient, but a number are gradually made so by the efforts made to train their growing faculties. A subject may easily be over-taught, or taught too exclusively and too laboriously. Teaching which is not fresh and lively is harmful, and in this book it is intended that the instruction shall ])e interesting. Nevertheless a great deal is purposely left to the enterprise of the student and the living voice of the teacher, and the examples given for practice are insufficient. The author has usually found that examples and illustrations are likely to be most serviceable, and least dull, when invented from time to time in illustration of the principles which are then being expounded ; but a supple- mentary collection of exercises for practice is necessary also, in order to consolidate the knowledge and establish the principles as an ingrained habit Wearisome over-practice and iteration and needlessly long sums should be avoided ; because long sums, other than mechanical moriey addition, seldom occur in practice, and es|)ccially because many kinds
Vlll PREFACE.
of future study, especially the great group of sciences called Natural Philosophy, will be found to afford plenty of real arithmetical practice; and even ordinary life affords some, if an open mind is kept. The cumbrous system of weights and measures still surviving in this country should not be made use of to furnish cheap . arithmetical exercises of preposterous intricacy and uselessness. There is too much of real interest in the world for any such waste of time and energy.
The mathematical ignorance of the average educated person has always been complete and shameless, and recently I have become so impressed with the unedifying character of much of the arithmetical teaching to which ordinary children are liable to be exposed that I have ceased to wonder at the widespread ignorance, and have felt impelled to try and take some step towards supplying a remedy. I know that many teachers are earnestly aiming at improvement, but they are hampered by considerations of orthodoxy and by the requirements of external examinations. If asked to formulate a criticism I should say that the sums set are often too long and tedious, the methods too remote from those actually employed by mathematicians, the treatment altogether too abstract, didac- tic, and un-experimental, and the subject-matter needlessly dull and useless and wearisome.
Accordingly, in spite of much else that pressed to be done, a book on arithmetic forced itself to the front. It is not exactly a book for children, though I hope that elder children will take a lively interest in it, but perhaps it may be con- sidered most conveniently as one continuous hint to teachers, given in the form of instruction to youth ; and it is hoped that teachers will not disdain to use and profit by it, even though most of them feel that all the facts were quite well known to them before. It is not intended to instruct them
PREFACE. IX
in subject-matter, but to assist them in method of presenta- tion; and in this a good deal of amplification is left to be done by the teacher. But it is of the first importance that the teacher's own ideas should be translucently clear, and that his or her feeling for the subject should be enthusiastic : there is no better recipe for effective teaching than these two ingredients.
For supplementary hints in connexion with the teaching of very small children, a subject which occupies the first four chapters, a couple of little books by Mrs. Boole recently pub- lished by the Clarendon Press may be mentioned ; and as a convenient collection of suitable examples for practice I suggest a set by Mr. C. O. Tuckey published by Bell and Sons. For supplementary information on the higher parts of the work such a book of reference as Chrystal's Algebra is probably useful.
The author has to thank Mr. T. J. Garstang, of Bedales School, Petersfield, Hampshire, and also Mr. Alfred Lodge, of Charterhouse, late Professor of Pure Mathematics at Coopers Hill, for reading the proofs and detecting errors and making suggestions.
PREFACE TO THE 1910 EDITION.
To this I have to add that several correspondents have kindly sent me little corrections and suggestions, which have been introduced into the book.
The only serious mistake detected wjis the rule for L.C.M. on page 104 ; for, by accident, little better than a parody of it appeared in previous editions. It is now corrected.
CONTENTS.
CHAPTER I.
The very beginnings. Counting. Extension or application of the idea of number to measuring continuous quantity. Intro<luction of the idea of fractions. Practical hints for teaching the simple rules. Addition. Subtraction. Multiplication. Multiplication of money. Division. Division of money. Origin of the symbols - jjp. 1-27
CHAPTER II.
Further considerations concerning the Arabic system of notation, and extension of it to express fractions decimally and duodecimally. Numeration pp. 28-.%
CHAPTER III.
Further consideration of division and introiluction of vulgar fractions. Extension of the term multiplication to fractions. Practical remarks on the treatment of fractions - pp. 37—14
cuAiTER rv.
Further consideration and extension of the idea of subtraction. Addi- tion and subtraction of negative quantities - pp. 45-50
CHAPTER V.
Ceneral tuition and extension of the ideas of multiplication and division to concrete quantity. First idea of involution pp. 51-57
CHAITER VI. Factors of simple numbers pp. 68-61
xn CONTENTS.
CHAPTER VII. Dealings with money and with weights and measures. Modern treat- ment of the rule called "practice." The practical advantages of decimalisation. Decimalisation of money. Typical exercises. Binary scale. Decimal system of weights and measures. Decimal measures. Angles and time. Further exercises - - pp. 62-85
CHAPTER VIII.
Simple proportion. Breakdown of simple proportion or *' rule of three " pp. 86-94
CHAPTER IX. Simplification of fractions pp. 95-101
CHAPTER X.
Greatest common measure and least common multiple. Rule for finding G.C.M. Algebraical statement of the process for finding G.C.M. pp. 102-108
CHAPTER XI.
Easy mode of treating problems which require a little thought
pp. 109-115 CHAPTER XII. Involution and evolution and beginning of indices - pp. 116-121
CHAPTER XIII.
Equations treated by the method of very elementary experiment. Further consideration of what can be done to equations
pp. 122-137 CHAPTER XIV.
Another treatment of equations. Introduction to quadratics
pp. 138-148 CHAPTER XV. Extraction of simple roots. Surds .... pp. 149-153
CHAPTER XVI.
Further consideration of indices. Fractional indices.
Negative indices pp. 154-161
CONTENTS. xiii
CHAPTER XVn.
Introdaction to logarithms pp. 162-164
CHAPTER XVIII.
Logarithms. Common practical base. Examples. Examples for practice. Fundamental relations .... pp. 165-176
CHAPTER XIX. Further details about logarithms pp. 177-183
CHAPTER XX. On incommensurables and on discontinuity • • . pp. 184-192
CHAPTER XXI.
Concrete arithmetic. The meaning of significant figures and practical accuracy pp. 193-197
CHAPTER XXII.
Practical manipulation of fractions when decimally expressed.
"Order" of numbers pp. 198-209
CHAPTER XXni. Dealings with very large or very small numbers - . pp. 210-218
CHAPTER XXIV. Dealings with vulgar fractions. Numerical verifications pp. 219-224
CHAPTER XXV. Simplification of fractional expressions - - - - pp. 225-229
CHAPTER XXVI. Cancelling among units pp. 230-234
CHAPTER XXVII. Cancelling in equations. Caution .... pp. 235-238
CHAPTER XXVIII.
Further cautions. Cautious of a slightly more advanced character
pp. 239-245
XIV CONTENTS.
CHAPTER XXIX.
Illustrations of the Practical Use of Logarithms.
(i). How to look out a logarithm, (ii). How to look out the number which has a giv^en logarithm. Examples. Logarithms of fractions, (iii). How to do multiplication and division with logs. pp. 246-257
CHAPTER XXX.
How to find powers and roots by logarithms. Exercises.
Roots of negative numbers pp. 258-264
CHAPTER XXXI.
Geometrical illustration of powers and roots. Further geometrical methods of finding square roots .... pp. 265-276
CHAPTER XXXII.
Arithmetical method of finding square roots - - pp. 277-282
CHAPTER XXXIIL
Simple algebraic aids to arithmetic, etc. Illustrations. Problems. Proof of square root rule. Cubes and cube root. Approximations
pp. 283-310
CHAPTER XXXIV. To find any power of a binomial. Exercise. Examples pp. 311-321
CHAPTER XXXV.
Progressions. Examples. Algebraic digression. General expression for any odd number. Arithmetical progression. Other series. Geometrical illustrations pp. 322-338
CHAPTER XXXVI.
Means. Examples. Mean or average of a number of terms. Weighted mean. General average. Geometric mean. Example pp. 339-351
CHAPTER XXXVIL
Examples of the practical occurrence of progressions in nature or art. Interest - - pp. 352-358
CONTENTS.
XV
PART II. MISCELLANEOUS APPLICATIONS AND INTRODUCTIONS.
CHAPTER XXXVIII.
Illustrations of important principles hy means of expansion by heat. Examples. Cubical expansion .... pp. 361-371
CHAPTER XXXIX.
Further illustrations of proportionality or variation.
Inverse variation. Summary pp. 372-400
CHAPTER XL.
Pumps and leaks. Leaks. Cooling. Electric leakage. Continuously decreasing G. P. Summary pp. 401-411
CHAPTER XU. Differentiation. Examples ....
CHAPTER XLII. A peculiar series. Natural base of logarithms
pp. 413-418 pp. 419-426
APPENDIX.
I. Note on the Pythagorean numbers (Euc. I. 47) pp. 427-430
II. Note on Implicit dimensions pp. 4.30-434
IIL Note on factorination pp. 434-43o
IV. Note on the growtli of population - - - pp. 435-436
CHArXER I.
The very beginnings.
Concerning the early treatment of number for very small children the author is not competent to dogmatise, but he offers a few suggestions, the more willingly inasmuch as he is informed by teachers that a great deal of harm can be and often is done by bad teaching at the earliest stages, so that subsequently a good deal has to be unlearnt. The principle of evolution should be recollected in dealing with young children, and the mental attitude of the savage may often be thought of as elucidating both the strength and the weakness of their minds.
Counting is clearly the first thing to learn; it can be learnt in play and at meals, and it should be learned on separate objects, not on divided scales or any other con- tinuous quantity. The objects to be counted should be such as involve some childish interest, such as fruit or sweets or counters or nuts or coins. Beans or pebbles will also do, but they should not be dull in appearance, unattractive as objects of property, and so not worth counting. The pips on ordinary playing cards will also serve, and they suggest a geometrical or regular arrangement as an easy way of grasping a number at a glance.
Counting should begin with quite small numbers and should not proceed beyond a dozen for some time, but there
L.E.M. A C
2 EASY MATHEMATICS. [chap.
is no object in stopping or making any break at ten. Several important fact's (the facts only, not their symbolic expression) can now be realised : such as that 3 + 4 = 7, that 7-4 = 3, that two threes are 6, and that three twos are the same, without any formal teaching beyond a judicious question or two. The lessons, if they can be called lessons, should go on at home before school age ; but, whether this initial train- ing is done at school or elsewhere, formal teaching at this stage should be eschewed, since it necessarily consists largely in coercing the children to arrive at some fixed notion which the teacher has preconceived in his mind — a matter usually of small importance. The children should form their own notions, and be led to make small discoveries and inventions, if they can, from the first. Mathematics is one of the finest materials for cheap and easy experimenting that exists. It is partly ignorance, and partly stupidity, and partly false tradition which has beclouded this fact, so that even influen- tial persons occasionally speak of mathematics as " that study which knows nothing of observation, nothing of induction, nothing of experiment," — a ghastly but prevalent error which has ruined more teaching than perhaps any other misconcep- tion of the kind.
As soon as small groups can be quickly counted, and dimple addition and subtraction performed with a few readily grasped and interesting objects — and the more instinctively such operations can be done the better, — the time is getting ripe for the introduction of symbols — for that arbitrary and conventional but convenient symbolism whereby !•! is de- noted by a crooked line, 5, and so on : a symbolism which the adult is only relieved from the necessity of elaborating and feel- ing difficult because of the extreme docility and acquisitiveness of childhood. It has already learned 26 symbols, it will patiently absorb nine or ten more, especially as they are soon
I.]
COUNTING.
found to be real conveniences; though if an adult wishes to realise tlie genuine difficulty of the process — always a most desirable thing to do — he should set to work to learn the Morse telegraphic alphabet, especially in the forms used for cable telegraphy.
I see no reason now why '.'..•'. should not be written 4 + 5 = 9, or soon afterwards why ; •(:) might not be written 5-2 = 3; but let no one suppose that these steps in nomen- clature are easy. The nomenclature introduced is just as hard as that of trigonometry or the calculus, only adult persons are accustomed to the one and are often unacquainted with the other. A set of little blocks, or some simple cheap squared paper lends itself to statements like the following :
5=
|
— |
I! H |
B
= 2+2+2
Fia.
3+3
4+2
5+1
I see no reason for troubling about the names " addition " and "subtraction," nor yet for artificially withholding them. If they come naturally and helpfully, let them come. Nothing is gained by artificial repression at any stage. Premature forcing of names is worse than artificial with- holding of them, but both are bad. If a gas, bubbling out of soda water and extinguishing a flame, is familiarly known as " carbonic acid," let it be called so : it is a help to have
4 EASY MATHEMATICS. [chap.
a label with which to associate observed properties, just as it is convenient to call a certain flower " daisy," or a certain star "Sirius." But to supply the label and withhold the object, to lecture about daisies or stars or numbers before they have been seen, is, let us politely say, unwise.
It seems to me that card games with counters may now be introduced, to enable the children to realise that their property may mount up beyond the smaller numbers that would be wholesome with sweets ; and they can learn how to group their counters into packets of six, or even into dozens, and then they will have simply to count their packets and the odd ones over. A child with four packets of six and three over would have a real idea of his wealth, though " twenty- seven " might still be a meaningless expression.
Diff'erently coloured counters are now serviceable to replace the packets, and thus the idea, but not the word, of different " denominations " will be imperceptibly arrived at : and it will be clinched by the at first unexpected discovery that even strangers will accept one white coin as equivalent to six much larger brown ones.
After this, some approach toward the admirable Arabic notation, whereby value is symbolised by place or position as well as by shape of digit, may be unobtrusively entered on. The idea of boxes or cases, or spaces of different value, in one of which odd counters or pennies are to be stored, another one in which packets, or silver coins, are to be kept ; and ultimately, but not too soon, a third one which is to be occupied by packets of packets, or gold coins ; if ever such wealth were attained.
While there is every advantage in thus emphasising atten- tion to the value or place of the digit, and so to a system of numeration, there are many reasons against concentrating attention on the particular number " ten " prematurely : it is
I.] COUNTING. 6
not a specially natural number, for one thing ; for another thing it is 80 large that ten packets of ten are unlikely to occur, whereas four packets of four, or six of six are quite possible. Another reason is that it is undesirable to suggest, what habit will subsequently only too erroneously enforce, that there is something special and divine about the number ten, so that the arrangement of digits 12 cannot help meaning a dozen. This false idea, due merely to habit, will not occur to a child, nor will he know intuitively that twelve pence make a shilling, or twenty shillings a sovereign ; indeed, strange to say, he is usually somewhat callous as to the importance of this pivot of human existence; and, though he soon gets to like coins, he attends chiefly to their number without much regard to their denomination, unless some are specially new and bright.
Having got so far, the conventional symbolism, in which practice has been quietly going on in the background during the few more formal school quarter-hours, may be extended, and the digit-symbols written in spaces drawn to represent the boxes, or on paper ruled into quarter-inch squares, which is cheaply and plentifully accessible, so that a 4 put in one box shall signify 4 counters, while a 4 put in another box shall signify 4 packets of say ten counters each, so that at the
end of a game 0 1 3 shall mean that the loser has no packets
and only three counters altogether, while another child may
have 3 I 0 ; that is, three complete packets and none over.
A third may have two packets and five over ; that is to say
2 I 5 , and another, the winner at the game, may possess , or in words, 1 packet of packets, 5 simple packets,
15 2
and 2 odd ones.
The packets may be represented by otherwise coloured counters, or the well known Tillich bricks or other Kinder-
6 EASY MATHEMATICS [ciia?.
garten devices can be employed for convenience ; the important thing is not prematurely (i.e. not until the under- lying reality has been essentially grasped) to proceed to the only partially expressive symbolism 25 or 152, which to us by mere habit looks so living and significant. Let the elementary teacher reflect that to a mathematician the symbol
I «"* dx looks equally living and significant, and be not hasty
with the children.
At the same time there is no need for artificial delay. A child brought along the right lines ^vill jump forward A\athout difficulty, will recognise the places without the boxes, will get accustomed to the savage's mode of reckoning by tens without being encouraged to go through the savage process of counting on his fingers, and before long will be able to interpret such a complicated symbolism as 50327, or .£175. 16s. lid. The last, indeed, is properly spoken of as " compound " instead of simple, for in it " scales of notation " are badly mixed up. The reckoning proceeds by tens, by dozens, and by scores, sometimes one and sometimes another, occasionally by quarters also.
The poor child who finds himself able to master this and the operations which arise out of it, need not be deterred by any legitimate obstacles in mathematics until he comes to its really higher walks, beyond simple differential equations : a step which he will not be called upon to take at all unless he is bom to be a mathematician, in which case difficulties of any ordinary kind will barely be felt.
The operations of addition and subtraction may now be extended. 7 + 5 may be done into a packet of one dozen, or into a packet of ten and two over, and denoted by 1/- or 1 2 according to which plan of grouping is adopted.
So alco 8 + 7 may be called either 1/3 or 15, the former
I.] COUNTING. 7
being the custom if they are pennies, the latter if they are nuts.
It is necessary to apologise to children for this needless complication; but they inherit some things that are good, to make up for several things that are stupid, and therewith they will have to be content : —
8 + 7 + 9, if shillings, will be grouped differently again, and be denoted by £1. 4s.; if pennies, they will be denoted thus, 2/-; if ounces, they will be written 1 lb. 8 oz.; if feet, they will be called 8 yards ; if farthings, they will be written 6d. ; if oranges, they will be called 2 dozen; but if boys, they will be \vritten 24.
I do not recommend anyone to confuse the minds of children by pointing out these anomalies, or by quoting a sample of them simultaneously as above. Children will not detect their true character, but will docilely receive them as if all this clumsiness were part of the laws of nature. This may account for their disinclination later on to make acquaintance with any more of those laws than they can help, but at this stage they are docile and assimilative enough : they can at this stage be taken advantage of with impunity. But I should very much like to confuse the minds of some teachers, and of some school inspectors — especially some varieties of school inspector and university examiner — and get them into a more apologetic and humble mood at having to insist on filling the mind of a child with any more of these artificial insular conventions than is absolutely necessary in the present stage of British political and commercial wisdom.
It is undesirable to hasten forward to numbers involving 3 digits too quickly ; they can bo mentioned and illustrated when convenient, but real work should for some time be limited to 2 figure numl>ers, because in these the real principles can be recognised and grown accustomed to in the simplest way.
8
EASY MATHEMATICS.
[chap.
The early operations in which practice can be given are such as the following: Suppose counters are employed and that little cases have been made which just hold six or ten or any convenient number, suppose ten ;
Then 13 will stand for one packet of ten and three counters over; 17 added to it will amount to two packets and ten counters over; which the child, if encouraged by the sight of an unused case available, may wish to make up into 3 whole packets, and so recognise the propriety of denoting the number by 30
Similarly 15 + 17 will make up into three packets and 2 over, which may be shown thus :
make which equals
while 25 + 37 will equal five packets and twelve over, or six packets and 2 over; 29 + 37 = 66, but it is equally per- missible to keep it as 5 packets and 16 counters over, if it should happen to be convenient — as it sometimes is.
To take 4 from 17 is easy, but to take 9 from 17 will involve emptying a case; and only
8 counters will be left.
To take 13 from 25 can be done by removing 1 case and 2 counters ;
to take 15 from 25 is also easy ; but to take 16 from 25 involves the breaking up of a packet.
|
tens 1 1 |
ones 5 7 |
|
2 |
12 |
|
3 |
2 |
1.] COUNTING. 9
After a time these operations can be followed when nothing concrete is present; but abstractions are not natural to children, and before calling upon them to follow a difficult conventional subtraction sum like
82 37 45
the operation of breaking up packets should be introduced into the symbolism which is employed faintly to shadow the concrete reality.
It is perfectly right to speak of 3 packets and 13 loose counters, although they may be more compactly grouped as 4 packets and 3 counters. So if we have to subtract say 7 from 43 we shall first break up one of the four packets, so as to turn 43 into 3 packets and 13, and then subtract the
7 without difficulty, leaving what is abbreviated into 36. Hence before doing the above conventional little sum,
8 packets and 2 should be expressed as 7 packets and 12, or
From this 3 packets and 7 have to be removed,
7 12
leaving obviously 4 packets and 5. AMierefore 82-37 = 45 without any argument.
The abbreviated form of the above breaking-up operation, called borrowing, will now gradually almost suggest itself, if many suras of the kind are given to be done. But the best and easiest method of subtraction is the complementary method, and if this is taught from the first, the complexity of lx)rrowing becomes unnecessary.
The adult cannot too clearly realise that many of the operations to which he has grown accustomed are lal)our- saving shorthand devices with the vitality and principle abbreviated out of them. Quite rightly so for practical pur- poses but not for educational purposes. The race invented
10 EASY MATHEMATICS. [chap.
them at first in more elaborate shape, and gradually abbre viated them into their present-day form. The child will likewise get accustomed to this form in due time, but he should not be over-hurried into it.
After adding two numbers for some time we may proceed to add more than two,
and find that 7 -f- 9 -t- 6 = 22, etc. ; also that 7 4- 7 4-7 = 21, and it is natural to speak of this as three sevens.
So also the fact that 54-5 + 5-f5 = 20 will naturally be quoted as four fives make twenty ; and thus the essential idea of multiplication will arrive, as a shorthand and memorised summary of the addition of a number of similar things, without any use of the narm multiplication or any feeling of a new departure. To find the value of three seventeens, that is, to group them into tens and ones, is a problem for an afternoon, and if it be done with counters in the first instance, and ultimately with symbols, the meaning of the operations having been realised beforehand with the counters, so much the better.
The operation of adding or multiplying means grouping the whole number into tens and ones, or into hundreds, tens, ones, etc., instead of in the given groups.
A child must not be expected to be able to formulate his conception of the operations, or to express them accurately in words, at this stage. It is a capital exercise later, but it is enough at first for him to realise the meaning of what he is doing in the back of his mind. From time to time he can be encouraged to interpret processes into words, but they must have become familiar first. To be able to apply a rule, from a precise statement in words of what has to be done, is an adult accomplishment, often not reached by adults. To dissect out and state a rule in words, from a knowledge of what the
1.] EDUCATION. 11
operation really is, is perhaps easier, and is a desirable gift, but it is a training in the use of language rather than in the subject matter of the craft. It is most appropriate and valuable prac- tice for children at the proper stage, a stage reached much earlier with some children than with others. Children who reach the word-expression stage late are usually called "stupid." If this adjective implies a stigma it is usually undeserved. There is a performance appropriate to each stage of develop- ment, and opprobrious epithets are generally employed by those who seek to force things several stages too soon. A highly trained and clever dog would soon prove himself " stupid " if tested by a formula, or by words even of only 3 letters. An adult who can hum or whistle an air may be told that he ought to be able to sit down and write it in the recognised musical notation. Similarly he ottght to be able to read off a piece of music handed to him. He might resent being called stupid if he found it difficult to do these, to some, so simple things.
" Badness " of many kinds may exist in spoiled children (and there are several ways of spoiling them), but badness in un- spoiled children is rare, and stupidity is almost non-existent unless they are physiologically out of order and therefore mentally deficient. Stupidity is however a product easily cultivated by improper feeding, especially improper mental feeding. The " badness " of children is largely the effort which nature makes at self-preservation ; for inattention and laziness are the weapons whereby an attack of mental indigestion can be warded off.
The only fault with very young children is that they are too good, and therefore too easily damaged. Later on, a spirit of rebellion acts as a preservative, but it would be better to dispense both with the rebellious spirit and with the causes which necessitate it.
12 EASY MATHEMATICS. [chap.
Keturning from this digression, which is either false or else of very extensive application, to our immediate subject, viz., the introduction of the fundamental operations to be performed on number, — and remember that what are called the first four simple rules are tremendously fundamental and important, more important than anything which follows, until involution, evolution, and logarithms are arrived at, — we must exercise children in Multiplication and teach them something of the multiplication table, at first experimentally, but afterwards by straightforward memory work, for it is one of the things with which the memory may be rightly loaded. We can next recognise that Division too can be unceremoniously introduced by trying to split up numbers into equal parts. The endeavour to share sweets or fruit or cards or counters is an obvious beginning. Then, since children are docile, they can be asked to split up 2 packets and 7 into three equal groups, or they can be asked to split up 2 packets and 4 into eight equal groups, and so on ; for no reason assigned. But it must be recognised that the operation of division in general is rather hard, and involves a good deal of tentative procedure or guess work. In other words it involves the rudiments of experiment and verification. Gradually, when the multiplication-table is fairly known over some little range, children can be encouraged to apply theory before practice and actually to think out the result before trying it ; but this is a lesson in deductive reasoning, and represents the nascent beginnings of a loftier mode of procedure than ordinary adults are accustomed to apply to their affairs. When asked to split 28 into four equal heaps, it is an application of pure theory to remember that 4 sevens are 28 and then to count out seven counters into each heap at once. The empirical mode would be a method of dealing out singly into four groups and then counting the result. It is easily done with ordinary plajdng cards, but
I.] EDUCATION. 13
its value as training is much enhanced if theory is applied lirst.
If for instance 30 cards were given, to be dealt to four players, the residue that will not go round to be put in the middle or pool, a decided effort is required for a child to perceive that there will be two for the pool and seven for each player : but if he could have time allowed him so to think it out, and then to make the experiment, he would be conscious that his powers were developing, and he would in reality be introduced to the first beginnings of a mode of comprehending nature such as is in the higher stages reserved for men of science, — using the term science in its most com- prehensive signification.
It is very often a mistake for teachers to suppose that some things are easy and other things are hard ; it all depends on the way they are presented and on the stage at which they are introduced. To ascend to the first floor of a house is difficult if no staircase is provided, but with a proper staircase it only needs a little patience to ascend to the roof. The same sort of steps are met with all the way, only there are more of them. To people who live habitually on the third floor it is indeed sometimes easier to go on to the roof than to descend into the basement. Educators should see that they do not forcibly drive children in shoals up an unfinished or ill- made stairway, which only the athletic ones can climb. It is extremely difficult in familiar subjects not to go too fast. The effort sometimes results in a process of going too slowly, which is wearisome and depressing and the worse fault of the two.
Extension or Application of the idea of number to measuring^ continuous quantity. So far we have been employing number to count discrete objects, and to perform simple operations of addition, and
14 EASY MATHEMATICS. [chap.
the like, among them. It is now appropriate to introduce the idea of multiples of a unit, so that one thing can be twice as long or twice as heavy as another, without being in another sense "two" at all. The lines on ruled paper enable one easily to draw across them a line twice or three times or six times as long as another. So also letter-scales can be used to show that a penny is twice as heavy as a half-penny, that a half-crown weighs how many sixpences, and the like.
Given a foot-rule they can measure the size of furniture, or of books. Given a few ounce weights they can make very rough estimates of the weights of things that have or might have to go by post.
It is desirable not to dwell on these things at this stage, but simply to accustom a child to recognise a rod 6 inches long, and such like, and to see instinctively and without formula or expression that number may he applied to con- tinuous magnitude hy the device of a unit of measurement. Adults may realise that there is a real step here, by remem- bering that if they were set to express the strength of an electric current, or the electric pressure on a main, or the strength of a magnet, numerically, they would be nonplussed, unless they knew something about the units which within a generation or two have been introduced for the purpose, — the ampere, the volt, and the line of force ; so that nowadays the British workman is able to speak familiarly of an electric current of so many amperes — (sometimes pronounced " hampers "). There is nothing really numerical about the length of a table or the height of a door or the weight of a sack or the brightness of a lamp or the warmth of a room or the length of a day ; and its numerical expression will depend entirely upon what conventional unit is em- ployed, and may vary in different countries accordingly. Do not assume therefore that a child is stupid to whom
J.] EARLY OPERATIONS. 15
an application of arithmetic to weighing and measuring is not obvious.
Introduction of the idea of fractions
In the same way the idea of fractions can naturally occur ; a halfpenny and a half ounce and a half inch being fairly easy examples : but not the easiest. There can be no doubt that just as niunbering ought not to begin with continuous quantity but with discrete objects, so fractions should be first displayed as actually cut and broken things.
The proper fractions to begin with are halves and quarters and eighths; and apples do admirably for that. Oranges suggest further modes of subdivision, except that the removal of the peel may constitute an unexpressed but felt complication.
Folding of a ribbon or paper easily leads to thirds and any other fractions wanted. Any child can be sent to cut off a quarter of a yard, or a yard and a half, or even a foot and three quarters, of tape. But again do not be surprised if this last mode of specification is found occasionally puzzling : it is of the nature of a problem, and requires time. The form of difficulty which may properly occur to some children is " a half of what " or " three quarters of what " : and if they bring the foot and the 3 quarters all separate, i.e. if they cut the tape into four pieces altogether, that is very well for a beginning. They should not be supervised or fidgeted during the solution of a problem. They cannot think if they are. These expres- sions, 6 miles and a half, etc., have a conventional ring, to which we have grown thoroughly accustomed, but they are shorthand terms not really fully expressive : it might possibly ambiguously suggest 9 miles.*
The measure of time in half and quarter hours may also be
*C£. George Meredith's " Rhoda Fleming," Chap. 3.
16 EASY MATHEMATICS. [chap.
appealed to as illustrative of fractions ; but in this form they are somewhat abstract. The divisions on a foot rule or metre scale are easier, and for further progress are indeed the easiest illustration to be borne in mind. Afterwards, the halfpenny, the half crown, the half sovereign, etc., and the other fractions of money may be brought in, whenever they appear to be natural.
Practical hints for teaching the simple rules.
Simultaneously with all this introduction of fresh concep- tions, mechanical practice in operations with symbolised num- bers can be proceeded with : —
Addition.
About addition there is little to be said : the idea of packets must have made everything concerning the carrying-figure easy.
The principle being understood, it is now only a question of practice in attaining quick and sure execution, as quick and sure as it is worth while to aim at at this stage.
Addition of money is a useful accomplishment, and since the packets into which it is to be made up are varied, it affords good practice, involving a certain amount of constant thought and care. It is wrong to try to force a child to acquire the facility of a bank clerk in adding up long columns : that will come in due time and is quite a useful faculty : it is clearly a thing to acquire in commercial schools, but not while still young and receptive.
It is well to begin thus :
£ s. d.
6 . 15 . 3
5.4.9
12 . — . —
I.] EARLY OPERATIONS. 17
where the packets to be carried forw^ard are complete. Then change the 3 into a 4 or 5 and get 1 or 2 pence over ; then change the 15 into 16 or 17 and get some shillings over, and so on, gradually. Always begin with what illustrates the procedure in the simplest form and gradvuUy complicate it.
There is one remark about addition worth making. In adding say 43 + 8, some beginners are told to bethink them- selves that 3 + 8 = 11, and so arrive at the digit 1 of the result ; while others are told to think of the sum as 43 + 7 + 1, stepping on to the intermediate stage of the complete packet en route to 51 ;
e.g, 77 + 9 = 77 + 3 + 6 = 80 + 6 = 86.
Perhaps it is permissible to introduce this aid as a temporary measure, but ultimately addition ought to proceed by instinct and without thought. It is a mechanical process, and a bank clerk who stopped to think, while adding, would be liable to make a mistake.
Subtraction.
There appears to be no doubt now but that the "shop method " of subtraction is the handiest and quickest : it may as well, therefore, be acquired almost from the first.
37 Three and four make seven.
15 One and two make 3.
24 Put down the figures in black type.
Verify by adding 13 to 24. Take another example :
174 Q<, Eight and six make fourteen.
—zz Nine and one and seven make seventeen. 7d
I do not think that children need find this method hard or
L.K.M. B
18 EASY MATHEMATICS. [chap.
unnatural, but practice will be needed before going on to money sums, such as :
£ s d
- - * ■ Four and seven make 11.
2j * g * 4 Eight and eighteen make 26.
"~^ 7^ ^ 12 and 5 make 17.
Verify by addition of the two lower lines. Get the children never to pass and hand in a result as finished unless they have taken pains to assure themselves that it is right. This does not mean that they are not to hand in a confessedly unfinished sum if they find they cannot do it without help.
Multiplication.
At good Kindergarten schools, a step beyond the first in multiplication is often introduced by some such questions as this :
How many stamps will three children haA^e if each has 14 ? They first add 14 three times, and they are allowed to do that till they find it quicker to use the phrase " three times," which, if they know the multiplication table, they can hardly help doing in the process of adding; and so they get to be able to give the answer "3 times 14" instantly, without necessarily having had time to realise what the operation would result in when executed. This kind of intermediate answer is to be encouraged.
In entering upon multiplication, employ a single digit as one factor, and do it first as an addition sum, e.g. :
142 142 142 142 142
710
I.] EARLY OPERATIONS. 19
then proceed
£. 8. d.
142 173 125 12 . 7 . 6
5 5 8 4
710 865 1000 49 . 10 . —
doing this latter also by addition first :
£. 8. d.
12 . 7 . 6
12 . 7 . 6
12 . 7 . 6
12 . 7 . 6
49 . 10 . 0 but it is well to lead up to the last type of sum by simple cases, e.g. 4 times 2/6 = 10/-; 4x5/- = £l; 4.x'JI^ = ZQI- = £1 10/- ; 4 X 3d. = 1/- ; 4 x 1/3 = 5/- ; 4 x 10/- = £2 ; 4x11/3 = £2. 5s.
Do not hurry. If the child can be allowed time to see a connexion between the three last statements, or the like, so much the better. The value of these trifles is when they are discovered ; there is hardly any virtue in them if they are pointed out, and none at all if they are laboriously emphasised. If they are not glimpsed let them pass. We all of us doubtless miss discoveries, most days, for lack of attention and insight.
Next comes multiplication with two digits : first by numbers like 10, 20, 70, etc.
Multiplying by ten means making every unit into a packet, every packet into a set of packets, and so on.
AVherefore
1|3|4 0
0 I I I 3 I 4 when multiplied by ten becomes the 1 being shifted into the empty compartment, and every other digit likewise moved ; the unit box, or box for single counters, being left empty.
20 EASY MATHEMATICS. [chap.
If we multiply by 20, the shift takes place similarly, and also every digit is doubled, yielding 2680.
So nowstart multiplying a number like 53 by 20, getting 1060. Then a number with a carrying figure from the units place,
^^^® 47 X 20 = 940 ;
then one involving two carryings, like
.r.A ar.r.r. 57 X 20 = 1140,
and so on. '
Next take multiplication by a number like 23. Let it be realised once more that 23 is short for 20 + 3, so that it may be felt to be natural to multiply by 20 and by 3 successively and add the results, which is what we do. At first let it be worked in this way ; for instance, to find 824 X 23
= 20x824 or 16480 and 3x824 or 2472
added together make 18952
but gradually get it abbreviated into the usual form
824 23
1648 2472
18952 without necessarily putting in the cipher after the digit 8.
There appears to be no doubt now that it is best in mul- tiplication to begin with the most important figure, so that sums look thus :
173 173 768
56 156 107
865 173 768
1038 865 5376
9688 1038 82176
26988
I.]
EARLY OPERATIONS.
21
a trivial matter to all appearance, but helpful in later stages, and therefore better practised from the first.
[In my opinion it is thoroughly unwise to reverse the digits of any factor before multiplying with them, though some teachers of experience think otherwise.]
Multiplication of money, at least of English money, is more difficult of course, because, in the specification of money, scales of notation are so mingled; thus, depicting the com- partments and labelling them when necessary :
|
£ |
8 |
• |
d |
• |
||
|
4 |
3 |
5 |
1 |
7 |
1 |
1 |
at the double line the scale is changed from ten to a dozen, and at the treble line it is changed again from ten to a score.
So if we have to double this sum, even doubling it is com- plicated, and results in
£871 . 15 . 10 Let no one suppose that this is an easy process, for a child or anyone.
It could in this case be performed more easily by simple
addition :
£ 8. d.
435 . 17 . 11
435 . 17 . 11
871 . 15 . 10 but that is hardly applicable to larger factors. Not only is doubling hard, but multiplying even by 10 is hard too. Take the amount £5. 17s. lid. and multiply it by 10; it becomes the totally different-looking amount
£58. 198. 2d. Multiplving by 12 will of course turn all the pence into shillings, and multiplying by 20 will turn shillings into
22 EASY MATHEMATICS. [chap.
pounds, but multiplying by any other factor is hard, and is probably best deferred for the present.
If multiplication of money by a number like 23 is wanted^ not only must the 23 be divided into two parts 20 + 3, and the multiplication done separately as usual, but it is generally needful to resolve the 20 into two parts also, say 10+10, and then add the three results together.
If however multiplication by 24 were desired, it v/ould be possible to split it into two factors 8x3, and to multiply first by one and then the result by the other, without any addition of results ; but there is great danger of confusion here, and there are plenty of what are considered and are really "higher" parts of arithmetic which are much easier than this. Low class or unskilled labour is not necessarily easy : it may in some cases be terribly laborious, like un- loading a ship. Another way of multiplying by 20 is to split up 20 into the two factors 2x10 or 4x5 and employ them successively. In that case the result of multiplying by 23 is ultimately obtained by multiplying the original sum by 2, the result by 10, the original sum by 3, and then adding the last two results.
The fact is that with money specified in the customary English way, the only operations that can comfortably be performed on it are addition and subtraction, and these are the only really frequent operations in practice.
To apply multiplication and division it is best to express the money differently, in fact to decimalise it before commenc- ing operations. This will be explained later (Chap. YII.), though of course to most teachers it is a process already well known. It ingeniously evades the difficulties caused by our currency, and converts its treatment into almost a worthy intellectual exercise.
L] EARLY OPERATIONS. 2S
Division.
First take simple sums to introduce the notation, such as
y = 3, or 214-7 = 3.
21 Let it be realised also that -^ = 7, and that 3x7 = 21.
There are a multitude of interesting things to be learnt before long about factors, and criteria for division, etc., but not yet; let the child learn how to perform the process on numbers of which he knows no factors. But at first do not trouble him with remainders : let him at first be given simple sums that divide out completely.
Thus we can tackle such sums as
71491036 .. , . I,, , .,, 491036 ^^,,^
' , which should be also written — = — «= 70148.
70148 7
The treatment of remainders is for subsequent consideration.
It is well to give the complementary sum 7x70148, especially since the teacher will thus have but little trouble in checking results — at least until the child finds out the dodge — a discovery which is to be encouraged like all other discoveries.
At good Kindergarten schools, a step beyond the first in division is often introduced by some such plan as the following :
To prove that 96 -r 4 = 24.
Take nine bundles and six sticks over, deal out into four places, two bundles in each place; and then deal sixteen sticks, four into each place, giving the result 24. And so on with other numbers.
As soon as short division is thoroughly understood, long division may introduce itself as an assistance when more difficult divisors are involved ; for instance 988 -f 1 9. This
24 EASY MATHEMATICS. [chap.
being difficult to do by short division, where the multiplica- tion and subtraction have to be done in one's head, it is permitted to write the operations down, at first both of them, thus:
19)988(5 95
3
Afterwards, perhaps, only the result of them, 3, which in short division would likewise not appear, nothing but the quotient being written in short division. Long division is therefore not harder than short division, but easier : it is the identical process, only written out more fully, so as to be applicable to harder sums. It is the largeness of the figures dealt with that makes it hard.
For long division it appears to be felt that by aid of the shop system of subtraction there is no undue strain on the brain by the use of the abbreviated method.
I would have it understood however that long division sums are among the moderately hard things of life, and that mathematicians seldom trouble themselves to do them. They can be deferred until many other things have been done and some familiarity with figures acquired. It is a gymnastic exercise to perform even so simple a long division sum as the following, and if attempted too early will involve strain.
72)5286456(73423 246 304 165 216 This is the process :
Sevens in 52 1 guess 7 times and write 7 as the first digit in the quotient, then 7 x 2 = 14, to which add 4 to make 18. Seven sevens = 49, say 50, to which add 2 to make 52 ; record only the figures here printed in black type; bring
t] EARLY OPERATIONS. 25
down the rest of the dividend 6456 or as much of it as is wanted ; only 6 is wanted so far, and we guess 3 for the next digit in the quotient. Three times 2 and 0 make 6, three times 7 and 3 make 24. Bring down more of the dividend, say 456, or at least 4, and guess 4 for the next
digit.
4x2 = 8 and six are 14. 4 X 7 = 28, say 29, and 1 are 30. Bring down the 5, and guess 2 for the next digit of the quotient ; twice 2 = 4 and 1 = 5, etc., and then finally bring down 6, and it goes 3 times exactly.
If the sum is neatly done the corresponding places are vertically under each other, a detail of appearance emphasised by the presence of a decimal point.
Let the result be written
5?^ = 73423.
Do not forget to set also the complementary sum
72 X 73423.
It will be well also to set the exercise whose result is
5286456 ^„ Mi^A.yo = 72, as a separate sum.
If the connexion is automatically noticed, it is well ; it will prepare the mind for the later-on extremely important and constantly occurring connected relations,
if T = c, then - = b. and be ^ <l be
but refrain from using this abstract language at present.
Watch for the time when it can without strain be naturally
introduced. It is a great help when that step is reached,
and it represents a vital stage of real mental progress. The
mind should be soaked with particular instances however
before generalisations can be usefully and permanently grasped.
26 EASY MATHEMATICS. [cuap.
Division of money is of course difficult, even when the divisor is a small number, because of our complex system of notation, unless the money is first expressed in decimal form.
To divide by 23 moreover it is not correct to divide by 20 and then by 3 and add the results, as it was with mul- tiplication. A long-division sum is necessary, and that is no joke with money as usually specified. Division by 24 can indeed be done in two stages, by help of its factors 3 and 8 consecutively applied, but that only masks the essential difficulty by a device applicable only to special cases.
My object in introducing these remarks about complex money-sums here (and the same thing applies to weights and measures sums) is to urge that they really belong to a later stage, and to beg teachers to defer them beyond the early years at which they are too often introduced. For their premature employment has often resulted in giving children an effectual and lifelong disgust with what they have docilely conceived to be arithmetic ; whereas much of what they had to do was really a mechanical and overstraining grind, having as much relation to mathematics as carrying heavy hods of bricks all day up a ladder has to architecture.
Origin of the symbols.
The real history of the symbols is complex, and stages of it are given in Dr. Isaac Taylor's learned work on the Alphabet, especially Vol. 11. pp. 263 et seq.
It appears that our digit symbols originated in India, and that several of them, especially 7, represent a corruption of the initial letters of the uwds previously employed to denote the numbers.
"They were introduced by the Arabs into Spain, from whence during the 12th and 13th centuries they spread over
I] COUNTING. 27
Europe, not, however, without considerable opposition. The bankers of Florence, for example, were forbidden, in 1299, to use them in their transactions, and the Statutes of the University of Padua ordain that the stationer should keep a list of the books for sale with the prices marked 'not by ciphers but in plain letters'. . . . Their use was at first confined to mathematical works, they were then employed for the paging of books, and it was not till the 15th century that their use became general."
The Roman symbol X for ten must have been the result of counting by strokes and crossing off every tenth stroke, thus:
1 1 1 1 1 1 1 1 III 1 1
a practice not unknown among workmen to this day.
Two such crosses would naturally mean 20, etc., while half a cross or V could conveniently be used to denote 5.
It has been suggested that the rounded M for 1000, CO, sometimes inscribed CIO, if halved, would give the D for 500, and that a square C for 100, if halved, would furnish an L for 50; but this may be fanciful. The symbol GClOO was used, it is said, for 100,000, and CCCIOOO for a million.
Multiplication Table.
I have been told that young children learn their multiplica- tion table quickly and without tears by the use of Mr. W. Stokes's "Pictorial Multiplication Table," published at a trifling sum by Houlston and Sons, Paternoster Buildings, London. I have no personal experience of its use, but the little pictures are simple and attractive, and the idea is ingenious though not in the least mathematical. The unrea-soning method of artificially jissisted memory, thus cultivated, may perhaps be practically etticient in the wirliest stages, and is worth a trial.
CHAPTEE 11.
Further considerations concerning the Arabic system of notation, and extension of it to express fractions.
Having become acquainted with the fundamental plan of the system of notation in use, and the mode of expressing any whole number of things by a combination of ten digits arranged in places of different value, not all places necessarily occupied — that is, by means of nine significant digits and a cipher to express emptiness in whatever place emptiness may occur, — it is permissible to elaborate it further, with a little repetition occasionally.
At the beginning of each chapter there is liable to be a little repetition of something that has already been explained, but in a slightly different form. This amount of repetition is purposely introduced and is useful : it is intended to link the new knowledge on with the old. A new subject should not be introduced as if it belonged to a perfectly distinct region of thought; its connexion with what is kno^vn should be indicated, and sufficient of the old should be reproduced to make the connexion secure. Eepetition of a judicious kind is by no means a thing to be avoided, though it is easy to overdo it ; and in every way the best kind of repetition is that which repeats the old idea in a different form of words, or which looks at something already known from a new aspect.
The beginnings of each new chapter should be easy, and the
CHAP. II.] NUMERATION. 29
steps tx) higher flights should be regular and moderate, like a staircase.
Now we know that the symbol 304 means usually that there are 3 jxickets of a hundred things each ,
no packets of tens and 4 single things, but the *' ten-system," though customary, is not an essential part of this plan of notation.
40 and 4/- are both constructed essentially on this plan, both are understood to signify 4 packets and no odd units, though the number in the packets is not the same in the two cases.
jC4 . — . — signifies again 4 of another variety of packet.
Three dozen and six pennies may be written either 3/6 or 42 pence. It would have been far more convenient if the human race had agreed to reckon everything in dozens, and so to express this number by the digits 3 6 instead of by the digits 4 2 ; but as they have in early semi-savage times arranged otherwise, we must now make the best of it. The general idea is the same, only that whereas in ordinary life things are commonly and conveniently reckoned by dozens, it is customary in arithmetic to reckon by packets of ten, the symbols being called digits because they used to be reckoned by actual fingers : which by some simple persons are so employed still. Thus whereas 7/6 is understood to mean seven dozen and six pence, it is customary to mean by 76, seven packets of ten and six units over ; that is to say, if the units were pennies, the same as 6/4. So also, instead of grouping dozens into a gross, as in ordinary life, in arithmetic we group tens into a large packet of ten tens, which we denote by 100. The symbol 346, therefore signifies six single units, 4 packets of ten each, and 3 packets of a hundred each. If there are as many as ten sets of 100, they are to be specified by 1000, and so on, as ordinarily learnt.
30 EASY MATHEMATICS. [chap.
This system of notation extends as far as we like to the left of the units place, and if six empty boxes follow the digit 1, it means a million. But we might suppose boxes added to the right of the units place ; can we find any use for them 1 Let us mark the unit box by a double line nearly round it, so that in a long row
1 1 Tj
there need be no hesitation about which is the unit box; then put the digit unity into each box. In the unit box it means one of some thing, in the next box on the left it means one packet of ten of those same things, and so on ; each digit to the left having ten times the value of the one immediately on the right. If this convention were extended to the box on the right hand of the units place, the 1 there would signify the tenth part of a unit, and a 1 in the next box on the right would signify the tenth of a tenth, and so on.
For we know that not only can we group things together into an aggregate, it is possible also to cut them up, or split them into fractions.
Thus the things counted may be bags of money, and each bag may be known to contain or to be worth 100 sovereigns. In that case the figure 6 might signify six bags, and so stand for 600 sovereigns. And each sovereign might be called a fraction of the contents of a bag, viz., a hundredth part. But in some of the bags the value might be made up with ten pound notes, and each of them would likewise be fractions of the contents of a bag, viz., the tenth part.
Three such notes may therefore be specified either as
3* ten pound notes or 30" pounds value or three tenths or '3 of the value of a bag;^
II.] NUMERATION. 31
the stop or mark or point being introduced whenever it is necessary for clearness. Any mark will do. In foreign countries a comma is commonly used, whereas we use a dot placed about the middle of the figure. In early days a | mark of this kind was used. Thus 346 [57 used to be %mtten where we should now write 346*57, or a Frenchman 346,57, the digits 5 7 being partitioned off to signify that they represent fractional parts of objects or units ; the digit 6 refers to whole objects or units, the digit 4 to packets of ten, the digit 5 to fractions of one-tenth, and the digit 7 to one-tenth part of tenths, that is to say, it signifies seven hundredths of a unit.
Suppose, for instance, the unit was a bag of sovereigns, as above specified, then the number written 346]57 or 346|57 or 346*57 would mean 346 complete bags of a hundred pounds each, with 5 ten pound notes and 7 sovereigns loose. The money specified would be equal in value to
3465*7 ten pound notes or to 34657* sovereigns or to 34*657 thousand pound notes or to -034657 million pound notes,
the position of the figures being changed according to the unit intended, and the dot or other mark being used to signify where whole numbers end and fractions begin.
The position of the above numbers relative to each other is constant, viz. the order 3, 4, 6, 5, 7 ; but their absolute position, or position relative to the unit place, is different in the different cases, and is specified by the dot, which is always and invariably placed after the units digit whenever it is inserted at all. It is not always necessary to insert it. For instance the number 3 might be written more completely and equally well 3* or 3*0 or 3*000, in which case it definitely
32 EASY MATHEMATICS. [chap.
signifies 3 units of something, and the 0 would indicate the fact that there was no fraction to be attended to. If the dot is placed thus 30-, it would mean 3 packets of ten units ; if placed thus 300*, it means three groups of ten packets each ; and any digit placed after the dot thus -3 means a fraction, viz. three-tenths of a unit. Whereas if a digit occurs 2 places to the right of the dot, as -03, it means three hundredths of a unit ; as for instance 3 sovereigns would be 3 hundredths or '03 of a bag in the above example, or -3 of a ten -pound note. Similarly a florin is one-tenth of a pound or £0-1. Again it is the hundredth, or -01, of a ten-pound note.
This use of the dot is only a matter of nomenclature, and its importance lies in its simplicity and convenience. It is always possible to write -03 as yf ^ if we please, just as it is possible to denote 1864 by mdccclxiv if we like; but it is not so simple.
It may be as well to observe that although there is no numerical difference between 6 feet and 6*00 feet, there is a practical and convenient difference of signification. In practice 6 feet would mean something approximately the height of a man, whereas 6 00 feet would be understood to signify either that you had measured a length accurately to the hundredth of a foot or something like the tenth of an inch, and found no fraction; or else that you wished some- thing to be made to that amount of accuracy.
Another way of reading the symbol '03 is three per cent., or three divided by one hundred. So also five per cent, is •05 ; twenty per cent, is -20 ; seventy-four per cent, is '74, and so on.
In the case of twenty per cent, it may obviously be written •2 or y\ or ^. So also -5 being 5-tenths or 50 per cent, is the and one-half is often the neatest way of speaking
n.] NUMERATION. 38
of it and writing it. Again twenty-five per cent., or '25, is the same as }, being 25-hundredth8 ; and -125 or 1 25-thousandth8 is the same thing as J. Sometimes one specification is handiest, sometimes the other.
Unfortunately it is not very easy to denote either J or J or f in any other very convenient way on our decimal system of notation, as it would have been if we had arranged to reckon in dozens.
One-third of 1/6 is easy enough, being sixpence, while two-thirds is 1/0: but one-third of 16 is an inconvenient number to write in the ordinary notation. It is -'/, ^^^^ Js 16 divided by 3, that is 5-333333... without end, as you find by simple division.
So also I of 16 is 10-6666....
These are called repeating or circulating decimals, and their frequent occurrence in ordinary transactions is caused by our unfortunate custom of reckoning in tens instead of in dozens. A simple circulating decimal may always be interpreted as so many ninths: thus whereas 3 means 3 tenths, •333 . . . means 3 ninths, which is the equivalent of one-third ; •6666 . . . means 0 ninths, and so on.
A third of ten is 3-333... A sixth of ten is 1-666... Two-thirds of ten is 6 666...
and even other fractions are not very convenient.
Thus a quarter of ten is 2b
an eighth of ten is 1 '25
a sixteenth of ten is *625 three-quarters of ten is 7*5
and the only simple things to specify are J of ten, which is not often wanted, viz. 2, and a half of ten, which is 5.
UK.M. C
34 " EASY MATHEMATICS. [chap.
This may be contrasted with the convenience of reckoning in dozens :
a third of a dozen is 4
a sixth of a dozen is 2
two-thirds of a dozen is 8
a quarter of a dozen is 3 half a dozen is 6
three-quarters of a dozen is 9
an eighth of a dozen is IJ
a sixteenth of a dozen is f .
Circulating decimals would not be avoided by the duo- decimal notation, but they would be rarer, for they would then in the simplest possible cases signify fifths or sevenths or elevenths, which are not the commonest fractions to come across in practice.
It should be remarked that in actual practice circulating decimals only occur in the translation of numerical fractions ; and then the decimals always either terminate or recur : but in real concrete measurement, or subdivision of continuous magni- tude, circulating decimals never occur, because such a specifica- tion would signify an infinite accuracy, which is impossible.
In all practical cases measurements can only be accurate to a certain number of significant figures, and though it may once in a lifetime happen that these figures are all the same by accident — as for instance 4*4444 — it cannot matter in the end whether the last figure is 3 or 5 or even some other digit. When the figures have expressed the actually attained accuracy, all subsequent ones are superfluous and even mis- leading, because they pretend to an amount of accuracy not really attained.
For this reason the doctrine of circulating decimals belongs rather to pure than to applied mathematics.
II.] NUMERATION. 35
In the duodecimal system the ordinary fractions would be
denoted as follows : , ^
i = -6
1 _ .4. 1 _ .Q
^ ~ A •
\ = '2497 or approximately '25 i = -2
\ = Iseos
i=16 J =14
^ = -12497 or approximately '125
5^=111111 ' = 1
twelve
Once we have realised the advantages of what is known as the duodecimal system, it is painful to have to return and use the decimal notation.
Nevertheless a change from one to the other would necessi- tate the uprooting of too deep-seated traditions. Among other things it would alter the multiplication table, that necessary but laborious thing to learn. In teaching children it should be realised by the teacher that the multiplication table is hard and tedious, and too much should not be ex- pected of them ; but for convenience of life it is one of those things that it is best to know thoroughly, and it is useful as a matter of discipline. Its rational basis should be understood, and experiment should be encouraged in the first instance to find out what, say, four sixes or seven nines are. It is fairly easy to see that four sixes will make two dozen, it is not so ea.sy to see that they will make two packets of ten and four over, but, the fact having been ascertained, it should be learnt that four sixes are 24, or four times six are 24— either way, whichever happens to be asked, )>ut not l)oth ways at the same time so as to spoil the rhythm.
36 EASY MATHEMATICS. [chap ii.
Similarly it can be ascertained that five sixpences amount to half-a-crown or 2/6 ; but that five sixes are 30, that is they just make three packets of ten.
It is a serious addition to the work of childhood in this country that they have to learn virtually two distinct multi- plication tables, viz. the duodecimal pence table and the decimal or ordinary numerical table. There is plenty of scope for discipline in these things, and so if it is possible to relieve the tedium in other places it is permissible.
The extent of multiplication table to be learnt is merely a matter of convenience, and it is handy to learn beyond 12 times 12, Especially is it convenient to remember that
13x13 = 169 17x17 = 289
14x14 = 196 18x18 = 324
15x15 = 225 19x19 = 361
16x16 = 256 20x20 = 400
Also that 9 X 16 = 12 X 12 = 144 = 1 gross.-
[The square numbers may with advantage be specially emphasised ; 1, 4, 9, 16, 25, 36, and so on ; and it is easy also as an exercise to ascertain and remember the powers of 2, especially that 32 is the fifth power of 2 : they are
2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, etc., the last written being the 10th power.
A few of the powers of 3 are also handy. 3, 9, 27, 81, 243, 729.
The cubes or third powers of the simple numbers are useful.
|
1x1x1 = 1 |
Cube of 7 = 343 |
|
2x2x2 = 8 |
„ 8= 512 |
|
3x3x3= 27 |
„ 9 = 729 |
|
Cube of 4 = 64 |
„ 10 = 1000 |
|
„ 5 = 125 |
„ 11 = 1331 |
|
„ 6 = 216 |
„ 12 = 1728 |
All this is to be arrived at merely by simple multiplication, and the phrase cube number need not yet be used.]
CHArTER III.
Further consideration of Division, and introduction of Vulgar Fractions.
Just as Multiplication is cumulative addition, so Division may be regarded as cumulative subtraction. Thus, for instance, when we say that 7 will go in 56 eight times, we mean that it can be subtracted from 56 eight times. From 59 it can like- wise be subtracted eight times, but there will be 3 over. This is the meaning of remainders.
To divide £748. 6s. lid. by £320. 2s. 4(1. wo can proceed if wo like by subtraction — it happens indeed to be the easiest way,— and having subtracted it t^vice, we find that that is all we can do, and that there is £108. 2s. 3d. over. So we say that the smaller sum goes twice in the bigger one, and leaves a certain remainder.
In general however it is more customary to regard division
as the inverse of multiplication ; and, so regarded, it leads
straight to fractions and to factors. Thus the fact that 3
multiplied by 4 equals 12, (3x4 = 12), may be equally well
/1 2 \ expressed by saying that 1 2 divided by 3 equals 4, ( -^ = 4 J,
(12 \ -^' « 3 j, or that 3 and 4
are corresponding factors of 12. Similarly 2 and 6 are other corresponding factors, since 12 -f 6 = 2 and 12 -r 2 = 6.
A number like 144, or one gross, has a large number of factors. It is a good easy problem-exorcise to suggest to a
38 EASY MATHEMATICS. [chap.
child to find them all. They are 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 36, 48, 72. The factors of 1728 are of course still more numerous. And even the number 60 has a fair number of factors, viz. 2, 3, 4, 5, 6, 10, 12, 15, 20, 30. These may be contrasted with the poor show of factors exhibited by 100, viz. 2, 4, 5, 10, 20, 25, 50.
Children can readily be set to find the factors of numbers, and will thus incidentally be doing many simple division sums.
Their attention must not however be too exclusively, i.e. for too long together, directed to integer or whole number factors; they must be prepared to write down the result of division when it is not a whole number, but a fraction, or a whole number plus a fraction. Thus ^4^ for instance will be found to be 28 and four over, the meaning of which should be carefully explained, being first thoroughly understood and led up to by the teacher.
To lead up to it, it may be pointed out that just as 28 oranges = 20 oranges + 8 oranges
so 28 half oranges = 20 half oranges + 8 half oranges and 28 halves = 20 halves + 8 halves
and 28 quarters = 20 quarters + 8 quarters ; just as much
as 28 farthings = 20 farthings + 8 farthings.
|
Now |
28 2 |
14, |
while |
20 8 2 "^2 ' |
= 10 + 4 |
= 14, |
|
|
28 4 |
7, |
while |
20 8 4 "^4 '' |
= 5 + 2 |
= 7, |
||
|
28 10 ~ |
20 10" |
8 ^10 = |
2 + -8 = |
= 2-8, |
|||
|
28 5 ~ |
20 5 ■ |
8 ^ 5 = |
-1- |
||||
|
but it |
is neater to write it |
||||||
|
= |
25 5 ■ |
3 ^ 5 = |
•4- |
»4- |
-■ 5-6. |
III.] FRACTIONS. 39
So now the child should realise that, since 144 = 140 + 4, so ij-*^ = -J^4-|; which indicates a division that can be done and a division that cannot be done. The division that can be done has the result 28 ; the division that cannot be done is 4 -j- 5, and it must be left, either in the form of ^, or in the form -Yij or -8. So the whole result is expressible as 28*8.
Accordingly a better way of saying that -j^ is 28 and four over, is to say that it equals 28 + 1, or 28*8.
To get it in the latter form directly and easily, the original 144 should be written 144 0, and then the sum will run
51144-0
28-8
quite naturally.
Take another example, because the mind of a child is often sadly fogged about this elementary and important matter.
3^ =104 = H:200:.= 10-333,..,
a result found by simple division, a process which in this case shows not the slightest sign of terminating but goes on for ever.
Again ^*- = 7 and 1 over, = 7 + ^, or as it is usually written 7h. But in thus writing it the question should occur, How then would one write 7 x J 1 and why does not 7 A mean seven halves, or seven multiplied by a half, or 3 J 1 It is a mere convention, and not a consistent one, that 7 J shall signify 7 + J and not 7 x J, and some confusion is thereby caused. By no means need the practice be altered : children must learn to accommodate themselves to existing practice, and must begin reform later in life if ever ; but the teacher should realise that the simplicity of 7 i^ to him is only because he has got accus- tomed to it, that it is a confusing thing in reality, and that a child who is confused by it is likely to be the bright child and not the dull one.
40 EASY MATHEMATICS. [chap.
Expression of vulgar fractions as decimals.
There is nothing new to be learnt about expressing a vulgar fraction in the decimal notation, it is only a question of practice. It is probable that beginners will find no diffi- culty, but will simply divide out. If any difficulty is felt it can be met by some such initial treatment as the following :
1 . ^, 2 3 4 5 - IS the same as - or - or - or -— ,
2 4 6 8 10
and each one of these may therefore be written 5, which means 5 things in the tenths place or compartment devoted to tenths. A florin for instance is the tenth part of the value of a sovereign, so 5 florins = J a sovereign. £7-5 means 7 pounds + 5 florins or £7. 10s. or £7 J.
1 9 1 91
So also i = :! = A = :12 etc.,
4 8 12 10'
so to express I in decimals we shall have to put 2^ in the tenths place ; but it is not customary to place fractions there, the I is best set down as 5 in the next place to the right, as •25. In that place 5 will mean j-J^ths, and that is the same thing as i a tenth, viz. -^jjth.
So J of a ten pound note = £2. 10s. = £2 J
= £2 5 = -25 ten pound note, and generally |- = 25.
So also f = -75, i = -125, etc.
The expression of any fraction as a decimal involves nothing more than simple division ; thus 4 can be written ready for operating 7;3"00000, and the quotient, written below, will be •42857 etc.
In this particular instance however there happens to be no simplification, so the operation is hardly worth performing in that case.
HI] FRACTIONS. 41
To prove that — f— = 57 57
work thus :
6 61345-42
57-57* To find ^V~ we do the simple division sum 8| 3475-000 434-375* Hence 3475 ^ ^.3^3^^ . 3-475 ^ .^3^3^^^ ^^
8 o
It is not really necessary to write it out in the division form : simple division can be performed on the fraction as it stands. In every case of writing decimal numbers one under the other, the rule is to keep the column of decimal points vertical ; in other words, adhere to your system as to which is the units place, which the tens place, and which the tenths, etc., throughout
Extension of the term multiplication to fractions.
The ordinary idea of multiplication involves the repetition of the same thing several times, as three times four, or seven nines.
The adding of seven nines together is what is called multi- plying nine by seven.
The payment of four £5 notes is not called multiplying £5 by 4 : but if a conjuror extracted ten apples out of a hat into which one had been put, he might be said to have multiplied it.
So also seed corn is multiplied into an ear ; and thus the notion of increase is associated with the notion of multiplying. But it is best to dissociate the notion of increase from the notion of technical multiplication, and to be prepared to multiply by I if need l)e, leaving it the same as before, or even by J, leaving it smaller than before. This phrase " multiply
42 EASY MATHEMATICS. [chap.
by a half " is not a simple and natural one : it is a permissible extension, such as we constantly make in mathematics, when any operation that has been found practically useful is applied over the whole range within which it is possible, and some- times a long way beyond where it appears possible at first sight.
Multiplication by J has some points in common with the addition of a negative quantity ; it results in diminution, and it is an operation that would not have occurred to us except as an extension of a straightforward process. To multiply by J and to divide by 2 is precisely the same thing. Why not call it then dividing by 2 ^ Well, we do very often, but not always, and a beginner must be content to be told that it is useful to extend the nomenclature of operations in this way. We shall speak of multiplication by J if we choose, when we mean division by 3. We shall occasionally speak of adding - 4 to a number when we really mean taking 4 from it. AVe shall do any of these things when we have good reason for doing so, and not otherwise.
Suppose we say that 2h sovereigns are equivalent to 50s., we arrive at the result by multiplying 20 by 2 J, that is first by 2 and then by J, and adding the separate results. It would be a nuisance to be obliged to say that we multiply 20 by 2 and divide 20 by 2 and add the results, though it would be quite true.
The fact that the half of 20 is 10 may be written if we like, thus : I X 20 = 10 ; or, of course, -^^ or 20 -f 2 = 10 ; J of 24 = 8 may be written J x 24 = %*- = 8 ; 1^ of ^ may be written J x |^ = J ; and that the half of the third of an apple or ribbon is a sixth of the apple or ribbon is easily verified by experiment. An experiment need not always be performed; after a time it can be vividly imagined, with advantages on the side of clear ness of apprehension.
in.] FRACTIONS. 43
The natural word to use for taking the fraction of a thing is the word " of," like the half of an orange or a quarter of a pound or one-sixth of the revenue ; and we shall gradually find that in all arithmetical cases the word "of" has to be interpreted as an instruction to perform the operation denoted by X , that is to say, the operation we have been accustomed to call multiplication.
Practical remarks on the treatment of fractions.
It so happens that the multiplication of vulgar fractions is easier than addition and subtraction, and so it may take precedence. One half of one quarter is one eighth : as can be found by concrete experiment, for instance on an apple, or by looking at the divisions on a 2-foot rule.
Infl — 1 — Ivl ^fOIy-^-^Xy
Joff = A
Such a statement as the last must be, and is, led up to; and gradually the empirical rule can be perceived, that in multiplication of fractions the numerators must be multiplied for the new numerator, and the denominators must be multi- plied for the new denominator.
[But initial difl&culties and confusion must be expected between this and the addition of fractions. Thus, for instance:
? I = 24 + 28 ^ 52 4"''8~ 32 "32* This is set down here as a warning.
The greatest difficulty in dealing with fractions is felt as long as they are abstract. "J of what?" is constantly or should constantly be asked by a child. In the above two sums the answer to this question would be different : —
44 EASY MATHEMATICS. [chap. in.
In one it is J of a fraction, viz. j of | of a unit, such as a foot, that has to be found. In the other it is | of one foot which has to be added to J of another.]
It is convenient to ascertain and remember that ^ + i = i [whereas J of J = -/J ; also that ^ + ^\ = ^^ or ^-^ = J.
Exercise.
Find the third plus half the third of eight. The answer is 4, but the decimal notation confuses the matter :
i of 8 is 2-6666... and half this is 1-333... so the sum is 3*9999..., that is 3^ or 4. So also a third + half a third of ten would seem troublesome, though it results simply in five. But a third + half a third of a dozen is simple enough, being 4 + 2 = 6. And always 1 . 1 _ 1
3" ^6 ~ 2'
Division of fractions may be exhibited thus :
Suppose we have to find what | -f § amounts to,
.. -, ,1 seven eisjhths
write it thus, — = pS_^ —
three nttns
seven x five fortieths 7x5 35 75
X
three X eight fortieths 3x8 24 8 3' wherefore instead of dividing by f , we find we may multiply
The idea underlying the above process is that things called eighths have to be divided by things called fifths, and that to make it possible they must be expressed in the same denomination, which in this case is fortieths. Thus we get the rule, invert the divisor and multiply. Or otherwise expressed : to divide by a number multiply by its reciprocal. Division by 4 is the same thing as multiplication by 2. The symbol -r J is equivalent to the symbol x 4.
CHAPTER IV.
Further consideration and extension of the idea of subtraction.
If a man gains £21. 6s. 5d. and loses £15. 4s. 4d., his nett gain is found by subtraction, and is called the "difference," viz. £6. 2s. Id. ; the total money which has changed hands being the "sum," viz, £36. 10s. 9d. A loss may be called a negative gain ; thus a gain of £10 minus £6, would mean a gain of £10 accompanied by a loss of £G, or a nett positive gain of £4. This leads us to discriminate between positive and negative quantities, and to regard subtraction as negative addition. Subtracting a positive quantity is the same as adding an equal negative one.
Geometrically it is sometimes convenient to discriminate between the journey A to B, or A By and the journey B to Aj or BAf just as a French-English dictionary is not the same as an English-French dictionary. When expressed numerically a length AB may be denoted by its value, say 3 inches, or 3 miles ; and the reverse journey may be denoted by - 3 inches or - 3 miles, because this when added to, or performed sulisequently to, the direct journey, will neutralise it and leave the traveller where he started. The two opposite signs cancel each other in this sense, and the two quantities added together are said to amount to zero algebraically— that is when their signs are attended t^), and as regards the end
46 EASY MATHEMATICS. [chap.
result only ; but the traveller will himself be conscious that although he is where he started from, he has really walked 6 miles; so that for some purposes such quantities may be added, and they are then said to be "arithmetically" or better " numerically " added ; for other purposes they are to be "numerically subtracted," or, as it is called, "algebraically added," that is with their signs attended to, and with "minus" neutralising an equal "plus."
If a height above sea level is reckoned positive, a depth below may be reckoned negative; so that a well may be spoken of either as 60 feet below or as - 60 feet above the sea level.
The latter mode of specification sounds absurd, but one should gradually accustom one's self to it, for practical pur- poses later on.
If children feel a difficulty with these negative quantities, as they have every right to, they can be accustomed to them gently, as a horse to a motor car. Mathematicians found some difficulty with them once upon a time, so the difficulty is real, though like so many others it rapidly disappears by custom. Debts, return journeys, fall of thrown-up stones, losings, apparent weights of balloons or of corks under water, dates of reckoning B.C., and many other things will serve as illustrations ; not, however, to be taken all at once.
Time is the one thing that never goes backwards; but nevertheless intervals of time may be considered negative if they date back to a period antecedent to the era of reckoning. In a race, for instance, it would be an ordinary handicap if one of the competitors was set 12 yards behind scratch, or if he was made to start from scratch 3 seconds late. In either case he could be said to have a negative start.
In golf handicaps it is customary to denote these positions behind scratch as positive, because they are added to the
IV.] NEGATIVE QUANTITIES. 47
score. This is because the object in golf is to get as low a score as possible, not a high one as at cricket.
Addition and subtraction of negative quantities.
Suppose a man inherited a lot of debts, his projierty would be diminished by their acquisition. The addition to it would be negative, and would be indistinguishable from subtraction.
A debt of £300 added to a possession of £500 would result in nett property of £200 ; which we might express by saying that -3 + 5 = + 2.
Or of course the debt might exceed the possession and leave a balance of debt. For instance -8 + 5 = - 3 ; where the unit intended by these digits might be a hundred or a thousand pounds. This may be taken as an illustration of the gain of a negative quantity. Take another.
An axe-head at the lK)ttom of a river weighs 3 lbs. Some corks, which, when submerged, pull upwards with a force equal to the weight of 49 ounces, are attached to the mass of iron. Its weight is thus more than counteracted, and it is floated upwards with a force equal to the weight of 1 ounce, because 48-49 = - 1.
A raisin at the bottom of a champagne glass, or a speck of grit in a soda-water bottle, can often be seen to accumulate bubbles on itself till it floats to the surface and gets rid of some, when it sinks again, and .so on alternately.
The negative or upward weight of the corks, or of the bubbles, counteracts and overlmlances the positive weight of the iron or of the fruit. It may Ix) said that we have sub- tracted more weight from it than it itself {assessed, and so left it with a negative weight — like a balloon. The weight of a balloon is not really negative, but it superficially appears to be ; because the surrounding air buoys it up with a force
48 EASY MATHEMATICS. [chap.
equal to the weight of the air it displaces, which represents a greater weight than its own.
When we have to subtract a bigger number from a smaller, we must not always merely say we cannot do it. It is con- venient in subtraction sums to say so, and to " borrow " from the digit in the next higher place {i.e. to undo one of the available packets and bring the contents one step down), so long as there is something there to be " borrowed," but if we perceive that at the end of the sum there will be a manifest deficiency we must proceed differently.
Suppose we were told to collect £S from a man who had only <£3, we could not really do it; but we might report to our chief, "if we do we shall leave him £5 in debt to somebody," which could be expressed arithmetically thus : 3 - 8 = - 5.
Suppose we were told to pull 5 feet of a gate-post out of the ground, and when we came to try we found that it had only 2 feet buried ; we might at first say that it could not be done ; but on second thoughts we could say that it was hard to do, and that the only plan we could see would be to pull it minus 3 feet out first, that is to get a mallet and drive it 3 extra feet in, before pulling at it at all.
Suppose a stone were 30 feet above the ground, and we were told to drop it 36 feet, that is to subtract 36 feet from its height of 30 feet. It would not be easy to do, but it could be done, for we might dig a hole 6 feet deep; or it might even be sufficient if we dropped it over a pond of that depth. In either case it would afterwards be 6 feet below the surface of the ground, for 30 - 36 = - 6 ; it would then be at an elevation of - 6 feet, which means the same as a depression of 6 feet.
To speak of a depth of 6 feet as a negative height, in ordinary conversation, would be absurd ; but to interpret an
IV.] NEGATIVE QUANTITIES. 49
arithmetical answer, which gives a height as - 6 feet, to mean that a thing is not elevated at all but is depressed 6 feet, would l>e quite right and in accordance ^\^th commonsensa Hence the following examples are correct : 4- 9 = - 5 17- 39 = - 22 546-827 = -281
But now here is a necessary caution. Take the last case. We see that it is light, for if we add 281 to 546 we get 827 ; ])ut suppose we had put it down like an ordinary subtraction sum and noticed nothing wrong with it, it would have looked like this
546
827 (example of the way wo/ to do it).
-319
We should have said in the old-fashioned way 7 from 6 we cannot, so borrow 10 from the next place; 7 from 16 is 9, put it down. Now we have either 2 from 3, or what is more ( ommonly said, and comes to the same thing, 3 from 4, leaving 1, which we put down; and then we have to take 8 from 5. There is nothing more to borrow, so we must set it down as - 3. Well that is not wrong, but it requires interpreting, and it is not convenient. The mmus sign only applies to the 3, which, being in the third place, means 300 ; I he other figures, the 19, were positive. Hence the meaning is -300+19, or in other words -281. It might be written 319^ with the minus sign above and understood to apply only to the digit 3, but it could not properly be written - 3 1 9. • The al)Ove is therefore a very troublesome way of arriving at the result. The convenient way is not to l)egin jxirforming the imiKMjsiblo subtraction, ))ut to perceive the thrwitening dilemma, and invert it at oii(;e; then .subtract the smaller
L.K.M. I>
50
EASY MATHEMATICS.
[chap. IV.
number from the bigger in the ordinary way, labelling the result however as negative. This is of course what we really do when we say 5-8 = - 3. We do not begin saying " 8 from 5 we cannot, so borrow" from nowhere, for there is nowhere to borrow from. We stop, invert the operation, and record the result as negative ; because a-h = - (b-a).
One more case we must take however, viz. where the quantity to be subtracted is itself negative : and its subtrac- tion therefore represents a gain. The loss of an undesirable burden was esteemed by Bunyan's Pilgrim to be a clear gain. A negative subtraction is a positive addition.
6-(-3) = 9; 7-(-9) = 16. This is sometimes expressed by saying that two minuses make a plus. The effect of a minus is always to reverse the sign of any quantity to which it is prefixed, so if applied to a negative quantity it turns it into a positive quantity. It is equivalent to more than the removal, or subtraction, of a debt, which would be effected by an equal sum added. A loss is more than neutralised by a negative sign, it is reversed.
Add -31 to 114, the result is 83; but subtract -31 from 114, and the result is 145.
No more words are necessary. Familiarity and practice will come in due course as we proceed. A surviving puzzle may occasionally be felt, and can from time to time be removed. It is a mistake to hammer at a simple thing like that till it becomes wearisome ; for trifling puzzles or foggi- nesses evaporate during sleep, and in a few years have automatically disappeared, from children properly taught. They continue to trouble too many adults at present.
CHAPTER V.
Generalisation and extension of the ideas of multi- plication and division to concrete quantity.
The iiiea of multiplication arose as a convenient summary of a special kind of addition, viz. the addition of several things of the same magnitude to each other. Thus four sixes added together, if counted, make 24, and so it is summarised and remembered as 4 sixes are 24, or 4 times 6 = 24 ; and 4 and 6 are called * factors ' of 24.
Originally therefore the two factors in multiplication signified, one of them the size of the quantity of which several are to be added together, and the other the number of times it was to be so added.
Thus 3x6, read 3 times 6, meant a summarised addition sum, 6 + 6 + 6. But if read 6 times 3 it meant the addition sum 3 + 3 + 3 + 3 + 3 + 3. That the result is the same may be treated as a matter of experience, and may be demonstrated by grouping, but it is not to be regarded as ......
self-evident. Nevertheless the diagram (fig. 2) ......
demonstrates that 3 rows of 6 each is the ••••••
same as 6 columns of 3 each. And the °' ^*
counting of window panes and postage stamps are illustrations of practically the same thing.
Thus we get led to the area of a
a
Fi«'. 3. rectangle of length a and breadth h as
a X A, or briefly written oh (fig. 3).
52
EASY MATHEMATICS.
[chap.
But the idea of multiplication soon generalises itself, and the expression ab gets applied to a number of things to which a simple numerical idea like 3 times 6, or a times b, would hardly apply.
It may be worth showing however that the numerical notion will apply further than might have been anticipated, for instance the rectangle (fig. 4) is built up of 5 equal staves each of them say 3 inches long and an inch wide. The area of each stave is thus 3 inches x 1 inch, or 3 square inches. And by adding 5 of the staves together (or multiplying one of them by 5) we get the total area.
Fio. 4.
Fig. 5,
And the same area could be equally well obtained by putting together 3 staves each of 5 square inches area (fig. 5).
The number 12 can be resolved into two factors 3 and 4, as is shown by the annexed group which consists of 3 rows of four dots each, or of 4 columns of 3 dots each, proving that 3 times 4 = 4 times 3.
Fio. 6.
Fio. 7.
A dozen can equally well be grouped as in fig. 7 : its large number of factors confers distinction on the number 12.
The number 10 has only two factors, viz. 2 and 5, since the name "factor" is usually limited to whole numbers. It is possible to say that 3 J is a factor of 10, because if it be
v.] CONCRETE QUANTITY. S3
repeated 3 times the iiuml)er ten results ; as is shown by the following set of 3J disks repeated 3 times, where the central sectors hiive eivch of them an angle 120° or q
J of a revolution, and so make up a disk q
when put together. But the name " factor " q
is not usually applied to fractions. O O O <B
Again, a slab of any given area and unit q
thickness will have a bulk which, measured O
in cubic inches, is numerically equal to its Q
area in square inches. If such a slab is mul- ^'"- ^•
tiplied or repeated, each slab being piled up on similar ones, say 7 times, then 7 times its bulk will give the volume of a rectangular block ; or the volume of a block may be said to be obtained by multiplying its length, breadth, and height. There is no reason to take one of these factors as numerical more than another, and the truth is that none of them need be numerical.
When we say volume = M, or length x breadth x height, we may and should mean by / the actual length, ^>y ^ M >» breadth, and by h „ „ height, — iiot the number of inches or centimetres in each — and the resulting product is then the actual volume, and not any numerical estimate of it. [If anyone disagrees Mdth this they are asked to withhold their disagreement for the present. This is one of the few things on which presently I wish to dogmatise. See Chap. XXVI. and Api>endix II.]
From this point of view the symbols of algebra are concrete or real physical cjuantities, not symbols for numbers alone, and algebra becomes more than generalises! arithmetic.
In such cases however the old original definition of multipli- cation requires generalisation, and a good deal can be written on it ; but no difficulty arises, and the question, being inter-
54 EASY MATHEMATICS. [chap.
esting chiefly from the philosophic point of view, does not in
this book concern us.
We may proceed without compunction to multiply together
all sorts of incongruous things if we find any convenience in so
doing. Thus, a linear foot multiplied by a linear foot giveb a
square foot,
6 feet X 3 feet gives 18 square feet,
4 feet X 3 feet x 2 feet gives 24 cubic feet.
In all these cases something real and intelligible results ; but if we multiply square feet by square feet, nothing intelli- gible results ; consequently such a process will never appear in a correct end-result, though we shall find that it often appears as a step in a process without any detriment.
Again we may multiply a weight by a length, say 3 lbs. by 7 feet, and get what is called 21 foot-lbs., where the unit has a meaning which can be interpreted, viz. the work done in raising a 3 lb. weight 7 feet high against gravity, or else the moment of a force round an axis. But if we try to multiply 3 lbs. by 7 lbs., we should get 21 square lbs., which has no intelligible meaning and is nonsense. There is nothing in the s3^mbols to tell us whether it is sense or not : operations can be consistently performed even on meaningless symbols. To discriminate sense from nonsense, appeal must be made to reality and to actual life and instructed experience.
Division is merely the inverse of multiplication, and similar considerations apply to it.
If we divide 1 by any quantity we get what is called the reciprocal of that quantity.
Thus J is the reciprocal of 2. y^ is the reciprocal of 10.
-p — is the reciprocal of a length, and could be read 1 per yard.
CONCRETE QdANTITY. 55
60
might represent the uumbcr of telegraph posts
1760 yards
per mile.
^ is the reciprocal of a time, and might be read
^^ * once every tenth of a second ' ; or it could be
simplified into a repetition of something ten times a second, or 10 per second. It is what is called a * frequency,' and is in constant use for vibrations.
Yq is a slow frequency, the frequency with which
^ a cycle of astronomical eclipses approximately recurs.
6000 revolutions ■ e r i. i.- e ^i. a
— - — -. is a frequency of rotation, as of the fly-
5 minutes ^ J J
wheel of a small engine, and may be read as 1200 revolutions per minute, or 20 revolutions per second.
If we divide a length by a time, as for instance ™ , we get a velocity ; e.g. the speed of an express l*"""- train.
^ is exactly the same velocity.
1 second
j"^ ^^ or approximately -^^- — ^ is a walking pace.
No hesitation must be felt at thus introducing the units into the numerator or denominator of fractions. If they are left out, the residue becomes a mere numerical fraction, the ratio of two pure numbers ; whereas with the units inserted they are real physical quantities with a concrete meaning, and are capable of vjtried numerical specification.
56 EASY MATHEMATICS. [chap.
Thus the velocity of sound in air at the freezing point is 1090 feet 33000 centimetres 1 mile
or , or
1 second 1 second 5 seconds
^ "33 kilometres 1 kilometre . , ,
or — — — — or ^; ^ — approximately
1 second 3 seconds
or ^Q"^^"^^tes'walk 240000 miles 3 seconds a fortnight
First idea of involution.
When a number of the same things were added together many times, the process was specially treated and called multiplication. When a number of the things are multiplied together several times, the process is likewise worthy of special treatment, and is called " involution " or the raising of a thing to a certain "power."
The raising to a power is compressed or summarised multi- plication. The expression 4x3 meant four added to itself 3 times (or 12), whereas 4^ is understood to mean 4 multiplied by itself 3 times (or 64).
So 25 = 32, 63 = 216, 103 _ 1000, 106 = a million,
12- = 144, and can be read 12 square, for short ; though really a sc[iiare number is an absurdity. It is called " twelve square" because if the 12 represented inches, 12^ would mean a square foot.
If a is a length, a^ is truly a square whose side is of length rt, and a^ is truly a cube whose side is of length a. So 4"^ is read " 4 square," and 6^ is often read " six cube," by analogy. It is also true that 2^ = 16, but here there is no geometrical analogy, and it is read " 2 to the fourth power " simply, the word "power" being often omitted in practice. Similarly a million is " ten to the sixth " or 10^.
v.] CONCRETE QUANTITY. 57
A length divided by a time is a velocity (»), and a velocity divided by a time is an acceleration (a).
V
« = -.
So in mechanics we find such an expression as
s = \aP,
where ^ is often read as the square of the time, although strictly speaking such an expression is nonsense. We can have a square mile, but not a square fortnight ; there is no meaning to be attached to the term ; time cannot be multi- plied by time with any intelligible result. Whenever such an expression occurs, it is to be understood as an abbreviation for something : in the above case for this
where the at is v, and is a real and simple physical quantity.
s is a velocity multiplied by a time, and the double reference to time is caused by the introduction of the specially defined quantity "acceleration," which is often expressed correctly as so many feet per second per second ; the two units of time in the denominator being conveniently spoken of as the square of the time — by analogy with geometry again — without thought and without practical detriment, though confusing to anyone who seeks a real philosophic meaning in the expression.
CHAPTER VI.
Factors of simple numbers.
A CHILD should be encouraged who notices that no factor is ever greater than half the number; for though there is nothing in that but what is obvious, yet that is the type of noticing which frequently leads to observations of interest. An even number always has this largest factor, but an odd number can never have a factor greater than a third its value ; and frequently its largest factor is less than this. Some numbers have no factors at all; like 7 and 11 and 13 and 29 and 131. These are called prime numbers, and a child should make a small list of them as an exercise. But do not attempt to make it learn them or anything of this kind by heart. Ease and quickness of obtaining Avhen wanted is all that is practically needed.
A child should be encouraged to discover criteria for the existence of simple factors; but is hardly likely to be able to notice the facts without aid.
Any number (written in the decimal notation) which is divisible by 3 (i.e. which has 3 as a factor) has the sum of its digits also divisible by 3. But this, though convenient as a rule, is in no sense fundamental : it depends merely on our habit of grouping in tens. In the duodecimal system every number ending in 0 would necessarily be divisible by 3 as well as by 4 and by 6 ; and extremely convenient the fact would be.
For instance, 1/- and 2/- and 4/- and 5/-, or any number of
CHAP. VI.] FACTORS. 59
shillings, can be divided by 3, 4, or 6 ; that is, can be parcelled out exactly into a whole number of pennies.
By reason of the system of reckoning 12 pence to a shilling, any sum of pounds can be subdivided into three or six equal parts without halfpence or farthings ; thus J of a pound is 6s. 8d., two-thirds is 13s. 4d., one-sixth is 3/4, one 8th is 2/6, and one-twelfth is 1/8.
In the decimal notation a number ending in 00 is certainly divisible by 4 ; and ending in 000 it is certainly divisible by 8. But the division is only worth doing when the cyphers are replaced by some number of two or three digits, itself divisible.
The number 5 in the decimal system has an artificial sim- plicity conferred upon it, but it is not often that we should naturally group things in 5, except for the accident of our 5 fingers : and one of them is a thumb.
The advantage of working in at least two different scales of notation is that it becomes thereby easy to discriminate what is essential and fundamental from what is accidental and dependent on the scale of notation employed. Thus the curious properties of the number nine or eleven are artificial, and in the duodecimal scale are transferred to eleven and thirteen respectively.
The well-known criterion for divisibility by 3 or 9, viz. whether the sum of the digits is so divisible, is accidental again, and disappears in another scale of notation — for instance when units are grouped in dozens instead of tens, — to give place however to a much simpler rule.
The rule about divisibility of the sum of the digits applies to eleven in the duodecimal scale, and indeed would always apply to the number which is one less than the group number artificially selected.
But the existence and identity of prime numbers is not
60 EASY MATHEMATICS. [chap.
accidental at all, but fundamental, and so also is the existence of any given numbers of factors to a number— however it be specified.
Thus one gross can be parcelled out into factors or equal groups in a given number of ways, whether it be denoted by 1/0/0 or by 144 or by any other system of notation.
So also the number one-hundred has only six factors whether it be denoted by 8/4 or by 100 (one nought nought), and its factors are (in the duodecimal scale) :
4/2 2/1 1/8 t 5 2,
that is these actual numbers, however they are denoted. In the duodecimal scale it is needful to have single symbols for ten and eleven ; and the initial letters serve the purpose.
An actual number is easily exhibited by means of counters or coins or marbles : its expression in digits is an artificial arrangement and is adopted simply for convenience : it is analogous to sorting the marbles into bags of which each must contain an equal number — whatever number may be chosen as suitable and fixed upon for the purpose.
It may be interesting to write down the numbers in the duodecimal scale which would be divisible by 5.
5, t, 1/3, 1/8, 2/1, 2/6, 2/6, 3/4, 3/9, 4/2, 4/7, 5/0, ..., and the even numbers in the above are divisible also by ten. The above numbers should be read five, ten, one and three, one and eight, two and one, two and six, two and eleven, etc , meaning one dozen and three, one dozen and eight, two dozen and six, two dozen and eleven, etc.
Numbers which have the factor 7 are 7, 1/2, 1/9, 2/4, 2/., 3/6, 4/1, 4/8, 5/3, 5//, 6/5, 7/0, ..., and the even ones are divisible also by fourteen.
Numbers which have the factor eleven {e) are e, 1//, 2/9, 3/8, 4/Y, 5/6, 6/5, 7/4, 8/3, 9/2, ^/l, e/O, ...,
VI.] FACTORS. 61
namely eleven, one and ten, two and nine, and so on : the last one written being read eleven dozen.
Numbers divisible by thirteen (1/1) are
1/1, 2/2, 3/3, 4/4, 5/5, 6/6, 7/7, 8/8, 9/9, tjt, e/e, 1/1/0, ....
In the last two cases a law or order among the digits is manifest, but in all four cases it may lie noticed that every digit makes its appearance in the units place, though only in the last two cases do they appear in a simple order.
Numbers divisible by 3 are
3, 6, 9, 1/0, 1/3, 1/6, 1/9, 2/0, 2/3, 2/6, 2/9, 3/0, .... and the even ones are divisible by 6. Every third one of the alx)ve series, viz. those in thick type, are divisible by 9.
Numbers divisible by 4 are 4, 8, 1/0, 1/4, 1/8, 2/0, 2/4, 2/8, 3/0, 3/4, 3/8, 4/0, .... Alternate ones are divisible by 8, and those in thick type are divisible by sixteen.
Numbers divisible by twelve, that is arrangeable in dozens, are of course,
1/0, 2/0, 3/0, 4/0, etc., 1/0/Q, ..., the last written being the symbol for a dozen dozen or one gross.
CHAPTEE VII.
Dealings with money and with weights and measures.
In the British Isles it is customary to count pennies by the dozen, the value of which when coined in silver is called a "shilling"; and shillings are counted by the score, the value of which is called a " pound sterling," or when coined in gold a "sovereign." Five dozen pence, or a quarter of a pound sterling, when in a single silver piece used also to be called a "crown." And these, together with the half-sovereign, half- crown, half-penny, etc., are the chief names in vogue ; except the "guinea" and the "farthing," neither of which need much concern us. The " florin " is an attempt at a decimal coinage, being the tenth of a pound; and the double-florin is an attempt at an international currency or equivalence ^vith the dollar and the five-franc piece.
The addition of money is a practical operation in constant use, and plenty of practice in addition is obtainable by its means. No other addition sums are worth attention for their own sake : but in addition of money it is worth while taking pains to acquire a fairly quick and accurate style. At the same time it is to be remembered that it is a purely mechanical process— one that in large offices is better, more rapidly and more accurately, performed by a machine, into which the figures are introduced by pressing studs, and then the addition performed instantaneously by turning a handle.
CHAP. VII.] MONEY. 63
Nothing that can be performed by turning a handle can be considered an element in a liberal education : it can only be a practical and useful art. That however it is ; partly because a machine is seldom available, partly because it is ignominious to be helpless without a tool of this kind, chiefly because addition of .money is an operation which is called for by commonplace daily life more often than any other.
Nothing much need here be said about it. The columns of an actual account book are the best addition sums to set for practice. Also, in waiting figures down, it is well to take care to place the unit digits under each other, leaving a place for a left-hand digit whenever such occurs in the pence and shillings columns, and to be equally careful to write the pounds with the corresponding places vertical. Also to write all figures very plainly. This last always, and for all purposes : A good clear style of figure- writing should be cultivated.
Subtraction of money is greatly facilitated by the use of the " shop " method : the old-fashioned process of " borrowing" was troublesome, and moreover only enabled one row of figures to be subtracted from one row, whereas with the shop or complementary method any number of rows may be sub- tracted from another row, and the process is practically only addition. For instance suppose it is wished to subtract all the smaller amounts from the larger in the annexed statement:
£ 8. d.
341 8 7
less 19 5 9
and 14 0 3
and 36 17 5
271 5 2
The process is, to say, 5 and 3 and 9 make 1/5 and 2 make 1/7, put down the 2d. and carry 1/-; then 18 and 0 and 5 make 23 and 5 make 28, put down 5/- and carry £\ ; then
EASY MATHEMATICS. [chap.
7 and 4 and 9 make 20 and 1 make 21, put down 1 and carry 2; then 5 + 1 + 1=7 and 7 make 14; and finally 1 and 2 make 3. In reading, emphasise all the black figures.
Verify by adding the four lower lines.
As to multiplication and division of money or of weights and measures we will deal briefly with them : the old- fashioned practice in such matters was tedious and was pushed in childhood into needless intricacies. Dulness is apt to line all this region, unless skill is expended on it and due care taken, and no more practice should be enforced in it than is required for ordinary life. Discipline and punish- ment lessons might possibly with advantage be confined to this region. Even for punishment it is however hardly necessary to inflict sums dealing with acres, furlongs, poles or perches ; or with bushels, pecks, scruples, quarters, pennyweights, and drams. Hogsheads, kilderkins, and firkins may perhaps at length be considered extinct, except for purposes connected with the study of folk-lore. There are plenty of real and living units to be learnt in Physics : we need not ransack old libraries and antique country customs for them. And, though the humanity involved in and represented by old names has been a relief to some children, during their dismal lessons, far too much has been made of the trivial and dull operations suggested by tables of British weights and measures. The sooner most of them are consigned to oblivion the better.
Real living aritlimetic is the same in any country ; and the most important of all is that which must necessarily be the same on any planet.
The units that are at present worthy of terrestrial attention are the following :
Units of length — inch, foot, 3"ard, mile, millimetre, centimetre, metre, kilometre.
vix.] UNITS. 65
units of time — second, minute, hour, day, week
year, units of area — square mile, square foot, square
centimetre, etc. units of volume — cubic foot, cubic inch, cul)ic yard, cubic centimetre, cubic metre ; occasionally also litres, gallons and pints. units of mass — pound, ton (ounce, grain, hundred- weight occasionally), gramme, kilogramme, milligramme, units of money — pence, shillings, pounds, francs, marks, dollars. But conversion from one to the other of the last-mentioned denominations should in every case be only approximate. Accurate work when wanted is done by tables, and the rate of exchange is constantly varying.
For division of money, and of weights and measures, the orthodox school rule is called " practice " ; and it sometimes happens that by excessive practice children are able to do this kind of sum much better than adults — better even than mathematicians; but since school time is limited, such extravagant facility in one direction is necessarily balanced by extreme deficiency in many others, and is therefore to be deprecated. The world is too full of interest to make it legitimate to exhaust the faculties of children over quite needless arithmetical gymnastics, which confer no mathe- matical facility, but engender dislike of the whole subject.
Modern treatment of the rule called " Practice." The practical advantages of decimalisation.
In old days some ver\' long sums used to be invented for British children whereby our insular system of coinage and of
L.K.M. B
66 EASY MATHEMATICS. [chap.
weights and measures was pressed into the service to make difficult exercises. The form was usually something like this :
Find the cost of 131 tons 5 cwt. 3 qrs. 24 lbs. 5 oz. at £i. 13s. 9id. a ton.
De Morgan led the way, I think, towards taking all the sting out of these outrageous problems, and reducing them to useful though unimpressive and essentially insular exercises, by introducing the chief advantage of the decimal system into the working, before it had been embodied by Parliament in a legal system of weights and measures and coinage itself.
If such sums have to be done, and a moderate amount of " practice " in that direction is quite legitimate, decimalisation of at least one of the quantities specified, that is, expressing it in terms of one denomination, is undoubtedly the proper initial step to take ; and then if we are asked the cost of so much goods at a given price, the matter becomes a mere straightforward multiplication ; while if we are asked to find the price of a given amount of goods which have cost so much money, or the amount of goods which can be obtained for a given sum of money at a given price, we have only a straight- forward division sum to do ; — once the complication of many denominations, that is to say the " compound " nature of the specification, with scales of notation mixed up, is by an initial process got rid of. It is always possible, and sometimes advocated, to reduce everything to the lowest denomination, e.g., in the sum above to halfpennies and ounces ; but that is terribly long and tedious. Expression in terms of the highest denomination is much neater. The initial process is as follows :
Decimalisation of money.
To express any sum of money in terms of a single unit, say £lf which is the best unit for the purpose, it is sufficient to
vn.] PRACTICE. CT
notice and remember a few simple convenient facts. They are all painfully insular, and are not an essential part of real arithmetic at all, but if properly and lightly treated they afford to British children an amount of easy practice which foreign children are destitute of. It is only when trivial facts and insignificant sums are laboured at, till they kill all interest of the British child in real arithmetic, that they become deadly and deserving of the harshest epithets.
The decimalisation of money in terms of a pound is easy, since a florin is the tenth of a sovereign ; so any number of shillings is easily expressed in decimals of a pound.
2/- = £% I/- = £05,
3/- = £-15, 6d. = £025,
4/- = £-2, 1/6 = £075,
5/- = £-25, 2/6 = £125,
6/. = £-3, 3/6 = £175,
7/- = £-35, 4/6 = £-225, and so on. etc.
A penny is ^^th of a pound, but that is not specially convenient when expressed as a decimal ; a farthing is ^J^th of a pound, and that is approximately y^V^r ^^ £'001.
Since money is never needed closer than to the nearest farthing, except in the price of cotton per lb. and a few rare cases, the approximation of £-001, sometimes called a mil, for 1 farthing, or the writing of a farthing instead of £001, often suffices ; especially in interpreting results.
The following expressions are all equivalent in value : J a sovereign = 10/- = 5 florins = £-5.
So also are the following, each row among themselves : a. lOs. = £7i = £7-5 = £7 + 5 florins. I of a ten-pound note = 25 ten-pound note =» £2*5 = £2. lOs. 15/. = £} - £-75.
|
So also |
||
|
£5. |
2s. |
6d. |
|
£3. |
Is. |
6d. |
|
£3. |
lis. |
6d. |
|
£3. |
lis. |
6h^d. |
68 EASY MATHEMATICS. [chap.
150/- = 75 florins = £7-5 = £7. 10s.
18/- = 9 florins = £-9.
12/- = 6 florins = £'6. £1. 12s. = £1-6. £4. 18s. = £4-9. £7. 19s. = £7-95.
All these expressions should be read backwards as well as forwards.
= £5'125. = £3-075. = £3-575.
= £3-577, almost exactly. Take a few examples of the interpretation of decimals of a pound into ordinary coinage : £1-2 = £1. 4s. £4-25 = £4. 5s.
£7*904 = £7. 18s. Id., the four mils being practically a
penny. £13-127 = £13. 2s. 6Jd., the -125 being 2/6, and the 2 extra
mils |d. £1-178 = £1. 3s. 6|d., the -15^^ being 3/-, -025 = 6d., and
there being 3 mils more. £•025 = 6d.
£-026 = 6Jd. almost exactly. £-027 = 6Jd. £-028 = QJd. £-029 = 7d.
£•030 = 7|d. „ „ or exactly 7id.
We are now ready to do any number of sums like the following :
Find the cost of 324 horses at £17. 9s. 6d. a horse. Now 9/6 = 8/--f 1/6 = £-4-1- £-075 = £-475,
VII.3 PROBLEMS. 69
so tho answer is merely
324 X £17-475 - £56619 = £5661. ISs.
Find tho cost of 900 things at £9. 78. 4jd. (Sonnenschcin.) Answer is £9-36875x900 = £8431*875 by simple multiplication = £8431. 17s. 6d.
How much a year is £31. 9s. 9d. per day?
Answer 365 x £31-4875 = £114929375 = £11492. 18s. 9d.
How much interest must be paid for 43 days' loan of a sum of £543. 17s. 6d. at the rate of 3 J per cent, per annum]
(Sonnenschcin.)
Here £3 J must be paid for each hundred pounds lent for a year, so for 43 days only ^jV^ths of that sum has to be paid.
Now 17/6 = 8 florins + 1/6 (or, otherwise, seven-eighths of a pound) = £-875 ; so the amount to be paid is :
iixJ|x £543-875;
7 V 4^ that is to say, L^^ x £5-43875.
This yields £2-243 = £2. 4s. lOJd., the answer.
Typical exercises.
There are certain time-honoured exercises of a type such as the following, in which a fair amount of practice is desirable. [Type only here given.]
If 3 peaches cost a shilling, what will 20 cost 1
If I have to pay 15 workmen at lOd. an hour for 8 hours, how much money do I need *?
If the butcher supplies 7J lbs. of meat for 58., what has ho charged per pound 1
70 EASY MATHEMATICS. [chap.
and so on. The last being a troublesome kind of sura fre- quently occurring to housekeepers, but usually and most easily done by tables.*
Examples like these are quite harmless and give needful practice, but when they become complicated a little of them is sufficient, except for discipline, and the more concrete and amusing they can be kept for ordinary purposes the better.
A slight further development, not quite so harmless, is of the following type :
Find the cost of 6 lbs. 11 oz. 9 dwt. at 17s. 8Jd. per ounce.
In British schools there is far too great a tendency to limit all exercises to pseudo-commercial matters. In real business this kind of sum hardly occurs ; and besides, greater interest can be obtained by opening up fresh ground for the sub- ject matter of examples.
A few specimens may be here suggested. A great deal of what has to be laboriously taught later as physics is nothing but simple arithmetic, and could easily be assimilated uncon- sciously while doing sums.
1. If the sound of thunder takes 10 seconds to reach our ears, how far has it come % (See p. 56 for velocity of sound : it travels approximately a mile in five seconds. For more accurate specification the temperature would have to be known.)
* Answers to these sums are as follows : Each peach costs the third of Is., so twenty peaches will cost the third of £1, or 6s, 8d.
The 15 workmen's wages will amount to £5, since 80d. is the third of a pound.
The price is 8d. a lb., since T^d. doubled three times makes 60d. or 5s.
But it will be observed that in each case some accidentally convenient relation is seized and utilised. That is the essence of mental arithmetic: it is a training in quickness and ingenuity, not in mathematics ; and its merits can be appraised accordingly.
VII.] PROBLEMS. 71
2. If a pistol shot is heard across an estuary 15 seconds after the pistol was fired (which can be told by observing the flash), how wide is the estuary 1
3. If light reaches the earth from the sun in 8 minutes, what is its velocity] (The distance of the sun being 93 million miles.)
4. How long does it take to come from the moon 1
5. How long would it take to travel a distance equal to seven times the circumference of the earth 1
6. If it takes 5 years to arrive from a star, how far off is that star ]
7. If a locomotive could be run 60 miles an hour day and night, how long would it take to go round the earth ?
8. How long to reach the sun ? etc.*
Answers should be given in weeks or years or whatever unit is appropriate and most suggestive. This is a good rule always, and is the real use of units to which people are accus- tomed. Conversion of miles into inches is tedious and use- less : but stating a big result in miles, a small result in inches, and a moderate result in feet or yards, is right and illuminating.
* Answers :
1. 2 miles.
2. About 3 miles.
3. 93 million miles -r 8 minutes
03000
8x60 3100
thousand miles per second
.„ — nearly 194 thousand miles per second.
4. About a second and a quarter.
5. About 1 second.
6. 5 X 365^ X 86400 x 194,000 miles.
7. Nearly 17 days.
8. About 180 years.
72 EASY MATHEMATICS. [chap.
9. If a pistol shot fired in a valley, at a spot which is distant from the summit of a mountain by an amount which is represented by a length of 4 inches on an ordnance map of scale 1 inch to the mile, is heard on that mountain top 25 seconds after the flash, how high is the mountain above the valley 1 (Ans. : 3 miles.)
This is perhaps hard : it can be done by drawing and measuring, after it has been perceived that the sound has travelled 5 miles in a straight line.
10. If a motor car is travelling 21 miles an hour, how long will it take to go 100 yards 1
Ans. : 9'74 or 9f seconds.
11. If the estimate of time were f second out, what error would be made in reckoning the speed from the measured distance 1
Ans. : f sec. is j^^th of -Y- sec, so the error in estimate of speed would be about
f J miles an hour, or about 7^ per cent.
12. If a volunteer corps of 84 members shoots 160 rounds a day each for 5 weeks, and if each bullet weighs f of an ounce, what weight of lead will they have expended ?
13. If each bullet needed one halfpennyworth of powder to propel it, and if lead cost 17/- a cwt., what would be the cost, in powder and shot, for a regiment of 12 such corps, in the course of 5 weeks 1
14. If an iron rod expands I per cent, of its length when warmed 200 degrees, what allowance must be made for the expansion of a bridge girder, J mile long, between a winter temperature of - 40° and a summer temperature of 110° ^
15. With the above data how much will an iron rod a foot long expand if warmed one degree *?
16. If a snail crawl half an inch each minute, how far will it go in 3 hours 1
vir.] PROBLEMS. 73
17. If sound goes a mile in 5 seconds, how long would it take tx) go a foot 1
18. If sound reverberated between two walls 10 feet apart, how many excursions to and fro will it make per second ]
19. If light takes 8 minutes to travel 93 million miles, how long would it take to go one yardi How many kilometres would it travel per second? How many centimetres per second 1 ♦
♦ Answers to the above :
12. 84 X 160 X 35 X 5 ounces = about 10 tons.
13. In shot, about £170 ; since a shilling per cwt. is a pound per ton. In powder, 84 x 160 x 35 x i pence = 7 x 80 x 35 shillings = 28 x 35 pounds per corps.
14. The range of tenijjerature is 150*"; for this range iron expands £ of i per cent, of its length ; that is,
w — 57v^ "li^e = ,^-''ru^ feet = ijTjp: = 1 '2375 feet, or nearly 15 inches. 16x800 16x800 800
15. 200*'^ °^ i T^^ ^^^^' °^ *^ length ; which is 0-07^-00*^ °^ * ^*****» or -0000125 expansion per unit length per degree ; which is about the right value for iron.
17. 1 second -7- 1056, or about the thousandth of a second.
18. In each excursion to and fro it will have to travel 20 feet ; but it can travel 1056 feet in a second, therefore it has time to make 52*8 excursions per second. If the walls were only 2 feet apart instead of ten, the rate of reverberation would be 5 times as rapid, and would
correspond to the note /k 1 ~ This therefore is the musical note
heard if a short sharp noise, like a blow or clap, be made between two walls two feet apart.
19. To travel 1 mile, light would take 8 minutes ^93 million ; there- fore to travel 1 yard it would take 1/1 760th part of this.
^"•- = m^m = srnro = m ""'"o"'*- »' " '^'"'-
Light travels 300,000 kilometres per second or 3 x lO*** centimetres per second ; as nearly as experiment at present enables ua to say.
74 EASY MATHEMATICS. [chap.
20. If you buy a large number of oranges at three a penny, and an equal number a-t two a penny, and then sell them all at five for twopence, how much have you lost on the transaction 1
(Ans. : a penny for every 5 dozen sold. The buying price per couple is |d. + ^d. ;
2 the selling price per couple is — -d.
^^ So the loss per couple is^ + J-^ = f-| = wu^-) There are many ways of doing this problem, and it should not be left till it is fully realised. Other problems depend on the same principle, which is an important one. For instance :
21. An oarsman rows a boat a certain distance up a river and back, and then across the river, or on a lake, the same distance and back. Which will be the quickest to and fro journey 1
22. If a steamer travels down a river at a rate of 19 miles per hour, and up the same river wdth the same engine-exertion at 7 miles an hour, what is the speed of the river 1 How long would the steamer take to go a journey of 65 miles and back ?
(Ans. : The speed of the boat in stagnant water is the half-sum, viz. 13, the speed of the river is the the half- difference, viz. 6 miles per hour. The journey of 130 miles would take ten hours in stagnant water, but up and down the river it will take nearly thirteen hours.)
The general principle is that, whereas (1 + a;) + (1 - a;) = 2,
+ does not equal 2
l+x 1
9
but does equal '^ ^, which is greater than 2 ; though not
much greater when x is small. This applies to (20) (21) and (22).
23. If a couple of travellers sharing expenses are found to
be out of pocket in the course of the day. A, £2. 4s. 6d., and
VII.] PROBLEMS. 76
Bj £\. 38. 4(1., what sum must be transferred from one to the other to equalise matters ]
(Ans. : Half the difference, viz. 10s. 7d. ; and the cost to each has been half the sum, viz. £\. 13s. lid.)
24. If three travellers on a tour have expended when they return
^ £17 . 4 . 6
B £4.3.2 C £7.5.4 how can they best arrange to share expenses equally 1
(Ans. Find the mean expenditure by adding the items together and dividing by 3 ; and then take the difference between this mean and the expenditure of each. B and C will then have to pay their respective differences to A. Their two deficiencies from the mean, added together, should equal A* 8 excess expenditure over the mean ; if this is not the case a mistake has been made.)
The same rule would apply to any number of travellers. Observe how it works for the couple of last question.
These exercises do not contain examples of so many quarts, pecks, pennyweights, and drams. Such sums have no business to occur. If artificial complexities of that sort are set, any way of dealing with them will do : the simplest way is the best way.
If a pupil is constrained to bethink himself of how the teacher intends him to do a sum, it destroys originality. His effort should always be devoted to find the best and simplest way. This a teacher can help him to find, but a self-found way is more wholesome in many respects than a coerced way, even though the latter is neater. Originality should always be respected : it is rather rare. Perhaps docility is made too much of, and budding shoots of originality are frozen.
76 EASY MATHEMATICS. [chap.
Binary scale.
Although the natural method of dealing with multiples of a unit is to employ the same system of notation as is in vogue in arithmetic, and although therefore it is natural to specify large numbers of things by powers of ten, there is a natural tendency also to deal with fractions on a different basis, viz. to proceed by powers of -J. AVe see this on a foot rule, where the inches are first halved, then quartered, then divided into eighths, then into sixteenths, and sometimes even into thirty- second j3arts of an inch.
The same method of dealing with fractions is found in prices, as for instance of cotton, or any commodity which requires a penny to be subdivided. Below the halfpenny and the farthing we find the eighth, sixteenth, thirty-second, and sixty-fourth of a penny in use for quotations; and these ungainly figures are, or used to be, even telegraphed and automatically printed on tape. So also a carpenter will understand a specification in sixteenths of an inch, while a decimal subdivision would puzzle him.
A thousandth of an inch is sometimes used however in fine metal fitting work, and the thickness of a rod wanted may be specified to a fitter as the thousandth of an inch greater than 2 j-^^ inch.
These peculiarities are insular and not to be encouraged, having originated in laziness and ignorance ; but they are not nearly so bad as the weights and measures which people who ought to know better still require that children shall be taught.
It is quite possible to word arithmetic itself on the binary scale, counting in pairs only ; thus 10 (read one nought) may be understood to mean 1 pair ; 100 may mean 1 pair of pairs (or 4), 1000 on the same plan will mean 2x2x2 or eight, and so on. And on this scale '1 would mean J, '01 a quarter.
VII.] COINAGE. It
•001 an eighth ; so that one and a quarter plus an eighth would he written 1011.
The natural tendency to this kind of subdivision is apparent in coins, even in countries with a decimal currency. For instance in America you find the half and the quarter dollar, beside the dime and the cent. In France you find the double franc, franc, half-franc, and quarter-franc. So in Germany we have as a drink-measure the halh-liter and viertel-liter. And in England we have half-sovereigns, half-crowns, also three- penny bits, sixpences, shillings, florins, and double florins, each double the preceding; the double florin being roughly equivalent to a dollar or to a five-franc piece.
So also the commonest gold piece in France is the Napoleon or 20-franc piece; not the ten-franc, or the hundred-franc piece, though they both exist.
This natural tendency is the chief difficulty in introducing a purely decimal coinage ; another is the convenience of the penny and the shilling. If a decimal system is to be intro- duced, one or other of these coins must give way. If the ehilling gives way, we can have an approximation to the franc, and much inconvenience or grumbling in connexion with cab fares, etc. If the penny gives way, and is made the tenth of a shilling, we approach closely to the German system; and many commodities used by poor people will automatically rise in price.
In Austria an attempt is being made to replace the gulden and kretUzer by their respective halves, called krone and heller^ which correspond approximately with the franc and centime ; but the older denominations persist, and it is quite likely that the two will co-exist and be convenient.
It may l>o asked "why mention these things in a book of this kind"? And the answer is because children can take an intelligent interest in them, and because it is instructive for
78 EASY MATHEMATICS. [chap.
them to realise that our present coinage is not a heaven-sent institution, but is susceptible of change, — change too in which, when adult, some of them can take their part, either in pro- moting or opposing. There is therefore a reality about these things, and arithmetical ideas can inculcate themselves in connexion with them without labour.
Decimal system of weights and measures.
Although the present division of money is so deep-rooted that decimal coinage is difficult of introduction, and although the decimal system in arithmetic is not the best that could have been devised ; yet its advantages over most other systems are so enormous that in connexion with weights and measures it undoubtedly ought speedily to be introduced.
The first and easiest place to introduce it is in connexion with weights. No one really wants to reckon in ounces and pennyweights and grains and scruples and drachms. Ounces used to be perpetuated and popularised by the Post Office regulations; but now that a quarter of a pound will go for a penny, and, under certain restrictions, an eighth of a pound for a halfpenny, the necessity for ounces has really disappeared. It would be quite easy to make the halfpenny postal regulations refer to a tenth of a pound instead of an eighth, and to construct ten-pound weights, hundred- pound weights, and their convenient' doubles and halves and quarters.
There is however this fundamental question to be considered : shall the British pound be adhered to, or shall we adopt the unit of our neighbours and employ the kilo (short for kilo- gramme) or the demi-kilo ^
The kilo is too big for many ordinary purposes. In France small marketing is still done by the demi-kilo, because it represents a reasonable and commonly-needed amount of stuff.
vn.] MEASURES. 79
It is altogether handier than the kilo. A demi-kilo might be introduced, and with us might still be called a pound, or, for a time, an " imperial pound," though its value would have to be increased by ten per cent, above our present pound. The kilo is approximatel}'' 2*2 lbs., so the new pound or demi-kilo would be one and a tenth old pounds. The gramme would be 002 new pound.
The disadvantages of any change are obvious. The advan- tage would be that we should then be using practically the same unit as our neighbours.
All other denominations could be swept away ; except, for occasional rough use, the ounce and the ton, which continue useful ; for the ton would be 2000 of the new pounds, and would correspond exactly with the French tonne ; and the ounce, slightly changed, would be yV of the new pound, or it might be changed so as to be one-tenth of it. The grain or ttjW P*^ o^ ^^® o^^ pound might easily give place to a new grain YTyh(ru P^^ ^^ *^® "®^ o"®-
These handy names are useful for common purposes and for gpeech. All accurate specifications should be made in terms of the pound, and of that alone. Thus 1*4903 lbs. would be a specification accurate to the nearest grain of a weighing of something like a pound and a half.
3*014 tons would be a statement, intended to be accurate to the nearest pound, of the weighing of a 3-ton mass.
Let me emphasise what may be regarded as one of the special advantages of this simple and easily introduced change. Children could then be practised in weighing at once : to the vast advantage of their education. At present an apothecary's scales are an al)omination, and no child can weigh satisfactorily with the weights of a letter balance, which are all in the binary scale ; though, as aforesaid, these serve as an introduction to ideas of weighing, etc., in quite early stages. lietter weights go
80 EASY MATHEMATICS. [chap.
down too rapidly ; there are not enough subdivisions ; and the result cannot easily and quickly be specified, except as an awkward series of vulgar fractions, or else in the binary scale of arithmetical notation.
The only way in which school weighings can be satisfactorily done now is by the use of grammes and kilogrammes : and there is a foreign feel about these things ; which those who learn chemistry indeed get over, but which gives it a flavour distinct from ordinary life.
What we want is that children shall weigh and measure all sorts of things, and do a large part of their arithmetic in terms of their own weighings and measurings : thus making it real and concrete and if possible interesting.
Weighings of plants and of growing seeds, of rusting iron and of burning candles, of dissolving salts and of evaporating liquids, can all be made interesting and instructive.
Weighings in air and water, and finding thereby the specific gravity or the volume of irregular solids, can easily be over- done and made tedious, but, short of this, such operations are quite instructive.
Gauging and measuring of regular solids is an equally in- structive way of arriving at their specific gravity, or, as it may be more scientifically called, " density." The approximate relative densities of such things as stone, lead, iron, gold, copper, platinum, cork, air, referred to water, are worth remembering : stone say 2*5, lead 11, iron 7, gold 19, copper 8, platinum 21, cork i, air -gj^.
Decimal measures. — Continued.
The introduction into commerce of "the decimal system" is a more difficult matter however. The admirable duo- decimal division of the foot into inches (like that of the shilling into pence) stands in the way. The foot and the
vn.] MEASURES. 81
inch and the yard seem ingrained in the British character, and will give place to the metre and the centimetre only with difficulty.
The fact is that the introducers of the " metre" made a great mistake by not adopting the yard or the foot or some other existing unit as its value : they would also have been wise if they had adopted the pound as their kilogramme, and left the dimensions of the earth alone. It is the magnitude of the human body which really and scientifically specifies and confers any meaning on absolute size : our bodily dimensions and time relations must be the basis of all our measures and ideas of absolute magnitude. To abandon the human body and to attend to the dimensions of the earth was essentially unscientific or unphilosophical : it has all the marks of faddism and self-opinionatedness. However these unwisdoms of sections of the human race we have to put up with, and at any rate the French evolved a better system on the whole than that which had come down to us by inheritance and tradition from uncivilised times.
If we were at liberty to adopt the foot as our standard, and to call its decimal subdivisions inches, or if a new foot were made ten inches long, the change would not be so very difficult. If it had been extensively customary to divide the inch too into twelfths (called lines) the change would be harder ; but divisions of the inch in the binary scale have been customary, and these are not really convenient : a decimal ystem is better than that ; and foot rules decimally divided and sulxiivided could easily be supplied and used.
But then, as in the case of our present pound, we should be using an insular measure different from all the rest of Europe, jind amid the stress of industrial and engineering competition this is a serious handicap.
A metre scale is a rather unwieldy thing: a half-metre
L.K.M. f
82 EASY MATHEMATICS. [chap.
scale is handier for many purposes, and might be made like a folding two-foot rule.
There is no help for it : we must get used to metres and centimetres, and the sooner we begin the better.
Angles and Time.
There are two things which have not yet been subdivided decimally with any considerable consensus of agreement : they are Angles and Time.
The division of the right angle into 90 equal parts is convenient. The subdivision of the degree into sixtieths and again into sixtieths (called respectively partes minutae and partes minutae secundae, now abbreviated into "minutes" and " seconds ") is peculiar and sometimes troublesome but not exactly inconvenient, though a decimal subdivision of the degree would be simpler.
As to time, the fundamental unit is the day or period of the earth's rotation (this being the most uniformly moving thing we know). Its subdivisions (into 24 parts, and then into sixtieths, etc.) are curious, but too deep-rooted for anyone to attempt to alter; and fortunately they are the same in all countries.* Legitimate practice in dealing with different denominations can therefore be afforded to children by our large admixture of universally understood measures of time; including weeks, months of different kinds, years of different kinds, and centuries. All other weight and measure complications, especially those of a merely insular and boorish character, should forthwith cease to be instilled into children.
Further exercises.
It is worth noticing and remembering that a kilometre = 10^ centimetres.
•'^A third subdivision, the sixtieth part of a second, is sometimes known as a "trice,"
vn.] MEASURES. 83
It is also ten minutes' walk, or very roughly two-thirds of a mile.
A cubic metre is a million cubic centimetres.
A cubic kilometre is a trillion cubic millimetres ; meaning by "trillion" a million million million, after the English custom. (But the French use the term " billion " to signify a thousand million ; and a million million they accordingly call a trillion ; while the above number would by them be desig- nated a quintillion : in any case it is 1 followed by eighteen ciphers).
A cubic centimetre is 1000 cubic millimetres, and is -^^j^ of a litre.
A gallon of cold water weighs 10 lbs., by definition of a gallon ; a pint therefore weighs a pound and a quarter.
A cubic metre of water is a tonney and very approximately, though accidentally, equals an English ton also.
A cubic centimetre of water, at its temperature of maximum density, weighs a gramme exactly, from the definition of a gramme.
The speed of an express train, 60 miles an hour, is only 15 times a walking pace.
The speed of a bullet, say 1800 feet a second, is twenty times that of a train.
The speed of sound is comparable with that of bullets.
The speed of light is a million times the speed of sound in air.
Four miles an hour is 2 yards a second, approximately, or accurately 60 miles an hour is 88 feet a second.
It is an instructive exercise to let a lx)y find out the sizes and distances of the planets of the solar system, and cal- culate a numerical model illustrating them on any convenient scale.
84 EASY MATHEMATICS. [chap.
I have myself found a local topographical scale the most convenient : one on which the earth was about the size of a football, and the sun the size of some public building a mile or two distant. The other planets distribute themselves about the town and county; some of them extending into more distant counties.
It is instructive to try to place the nearest fixed star in such a scale, and to find that it will not come on to the earth at all.
The price of a railway ticket to the nearest fixed star, at Id. per hundred miles, can also be calculated; and found to approach or exceed the National debt.
The earth takes a year to. go round the sun in a circle of 93 million miles radius : how fast does it go 1
(Ans. : About 19 miles a second.) Light goes 10000 times as fast as this.
How fast would a train have to run on the equator if it were to keep up with the apparent motion of the sun, so that it should continue the same time of day 1
(Ans. : About 1000 miles an hour.) How far from the North Pole could the same thing be accomplished by a man walking 4 miles an hour ?
(Ans. : About 30 miles away.) If a man walked 30 miles South from the North Pole, and then walked 40 miles due AVest, how far, and in what direc- tion, would he have to go to get back to the Pole 1
(Ans.; 30 miles due North.) What is the density of a rectangular block whose height is 5 inches, length 11 inches, breadth 8 inches, and weight 821 lbs. 1
(Ans. : 3 ounces per cubic inch.)
VII.] EXERCISES. 85
Directly the elements of mechanics and of heat and of chemistry have been begun, any number of useful and fairly interesting examples can be constructed. They afford practice in arithmetic of the best and most useful kind; quick and ingenious computation being what is wanted, not laborious dwelling upon long artificial sums. Long sums are never done in adult practice : there are always grown-up methods of avoiding them.
It is cruel to subject children to any such disciplinary process, as part of what might be their happy and stimulating education. Before they have been subjected to it they are often eager to have lessons; but experience of the average lesson, as often administered, soon kills off any enthusiasm, and instils the fatal habits of listlessness and inattention which check the sap of intellectual growth for a long time.
If the teacher of arithmetic knows arithmetic and nothing else, he is not fit to teach it. His mind should be alive with concrete and living examples ; and it is well to utilise actual measurings, weighings, surveyings, laboratory-experiments, and the like, to furnish other opportunities for arithmetical exercises.
Arithmetical exercise can be obtained unconsciously, as bodily exercise is obtained by playing an outdoor game. The mechanical drill or constitutional-walk form of exercise has its place doubtless, but its place among children is limited. There used to bo too much of it, and too little spontaneity of bodily exercise, in girls' schools. Now the spontaneity and freshness is permitted to the body, but too often denied to the mind.
The same kind of reform is called for in both cases. The object of this book is to assist in hastening this vital reform.
CHAPTEK VIII.
Simple proportion.
Any number of sums are of the following character :
If 3 sheep cost £20, what will 100 cost ?
Now the so-called "rule of three" method of dealing with sums of this kind, though permissible, is not really a good method, because it leads to nothing beyond, and employs an antiquated system of notation.
The answer is one hundred thirds of twenty pounds = i§^x£20 = 20^00 _ £666-6 = £666| = £666. 13s. 4d.
If the answer is not obvious, it can be arrived at by the intermediate step of considering one sheep, which will cost the third of £20, namely, £6. 13s. 4d.*
And so a hundred sheep will cost 600 pounds, 1300 shillings, and 400 pence.
The 1300 shillings reduce to 65 pounds, since 100 shillings is five pounds; and the 400 pence make £1. 13s. 4d., since 240 pence is a pound, and so 400 pence is thirty shillings and 40 pence (or 3s. 4d.) over.
This is not an orthodox way of doing the sum, but it is just as good as any other, and it is one that a boy might
* [It would not come out even so well as this but for the fortunate duodecimal division of the shilling into pence, so that one-third of a pound, viz. 6s. 8d., and two-thirds, viz. 13s. 4d., can be exactly specified without fractions. These amounts arc worth remembering as one-third and two-thirds of a sovereign.]
CHAP. VIII. i SIMPLE PROPORTION. 8t
scheme for himself. There would be no need to snub him for it. Everything which is troublesome about such a sum results from the miserable property of the number ten, that it is not divisible by 3.
If we had set the following very similar question :
If 3 sheep cost £24, what would 100 cost ? An infant could answer £800, doing it in its head. But it would clearly do it by the same process, viz. the process of considering the price per single sheep, and that is therefore the natural and simplest method.
To summarise : The childish method is the method by units, and may be written out at length ; the adult method is the method by ratio; what place is there for the rule of three 1 The rule of three with its symbols : : : : is reserved for antiquated school instruction.
Observe, there is no harm in writing a ratio as 2 : 3 or a : 6,
and occasionally it may be convenient to do so, though 2 -f 3,
2 or a -r 6 is precisely the same thing, and usually the form ^ or
o
j-j or rt/6, is in every way better. So the symbol : : is needless,
because replaced by =. The fact is that : connotes the theoretical idea of ratio, while -f indicates the practical operation of division, which is the actual means of working a ratio out. The vulgar-fraction form may be used instead of either of these signs and is usually best. The division may or may not bo actually performed, as we please.
I feel inclined to illustrate good and bad methods at this stage a little further, by taking a few more very simple examples. For instance :
If twenty dogs pulling equally at a sledge exert a hori- zontal force of 1 cwt., what force do any three of them exert?
EASY MATHEMATICS. [chap.
Adult method
3., ., , 3x112 lbs. ICQ,,
-— ths of 1 cwt. = ^^ = 16 "8 lbs.
20 20
Good cMldish method :
20 dogs pull 112 lbs. 10 dogs pull 56 „
1 dog pulls 5 "6 ,,
3 dogs pull 3 X 5-6 = 16-8 lbs.
If it be asked why not stop at /^ths of a cwt. and give the answer as '15 cwt., I reply, no reason against it at all ; but children should be accustomed to realise forces and other things, in actual homely units that they can feel and appre- ciate ; and a cwt. is too big for them.
Mechanical method :
20 : 3 :: 112 : the answer. Rule. Multiply the means and divide by one extreme and you get the other extreme.
.'. the answer is, etc.
British Method:
There is indeed a barbarous way of complicating the sum, which is typical of much that goes on in these islands :
lbs. oz. drachms 20 1 336. 0. 0
16 . 12 . 12| which is done thus :
Twenty into 336 goes 16 and 16 over, that is 16 lbs. over, which equals 256 ounces. Twenty into this goes 12 times and 16 over, that is 16 ounces or 256 drachms ; into which twenty again goes 12 times and "2 oths over, which last equals -|ths, that is g^ths of a drachm.
So the answer is 16 lbs. 12 oz. 12y drachms.
On this one has to remark that since the unfortunate | has to appear (as it happens) sooner or later, why should it not appear at first "? Why is 4ths of a drachm easier to understand than ^ths of a pound*? The fact is that it is not easier to understand, and by children is not understood : the " 4 over " which remains at the end is a continual puzzle to them.
vui.] SIMPLE PROPORTION. 89
They have been so accustomed to getting rid of fractions by reducing to a lower denomination, that at the end, when lower denominations unaccountably fail them, they are non- plussed. Quite rightly so ; the fault i^ not with the children. Whenever an attentive child finds a persistent difficulty, teachers should be sure that there is something wrong with their mode of presenting it, probably with their own compre- hension of it. Nothing is difficult when properly put. The whole art of teaching should be so to lead on that everything arrives naturally and easily and happily, like fruit and flowers out of seeds.
Another British method. Usually however the sum is not recorded so briefly as this, but is written out in what is known as the long- division plan ; and it is perhaps the safest mode of getting the right answer if the answer is required to be thus barbarously specified, for it certainly shirks nothing. This is the way of it :
To divide 336 lbs. av. into 20 equal parts
lbs. oz. dr.
2 0)33,6(16 . 12 . 12^ 20
196 120
16 16*
96
15-
256 oz. 240
16 _I§*
06
15_
256 dr. 240
16 remainder, and ^ = $ dr.
•If any mathematician glances through this book, as I hope he may, he will require at these stages to be reminded if British, to be informed
90 EASY MATHEMATICS. [chap.
This may look like a parody, but it is soberly the way in which innumerable children have been taught in the past to do such a sum. And the fact that they have been so taught can easily be tested by setting it to people who were children a few years ago.
Another method. If the factor plan of division is adopted there is great danger of confusion and error about the carrying figure. For instance, in dividing 336 lbs. into 20 equal parts, a child as sometimes now taught will proceed thus :
2 [336 lbs. 1 0 1 16,8
16 and 8 over.
8 what over ? They are apt to take it as 8 lbs. over, and so interpret it as 128 ounces, and proceed to divide these again by 20 by the same process
2 1 128 10 1 64
6 and 4 over
apt to be called 4 ounces over, which are interpreted as 64 drachms, and so on.
if Foreign, that in these islands a drachm is defined to be the sixteenth of an ounce, and that an ounce avoirdupois is one sixteenth of an avoirdupois pound ; moreover that a drachm is the lowest recognised denomination of avoirdupois weight : after that fractions are permitted. Pennyweights and grains belong to a system of measures to which the name of "Troy" is (for some to me unknown reason, perhaps from Troyes in France) prefixed. There is a ** Troy pound" and a "Troy ounce," for "metallurgical" use, but they difler from their "grocery" cousins which are explicitly asserted "to have some weight." Then between grains and Troy ounces there are other denominations used by "apothecaries," called scruples and drams. This dram is not the same as the grocery drachm. There appears however to be only one kind of "grain," and 7000 of these make 1 lb. avoirdupois, while 5760 of them make 1 lb. Troy.
viii.] SIMPLE PROPORTION. 91
This is quite wrong. The 8 over in the first little sum was really 8 double-pounds, and so the second little sum is all wrong. If it had been right, the 4 over could not have been 4 ounces, but 4 double-ounces ; but what needless trouble and risk of error is introduced by having to perceive this !
Again let many children be asked to divide JC336 by 25, they will few of them have been taught to proceed thus :
%^/=3-36x4 = £13-44
= £13. 8-88.
= £13. 8s. 9'6d. or about O^d.
but they will proceed, either by long division on much the same lines as in the last example, which is long to write, oi* else by short division, dividing by 5 twice over, which is not too long to write,
£ s. d. 5 I 336 . 0 . 0
5|67 .4.0 13 . 8 . 9f
short to write, but rather hard to do. Such trivial sums should not call for so much brain power as is involved in various and complicated carryings.
Money sums however are the best examples of the kind. If it was 336 tons that had to be divided into 25 equal parts, grown people would be satisfied to say that each part must be 1344 tons; but at some schools it would have to be done thus, — if not by a still longer process equally liable to acci- dental error :
tons, cwt qra. lb*, cm. dr. 51336 .0.0.0.0.0
5|67.4.0.0.0.0
13 . 8 . 3 . 5 . 9 . 9J Ans.
92 EASY MATHEMATICS. [chap.
Breakdown of simple proportion or "rule of three."
Simple proportion, or the- rule of three, is by some teachers regarded as a kind of fetish ; moreover its extreme simplicity makes it ^ rather favourite rule with children and they will naturally do many exercises in it. Not always, it is to be hoped, by the same mechanical method.
But there is all the more necessity for bringing home to them the fact (strange if it is unknown to any teacher), that it does not always work. For instance :
A stone dropped down an empty well 16 feet deep reaches the bottom in one second. How deep will a well be if a stone takes two seconds to reach the bottom 1
The answer expected is of course 32 feet; but it is not 'right. The correct answer is 64 feet.
If a stone drops 16 feet in one second, how far will it drop in J second 1 (Ans.: 12 inches.)
Again, if a stone dropped over a cliff descends 64 feet in 2 seconds, how far will it drop in the next second ]
(Ans. : 80 feet.)
A steamer is propelled at the rate of 8 knots by its engines exerting themselves at the rate of 1000 horse power. What power would drive it at 12 knots ^
Probably no one would expect the answer 1500 to this ; for on that principle 10000 horse power would propel it at 80 knots.
An initial velocity of 1600 feet a second will carry a rifle bullet 3 miles. What velocity would carry it 6 miles 1
An ounce weight drops 4 feet in half a second. How far will a pound weight drop in the same time ?
(Ans. : By experiment, 4 feet likewise. A most important fact, discovered by Galileo, and illustrated from the tower of Pisa.)
VIII.] NON-SIMPLE PROPORTION. 93
Let it not be dogmatised on, but illustrated by dropping things together ; and if it appears puzzling, so much the better. Ignoring or eliminating the resistanco of the air everything fulls at the same pace. The air has very slight influence on the drop of smooth spheres through a moderate height. Cotton wool and feathers and bits of paper will drop more slowly, but the reason is obvious : a bullet will drop more slowly in treacle than in air. That is because the air resistance is small : it is not zero, and if a bullet and a pea were dropped from too great a height, air friction would begin perceptil)ly to retard the lighter body. So it is that big rain-drops fall quicker than little ones ; and these small drops quicker than mist and cloud globules. So also does heavy fine powder, even gold powder, fall slowly in water, not because it is buoyed up, but because it is resisted. Remove the air, and in a vacuum a coin and a feather will fall at the same rate. The statement does not explain the fact. The full explanation of the fact is not even yet known. But a very great deal more is known about the whole subject than is or can be liero expressed. That is characteristic of elementary books through- out, and the object of the learners should be to get through all this easy stuff, and get on into more exciting matters l>eyond : matters which the majoiity of the human race never have the least knowledge of, because their early education has ])ee!i neglected.
A balloon 18 feet in diameter can carry a load equal to one man. What load can a similar balloon Ciirry which is 36 feet in <liametcr. (Simplest rough answer, 8 nuni.)
A rope stretches half an in(;h wlnni kmdcd with an extra hundiedweight.
How much would it stretch if loaded with an extra ton f
94 EASY MATHEMATICS. [chap. viii.
A half crown is ten times the value of a threepenny bit. How many threepenny bits can lie flat on a half-crown without overlapping the edge *? (Ans. : By experiment, one.)
A boy slides 20 yards with an initial run of 10 feet. What initial run would enable him to slide half a mile ?
If 2 peacocks can waken one man, how many can waken six?
If a diamond is worth ten thousand pounds, what would 950 similar diamonds be worth 1
If a camel can stand a load of 5 cwt for 6 hours, for how long could he stand a load of ten tons ?
These things cannot be done by simple proportion. They require something more to be known before they can be done at all ; and accordingly it would appear as if generations of teachers had discreetly shied at them all, indiscriminately, and had excluded them from arithmetical consideration altogether. It is just as if in geometry, finding straight lines simpler than curves, they had agreed to found all their examples upon straight lines.
CHAPTER IX.
Simplification of fractions.
VuuJAR fractions are much harder to deal with than decimals ; but as sometimes several have to be added together it is desirable to know how to do it. Besides, the exercise so afforded is of a right and wholesome kind.
Consider the following addition : h + i- Small children can see (by experiment on an apple) that the result is |, and they can also be taught to regard it as j + J = J, which should be read in words — two quarters added to one quarter make three quarters.
Thus, it can be realised that when the denominators are all the same, addition of fractions becomes simple addition of the numerators.
For just as 5 oranges + 6 oranges =11 oranges, so
ir + TT - 11 f reading " seventeenths " instead of " oranges."
When denominations differ, therefore, the first thing to do is to make them the same.
Thus, for instance, 3 apples + 4 oranges, is an addition which can only be performed by finding some denomination which includes both, say " pieces of fruit."
So also 7 horses + 3 pigs = 10 quadrupeds. 5 copies of Robinson Crusoe + 3 copies of fvanhoe = 8 prize-books, perhaps.
96 EASY MATHEMATICS. [chap.
Reduction to the same denomination cannot always be done, when denominations are anything whatever, except by using the vague term "objects" or "things"; but with numerical denominators it can always be done, and the method of doing it has to be learnt, y'^tt == \h ^"^ ^^^^ ^i^^* ^^6 easy examples. ^ + T2 ^ |^ is a slightly harder one.
It is done by saying j% + ^V = i% = h
So also J + ^ = fi, being equal to 4.
A harder example is ^ + f , which can be written
21.10_31_F;1
In the decimal notation this would appear thus :
3-5 + 1-666... = 5-1666....
A still harder example can be worked out thus : «L 4- 5 _ fi. 3 I j4 p _ i_o_3 _ 1 47
»'7"~50~56 56 ^5C»
though the final step is one that need not always be made.
Now it is evident, or at least it will gradually be found true, that in a mechanical process of this kind there is always some simple rule by which the result can bef obtained without thought. What is that rule *? If the child can find it out for himself, by experimenting on lots of pairs of fractions, so much the better. A week is none too much to give him to try, for if he finds it out himself he will never forget it.
The rule is : cross-multiply for the numerators, and multiply the denominators.
1 1 _ 6 + 2 _ 8^2
1 l_h + a
a b ~ ab '
3 4 _ 27jf 28 _ 55
7 "^9" 63 " ~ 63'
a c _ ad + bc
b'^d~ bd '
but it would be a pity to spoil this by premature telling.
IX.] FRACTIONS. 97
The fact that the sum of two reciprocals is the sum of the numbers divided by their product, is worth illustrating fully and remembering: remembering, that is, by growing thoroughly accustomed to it, not exactly learning by heart. There is hardly any need to learn easy things like that by heart: nevertheless it is a very permissible operation, whenever the fact to be learnt is really worth knowing.
5 + 7 = Y5, that is the
3 4 12' product
11 a + b
J_ 1_ 28 23'*'5~Il5*
1 1 _51 52 _ -„
2 + 49~98'^l00- '^'^'
the symbol «£fe meaning "approximately equals."
[The approximation is seen to be true because adding 1 to
50 makes the same proportional difference as adding 2 to 100.
If this is too hard, it can be postponed. It is unimportant,
but represents a kind of thing which it is often handy to
do in practice.]
But this rule of cross-multiplication hardly serves for the (Idition of three or more fractions, at least not without
iiiodification. Take an example.
|
^4.24.7 14 21 26 , |
|||
|
Take another, |
111 4+2+1 7 2'''4"^8~ 8 8' |
||
|
where the three |
fractions J, |, and J, all having |
the |
same |
|
denominator, are |
written all together, with the addition of the |
||
|
numerators indicated, and subsequently performed. |
|||
|
One more, |
1.1.1 3 + 4i + l 8\ 17 3-*-2'*-9^ 9 -9"18- |
This might hardly bo considered a legitimate procedure, but
Xi.B.M. G
98 EASY MATHEMATICS. [chap.
there is nothing the matter with it. You might, instead, proceed thus :
1 1 1_18 27 A_^l-iZ
3"^2"^9 - 54 "^54"^ 54 " 54 ~ 18'
and that is equally a correct method.
But neither of these plans is quite the grown-up plan. Let a better plan be found; but first let the above plans be formulated and expressed. Is it not plain that the numerator of each particular fraction is found by multiplying two of the denominators together, while the common denominator of all the fractions is found by multiplying all the denominators together 1 Apply this rule :
111 _ 20 + 24 + 30 _ 74 37 6 5"^4 120 ~i20~60'
For instance, a sixth of an hour + a fifth of an hour + a quarter of an hour = 37 minutes, a minute being the sixtieth of an hour. Now a sixth of an hour is ten minutes, a fifth is 12 minutes, and a quarter of an hour is 15 minutes: conse- quently the neatest way of doing the sum would be
1 1 1 _ 10+12+15 37 6"*"5"^4~ 60 "60*
. ^, 1 111 180 + 36 + 720
Another example, 1 f- - = — — .
^ ' 12^60 3 720x3
but here every term in numerator and denominator can be
divided by 3 and by 12, so that the above may be written
J_ 1 1 _ 5+1 + 20 _ 26 _ 13 ^ ^ ^ 12^60^3" 60 "60 3d "
And it would have been neater to write it so at first — neater but not essential, and sometimes not even the most rapid plan.
To illustrate the above example :
•joth of a day is 2 hours, •j^^th of a day is 24 minutes, ^rd of a day is 8 hours.
IX.] FRACTIONS. M
Consequently the sum of these fractions of a day is 10 hours and 24 minutes,
which is \0l J of an hour [ = lOiV = 10'4 hours] or ^ J + uV o^ * day, which again may be written f J + vV = ii = ijths = KMS, as before. The form of the general rule, then, is given by J. 1 1 _ bc + ca + ab , a ^ c ~ abc ' but in practice it is possible to abbreviate this in some cases, when one of the denominators contains the others as factors, or when some simple relation of the kind exists between them. This is what was made use of in the early simple cases,
such as tV + AJ ^® *^^^ '*°* proceed to write ^^r^^— and
then simplify it, but we wrote at once /j + tjV = irj J ^^^^ i^ to say we perceived that 24 would do for the new denomin- ator, and we adjusted the numerators accordingly.
Perhaps we had better display this algebraically. Let each denominator contain a common factor, say n, so that the
reciprocals to be added are — + -r + - j t,hen if we applied the * na no nc ^^
II u ij -^ ri^bc + n^ca + n^ab , ^ , mere general rule we should write g-r , but the
repetition of the powers of n is manifestly needless, since they
cancel out ; and it is much neater to write for the new
denominator an expression which contains the common factor
, . bc-i-ca + ab n only once, thus : ^
The denominator so obtained is called the least common multiple of the three denominators ; and it is frequently, in examination papers, denoted by the letters L.G.M. It is not an important idea at all. Sums can be done quite well without it, but its introduction affords some scope for neat- ness and ingenuity. Easy processes can be given for finding
100 EASY MATHEMATICS. [chap.
it, but they are hardly worth giving, as in real practice they are seldom used : they are of most educational service if employed as an exercise for the student's invention. They will be dealt with sufficiently in the next chapter. Now take a numerical example :
Add together | + t + 1 + TF + /a*
Here 32 is evidently the l.c.m. of the denominators, since it contains all the others as factors. So that will serve as the simplest common or combined denominator. The first numerator accordingly will be 16, the second 8, the third 4 but taken 5 times and therefore 20, the next 2 taken 3 times, and the last 1 taken 7 times.
Consequently the sum is written as follows :
115 3 7 _ 16 + 8 + 20 + 6 + 7 _ 57 2'^4"^8^16"^32"" 3Q ~ .32*
Take another example of addition :
1 1 1 2. _ 72 + 9 + 56 + 8 _ 145 7'^56"^9'^63~ 504 "504
Here 7 is plainly a factor of both the larger denominators, and 8 and 9 are the other factors, so the least common denominator will only contain 7 ard 9 once, and will equal 7x8x9 = 504, and this being the smallest common multiple possible, no further simplification can be effected ; beyond of course expressing the result as a decimal if we so choose. To express it as a decimal we must effect the division indicated ; the result happens to equal "2877 almost exactly.
It is worth noticing that the series of powers of v, viz. :
^+4'+8+ TF +32 + 64 + •••
add up very nearly to 1 ; and the more nearly the more terms of the series are taken.
It can be shown, not by trial indeed, but by simple reason- ing, that if an infinite sequence of this series are added together the result is exactly 1. Thus the first term con- stitutes half of the whole quantity, say a loaf, the second term added to it gives us three quarters, the third term gives us Jth more, and we only need another eighth to get the
IX.]
FRACTIONS.
101
whole. The next term gives us half of the deficiency, and now we need the other sixteenth to make the whole. Wo do not get it however : we get half of it in the next term, and thus still fall short, but this time only by ^V ; and so at the end of the above series, as far as written, our deficiency is ^\ th. Each term therefore itself indicates the outstanding deficiency, and as the terras get rapidly smaller and smaller, so does the deficiency below 1 get rapidly diminished till it is imperceptible. (Ck)mpare p. 325.)
It is convenient to plot these fractions as lengths (setting them up at equal distances along a horizontal line), say half a foot, then a quarter, then an eighth, and so on. Then joining their tops we get a curve which has the remarkable property of always approaching a straight line, but never actually meeting or coinciding with it, or at least not meeting it till infinity ; when at length it has become quite straight.
pio, a.
There are many curves with such a property, but fig. 9 may bo the first a child has met. Ho can of course continue the curve in the other direction — the direction of whole numbers, or powers of two, and observe how rapidly it tilts upwards ; but there is no straight line in this direction to which it tends to approach ; this end proceeds to infinity l>oth upwards and sideways, not only upwards, though it proceeds far more rapidly in the vertical direction than in the horizontal ; and this end of it never becomes strai^^ht.
CHAPTER X.
Greatest Common Measure and Least Common Multiple.
Another name of slight importance, which is usually paired off with Least Common Multiple (page 99), is Greatest Common Measure or Highest Common Factor : often denoted by G.c.M. or by H.C.F.
The two numbers 24 and 16 have several factors common to both of them, for instance 8 ; and this as it happens is the greatest common factor, the others which they possess in common being 4 and 2.
The numbers 20 and 35 have 5 as the largest factor common to both of them. The numbers 72 and 84 have 12 ; while 72 and 96 have 24 as their G.c.M.
The numbers 23 and 38 have no factor, above unity, common to both. In fact 23 has no factor at all.
The word "common" so used does not mean "ordinary," as children sometimes think, nor does it mean vulgar, but it has the signification which it possesses in " common friend," or in vulgar phrase " mutual friend," or when people are said to own property "in common."
To find common factors of two numbers, one way is to arrange all the factors of each in two rows one under the other and see how many correspond. Inspection will then readily show which pair is the biggest.
Suppose the two numbers given were 40 and 60 ; the following are the factors of 60,
2, 3, 4, 5, 6, 10, 12, 15, 20, 30, and the following are the factors of 40, 2, 4, 5, 8, 10, 20.
CHAP. X.] GREATEST COMMON MEASURE. 103
Of these, the 2, 4, 5, 10, 20 are common to both, and 20 is the largest of them.
In old-fashioned language, factors were called "measures," and the largest common factor was called the "greatest common measure," and abbreviated into G.C.M.
What is the use of it 1 Very little ; but the meaning is perfectly simple and should be understood. It can be utilised for finding the Least Common Multiple of a set of numbers, that is to say the smallest number which contains them all as factors ; for the Ci.c.M. represents what may be struck out, once at least and sometimes more than once, from the product of a set of numbers, in order to leave behind the smallest number which they are able to divide without a remainder.
Thus take the numbers 40 and 60, their product is 2400, and of course they will both divide that ; but their G.C.M., 20, may be cancelled out of it, leaving 1 20 ; and both 40 and 60 will divide that too. It is the least number which they can both divide exactly, i.e. it is the least number of which they are both factors, it is in fact their least common multiple.
Example.— Of the numbers 12, 20, 24 what is the Ci.C.M. and LC.M.1 Of these, 12 need not be attended to, because it is itself a factor of 24.
Of the numbers 20 and 24, 4 is a common factor ; so divide all by that, and we get left with 3, 5, 6.
No factor will divide all these, so 4 was the largest common factor, or g.c.M., of the original numbers.
Their least common multiple is not 5 x 6 = 30, because that would have omitted the factor 4 which they possess in common. The common factor need not be repeated more than once, (for if it is, though you get a common multiple, you do not get the least common multiple), but it must not be omitted altogether, or you will not get a common multiple at all.
104 EASY MATHEMATICS. [chap.
The L.C.M. accordingly is 5x6x4 = 120, and of that it will be found that the given numbers 12, 20, 24, are factors.
A process for finding the L.C.M. of a group of numbers may therefore be described thus : Keduce the size of each member of the group as far as possible by striking out every common factor which any pair or set of them possesses, at the same time keeping a careful record of each cancelled factor; then multiply together all the reduced numbers and all the factors that have been struck out. If the reduction has been carried to the uttermost this product is the L.C.M. ; but there must be no cancelling among the separated factors.
A rule for finding the g.C.m. is by no means so easy to invent: it is an ingenious process, and the whole subject is essentially a little bit of rudimentary pure mathematics; it has no practical importance or application except when dealing with the properties of numbers.
The proof of the rule is an interesting and easy exercise in the application of reason and commonsense to arithmetic, but beginners can defer it.
Rule for finding G.C.M.
The rule depends on the demonstration that any factor of two numbers is likewise necessarily a factor of the remainder left when one is divided by the other.
Thus consider the two numbers 40 and 24. Divide one by the other, we get 1 and 16 over. The above sentence in black type assumes or asserts that every factor of 40 and 24 must also be a factor of 16. In this case, as a matter of fact,
40 = 24 + 16 and it is manifest that a number which divides 24 and does not divide 16 cannot divide 40.
AVell that is the whole idea.
X.] GREATEST COMMON MEASURE. 105
If wo were told to find the G.c.M. of 40 and 24, we could by this means reduce the problem to finding the G.c.M. of 24 and 16. And then, repeating the division process, we should observe that 24=16 + 8,
so that the problem becomes reduced still further into finding the G.c.M. of 16 and 8. There is no question but that this is 8; — as indeed we might have guessed at first if our object had been attainment of a result, instead of explication of a process — and the way to clinch that is to perform the division again and to find that there is now no remainder at all.
The matter can be stated algebraically, but beginners can skip the algebra and come to the " illustration " which follows.
Algebraic proof of the process for finding (LC.M.
To find a common factor of two numbers P and (2, of
which P is the bigger,
let X be one common factor,
P Q
then — and - will be the complementary factors.
X X
An extreme case is when P is divisible by Q without a remainder, in that case x = Q. Suppose however that when P is divided by Q the remainder is L\
Q)P(n nQ
li 80 that P = nQ + R\ then if 7? is a factor of Q it must be one of P also (because P equals a multiple of Q plus li)^ so try if li is a factor of Q.
If it is, it is the common factor required ; but if not, work out a division again, and let the remainder be -S',
R)Q{m mil
S so that Q = mli + S,
106 EASY MATHEMATICS. [chap.
Then if /S' is a factor of R it must be one of Q too, and so also of P, and in that case *S' will be the common factor required.
But if not, we must repeat the process and sec what the remainder is when R is divided by S. Call it 1\
S)R(l IS
so that R = IS+T. ^
Now once more if 7" is a factor of S it is necessarily a factor of Rj and therefore of Q, and therefore also of F, and so 2' is the common factor required.
If not, the process must go on until there is no further remainder; and then the last remainder (or divisor) is a common factor of the two original numbers P and Q, Let us assume that T divides S without a remainder, then T is the common factor of all the numbers P, Q, R, S, T.
It is likewise the largest common factor which exists. Why 1 because it has to be a factor not only of P and Q but also of R, of Sj and of T ; and certainly T is the largest factor of 2\ therefore it is likewise the largest common factor of the others.
Statement in another form.
The whole process can be written thus :
To find the G.C.M. of P and Q, work successive division sums thus : P R
Q S
R^'^'-PC
R _ T T
S ~ ^"^S'
P 1 1 1
or -^ = 7t 4- TTTT. = n-\ r- = 71 +
Q/R ' 1 ■ 1 '
"^"■r/s '^"+— r
''^Sjf the process terminating only when S/T is an integer.
x] GREATEST COMMON MEASURE. IQJ
The r is a factor of all the numbers P, Q, R, S, T\ and since it must satisfy this condition if it is to be a factor of P and Q at all, it is necessarily the greatest common factor of P and Q, and indeed of the others too.
Or the whole process may be written (as usually performed)
in one sum thus :
Q)P{n nQ
R)Q{m mR
S)R(l IS
T)S(k kT
Then the last remainder (or divisor) T is the g.c.m. of P and Q.
Illustration (modified from Kirkman and Field).
Let the two numbers be 492 and 228. Go through a process of successive divisions.
228)492(2 456
36)228(6 216 12)36(3 36
Hence 12 is the G.C.M. of the two original numbers, and it likewise is a factor of the intermediate divisor, viz. 36.
The argument runs as follows :
Tlio common factor of 492 and 228 must also be a factor of the remainder when 492 is divided by 228, for in fact 492 - (2x228) + 36,
108 EASY MATHEMATICS. [chap. x.
so that anything which divides 228 and fails to divide 36 cannot possibly divide 492.
Hence the problem reduces itself to finding the common factor of 228 and 36.
But now 228 = (6x36) + 12,
hence the factor required must likewise divide 12, as well as 36. The numbers 2, 3, 4, 6, 12 all satisfy that condition, and hence all these are factors of both the original numbers, but of them 12 is the biggest.
Therefore 12 is the G.C.M. of the two given numbers 492 and 228. (Verify this by actual division.) The quotients are 41 and 19, and neither of these has any factors at all, as it happens, i.e. they are what are called "prime" numbers, being not divisible without a remainder by anything but unity. It is not necessary that the residual numbers be absolute primes, like this ; all that is essential is that they shall be prime to each other, that is they must have no common factor ; and to make sure of this in the case of a pair of large numbers it is safest to go through the process of finding their G.C.M. before you can be sure that they have none.
But, in addition to being relatively prime, they may happen to be absolute primes too, that is may have no factors at all, as in this case.
The following is a list of the early " prime numbers," and a beginner can easily extend the list as an exercise :
1, 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, etc.
The prime numbers, and indeed the factors of other numbers when there are any, have all been picked out and tabulated through the first nii^e million natural numbers. The law of their succession is remarkably complicated, and the enumera- tion of prime numbers is a subject which has engaged the attention of advanced mathematicians.
CHAPTER XL
Easy mode of treating problems which require a little thought.
Many of the problems set for purposes of arithmetic are best done in the first instance by rudimentary algebra, that is by the introduction of a symbol for the unknown quantity, so that it can V)e tangibly dealt with. This introduction and manipulation of a symbol for an unknown quantity need not be discouraged, even from the first. It confers both power and clearness. Many arithmetical sums are needlessly hard because x is forbidden. There is a certain amount of sense in the artificial restriction, but in complicated sums and in physics the symbolic treatment of unknown quantities is essential, and the sooner children are accustomed to it the better.
The introduction of a symbol for an unknown quantity is a device to enable a sum to be clearly and formally stated. After the sum has l)een solved by this aid, it is well to try and express it so that it can be grasped and understood without such assistance. The fear of those who object to x in arithmetic is that this final step may be omitted. The grasp is clearer when an auxiliary symlx)l can be dispensed with ; but that is not possible always at first. The 2 is to be thought of as a kind of crutch : but sometimes it is like a leaping-pole and enables heights to be surmounted which without it would be impossible.
110 EASY MATHEMATICvS. [chap.
Example, — How soon after twelve o'clock will the hour and minute hand of a clock again be superposed 1
It is plain that it is soon after 1 o'clock, and that it is an amount which has been traversed by the hour hand while the minute hand, travelling twelve times as quickly, has gone that same distance and 5 minutes more ; but it is not easy to think out the required fraction in one's head, though ex- ceptional children can do it.
But let it be postulated as n minutes after 1 ; the hour hand travels, starting from mark I, a distance n, while the minute hand, starting from mark XII five minutes further back, has to travel b + n in order to catch it up ; so the relative speeds of the two hands are as (n + b) : n, and are also as 12 : 1 ; wherefore ^ + 5 12
~^ ^T' or 12w = 71 + 5,
or IIti = 5,
5
^ = H'
and so the time required (or the answer) is five minutes and five elevenths of a minute (i.e. jj hour) past one o'clock.
Take another question. — Start with a clock face indicating 9 o'clock, and ask when the hands will for the first time be superposed.
The slow-moving hand has forty-five minutes' start; so, how- ever many minutes it goes, the quick one has to go 45 minutes more, at twelve times the pace. Wherefore x + ib = 1 2a:, or the meeting point is -jt == '^tt minutes after the mark IX; or yf ths of 45 minutes, i.e. j^ths of an hour, since 9 o'clock. The start in this case is nine times as great as was allowed after one o'clock, in the previous question, and accordingly the distance before overtaking occurs is likewise nine times as great : in accordance with common sense.
XT.] PROBLEMS. Ill
The constant occurrence of 1 1 in such sums shows that 1 1 must have a decipherable meaning : it means the excess pace, or relative velocity, of the quick hand over the slow. And when this has been perceived, the easiest way to do such sums in the head is self-suggested, viz. to treat it as a case of relative velocities, with the hour hand stationary, and simply ask how soon the minute hand will move to where the hour hand tons, if it (the minute hand) went at jiths of its real speed.
The interval between successive overlaps is therefore always yfths of an hour, or 65y\ minutes.
Exercise. — The hands make a straight line at 6 o'clock, when will they be at right angles % Ans. : One has to gain relatively 15 minutes on the other, and since its relative speed is {4ths of an hour per hour, the time required is 15x|J minutes, that is to say ly*y minutes more than a quarter of an hour.
Pains should always be taken to express an answer com- pletely and intelligibly. If any joy is taken in work, it should be decorated and embroidered, so to speak, not left with a minimum of bare necessity.
Moreover, never let it be taught (as Todhunter taught) that the x or other symbol so employed is always necessarily only a pure number. When we say " let x be the velocity of the train," or "the weight of the balloon," etc., we should mean that a; is to stand for the actual velocity, the actual weight: however they be numerically specified. (Appendix II.)
Some teachers of importance will demur to this. I assert with absolute conviction that it is the right plan, and will justify it hereafter. But it is a matter for adults to consider, and is only incidentally mentioned here.
The dislike felt by teachers of arithmetic to the intro- duction of X prematurely, is because there is a tendency
112 EASY MATHEMATICS. [chap.
thereafter to do arithmetical problems so easily that their features are not grasped, and so some useful perceptions are missed. If this were a necessary consequence it would bo a valid argument against the introduction of an algebraic symbol, but it is not a necessary consequence.
For instance, in examples about the supply of a cistern by pipes, or the work of men per day, it is admittedly desirable to realise that we are here often dealing with the reciprocals of the specified quantities ; and this may be masked by the use of algebra, possibly, but it need not. I suggest that algebra is the right way of discovering the fact, but that after its discovery the fact itself may be properly dwelt on, and thereafter directly applied. There is indeed too much ten- dency to hurry away from an example when its mere "answer" has been obtained, without staying to extract its nutriment and learn all that it can teach : sometimes without even trying whether the answer found will really fit or satisfy the data in question. That is altogether bad. The full discussion of a sum, in all its bearings, after the answer is known, is often the most interesting and instructive part of the process.
Children should always be encouraged to do this, and to invent fresh ways of putting things, or detect or devise a generalisation of their own for any suitable special case. Here is afforded a first scope for easy kinds of originality of a valuable kind.
Girls especially would find the benefit of being encouraged to seek the general under the mask of the special. It seems to fail to come to them naturally.
Illustrative Examples, showing the advantage of intro- ducing symbols for unknown quantities.
Three pipes supply a cistern which can hold 144 gallons.
XI.] PROBLEMS. 113
One supplies a gallon a minute, another 2 gallons, and the third 3 gallons per minute. How soon will the cistern be full ? Let t be the number of minutes before the cistern is full after the pipes are all turned on simultaneously ; then in i minutes the first pipe will have supplied / gallons, the second 2t gallons, and so on, hence / + 2^ + 3/ = 144.
So t = 2i.
This is easy enough, but I think even this is made easier by the introduction of a symbol for the unknown quantity. Take however the following variation of the same problem : A cistern is to be filled by three pipes labelled A, B, and C; Pipe A alone would fill the cistern in 2 hours 24 minutes. Pipe B alone in 1 hour 1 2 minutes. Pipe C alone in 48 minutes. How soon would they all three fill it t
This form of statement evidently makes the problem harder, and it is clearly desirable to simplify it by ascertaining the rate of supply of each pipe. This can be done at once if we say, let n be the number of gallons corresponding to the contents of the cistern, for then the data give us that
Pipe A supplies at the rate of n gallons in 144 minutes
or — — gallons per minute, 144
B supplies at the rate — gallons per minute,
and C supplies at the rate ~ gallons per minute.
So the set of pipes together supply, at the combined rate, n n n n T44'^72"''48 " ? that is to say, n gallons in the unknown time /, which time is the thing to be found.
L.E.M. H
114 EASY MATHEMATICS. [chap.
We now see that the contents of the cistern is immaterial, when the data are thus specified, for n cancels out of the equation, and leaves us with the relation 1^1 1 1
We have thus discovered the