ARITHMETIC (Gr. apeOµ7run7, sc. TEXVn, the art of counting, from (ipLBµos, number), the art of dealing with numerical quantities in their numerical relations.
1. Arithmetic is usually divided into Abstract Arithmetic and Concrete Arithmetic, the former dealing with numbers and the latter with concrete objects. This distinction, however, might be misleading. In stating that the sum of rid. and 9d. is Is. 8d. we do not mean that nine pennies when added to eleven pennies produce a shilling and eight pennies. The sum of money corresponding to rid. may in fact be made up of coins in several different ways, so that the symbol " rid." cannot be taken as denoting any definite concrete objects. The arithmetical fact is that 11 and 9 may be regrouped as 12 and 8, and the statement " 11d.+ 9d. = Is. 8d." is only an arithmetical statement in so far as each of the three expressions denotes a numerical quantity (§ II).
2. The various stages in the study of arithmetic may be arranged in different ways, and the arrangement adopted must be influenced by the purpose in view. There are three main purposes, the practical, the educational, and the scientific; i.e. the subject may be studied with a view to technical skill in dealing with the arithmetical problems that arise in actual life, or for the sake of its general influence on mental development, or as an elementary stage in mathematical study.
3. The practical aspect is an important one. The daily activities of the great mass of the adult population, in countries where commodities are sold at definite prices for definite quantities, include calculations which have often to be performed rapidly, on data orally given, and leading in general to results which can only be approximate; and almost every branch of manufacture or commerce has its own range of applications of arithmetic. Arithmetic as a school subject has been largely regarded from this point of view.
4. From the educational point of view, the value of arithmetic has usually been regarded as consisting in the stress it lays on accuracy. This aspect of the matter, however, belongs mainly to the period when arithmetic was studied almost entirely for commercial purposes; and even then accuracy was not found always to harmonize with actuality. The development of physical science has tended to emphasize an exactly opposite aspect, viz. the impossibility, outside a certain limited range of subjects, of ever obtaining absolute accuracy, and the consequent importance of not wasting time in attempting to obtain results beyond a certain degree of approximation.
5. As a branch of mathematics, arithmetic may be treated logically, psychologically, or historically. All these aspects are of importance to the teacher: the logical, in order that he may know the end which he seeks to attain; the psychological, that he may know how best to attain this end; and the historical, for the light that history throws on psychology.
1 The logical arrangement of the subject is not the best for elementary study. The division into abstract and concrete, for instance, is logical, if the former is taken as relating to number and the latter to numerical quantity (§ I1). But the result of a rigid application of this principle would be that the calculation of the cost of 3 lb of tea at 2S. a lb would be deferred until after the study of logarithms. The psychological treatment recognizes the fact that the concrete precedes the abstract and that the abstract is based on the concrete; and it also recognizes the futility of attempting a strictly continuous development of the subject.
On the other hand, logical analysis is necessary if the subject is to be understood. As an illustration, we may take the elementary processes of addition, subtraction, multiplication and division. These are still called in textbooks the " four simple rules "; but this name ignores certain essential differences. (i) If we consider that we are dealing• with numerical quantities, we must recognize the fact that, while addition and subtraction might in the first instance be limited to such quantities, multiplication and division necessarily introduce the idea of pure number. (ii) If on the other hand we regard ourselves as dealing with pure number throughout, then, as multiplication is continued addition, we ought to include in our classification involution as continued multiplication. Or we might say that, since multiplication is a form of addition, and division a form of subtraction, there are really only two fundamental processes, viz. addition and subtraction. (iii) The inclusion of the four processes under one general head fails to indicate the essential difference between addition and multiplication, as direct processes, on the one hand, and subtraction and division, as inverse processes, on the other (§ 59).
6. The present article deals mainly with the principles of the subject, for which a logical arrangement is on the whole the more convenient. It is not suggested that this is the proper order to be adopted by the teacher.
I. NUMBER
7. Ordinal and Cardinal Numbers.—One of the primary distinctions in the use of number is between ordinal and cardinal numbers, or rather between the ordinal and the cardinal aspects of number. The usual statement is that one, two, three, . . . are cardinal numbers, and first, second, third, . . . are ordinal numbers. This , however, is an incomplete statement; the words one, two, three, . . . and the corresponding symbols 1, 2, 3, .. . or I, II, III, . . . are used sometimes as ordinals, i.e. to denote the place of an individual in a series, and sometimes as cardinals, i.e. to denote the total number since the commencement of the series.
On the whole, the ordinal use is perhaps the more common. Thus " loo " on a page of a book does not mean that the page is loo times the page numbered 1, but merely that it is the page after 99. Even in commercial transactions, in dealing with sums of money, the statement_ of an amount often has reference to the last item added rather than to a total; and geometrical measurements are practically ordinal (§ 26).
For ordinal purposes we use, as symbols, not only figures, such as 1, 2, 3, ... but also letters, as a,b,c, ...Thus the pages of a book may be numbered 1, 2, 3, . . . and the chapters I, II, III, ... but the sheets are lettered A, B, C, .... Figures and letters may even be used in combination; thus 16 may be followed by 16a and 16b, and these by 17, and in such a case the ordinal xoo does not correspond with the total (cardinal) number up to this point.
Arithmetic is supposed to deal with cardinal, not with ordinal numbers; but it will be found that actual numeration, beyond about three or four, is based on the ordinal aspect of number, and that a scientific treatment of the subject usually requires a return to this fundamental basis.
One difference between the treatment of ordinal and of cardinal numbers may be noted. Where a number is expressed in terms of various denominations, a cardinal number usually begins with the largest denomination, and an ordinal number with the smallest. Thus we speak of one thousand eight hundred and seventysix, and represent it by MDCCCLXXVI or 1876; but we should speak of the third day of August 1876, and represent it by 3. 8. 1876. It might appear as if the writing of 1876 was an exception to this rule; but in reality 1876, when used in this way, is partly cardinal and partly ordinal, the first three figures being cardinal and the last ordinal. To make the year completely ordinal, we should have to describe it as the 6th year of the 8th decade of the 9th century of the 2nd millennium; i.e. we should represent the date by 3. 8. 6. 8. 9. 2, the total number of years, months and days completed being 1875. 7. 2.
In using an ordinal we direct our attention to a term of a series, while in using a cardinal we direct our attention to the interval between two terms. The total number in the series is the sum of the two cardinal numbers obtained by counting up to any interval from the beginning and from the end respectively; but if we take the ordinal numbers from the beginning and from the end we count one term twice over. Hence, if there are 365 days in a year, the moth day from the beginning is the 266th, not the 265th, from the end.
8. Meaning of Names of Numbers.—What do we mean by any particular number, e.g. by seven,' or by two hundred and fiftythree? We can define two as one and one, and three as one and one and one; but we obviously cannot continue this method for ever. For the definition of large numbers we may employ either of two methods, which will be'called the grouping method and the counting method.
(i) Method of Grouping.—The first method consists in defining the first few numbers, and forming larger numbers by groups or aggregates, formed partly by multiplication and partly by addition. Thus, on the denary system (§ 16) we can give independent definitions to the numbers up to ten, and then regard (e.g.) fiftythree as a composite number made up of five tens and three ones. Or, on the quinarybinary system, we need only give independent definitions to the numbers up to five; the numbers six, seven,. . . can then be regarded as five and one, five and two, . . . , a fresh series being started when we get to five and five or ten. The grouping method introduces multiplication into the definition of large numbers; but this,.. from the teacher's point of view, is not now such a serious objection as it was in the days when children were introduced to millions and billions before they had any idea of elementary arithmetical processes.
(ii) Method of Counting.—The second method consists in taking a series of names or symbols for the first few numbers, and then repeating these according to a regular system for successive numbers, so that each number is defined by reference to the number immediately preceding it in the series. Thus two still means one and one, but three means two and one, not one and one and one. Similarly two hundred and fiftythree does not mean two hundreds, five tens and three ones, but one more than two hundred and fiftytwo; and the number which is called one hundred is not defined as ten tens, but as one more than ninetynine.
9. Concrete and Abstract Numbers.—Number is concrete or abstract according as it does or does not relate to particular objects. On the whole, the grouping method refers mainly to concrete numbers and the counting method to abstract numbers. If we sort objects into groups of ten, and find that there are five groups of ten with three over, we regard the five and the three as names for the actual sets of groups or of individuals. The three, for instance, are regarded as a whole when we name them three. If, however; we count these three as one, two, three, then the number of times we count is an abstract number. Thus number in the abstract is the number of times that the act of counting is performed in any particular case. This, however, is a description, not a definition, and we still want a definition for " number " in the phrase " number of times."
I0. Definition of " Number."—Suppose we fix on a certain sequence of names " one," " two," " three," . . . , or symbols such as I, 2, 3, . . ; this sequence being always the same. Ifwe take a set of concrete objects, and name them in succession " one," " two," " three," . . . , naming each once and once only, we shall not get beyond a certain name, e.g. " six." Then, in saying that the number of objects is six, what we mean is that the name of the last object named is six. We therefore only require a definite law for the formation of the successive names or symbols. The symbols I, 2, . . . 9, 10, . . . , for instance, are formed according to a definite law; and in giving 253 as the number of a set of objects we mean that if we attach to them the symbols I, 2, 3, ... in succession, according to this law, the symbol attached to the last object will be 253. If we say that this act of attaching a symbol has been performed 253 times, then 253 is an abstract (or pure) number.
Underlying this definition is a certain assumption, viz. that if we take the objects in a different order, the last symbol attached will still be 253. This, in an elementary treatment of the subject, must be regarded as axiomatic; but it is really a simple case of mathematical induction. (See ALGEBRA.) If we take two objects A and B, it is obvious that whether we take them as A, B, or as' B, A, we shall in each case get the sequence I, 2. Suppose this were true for, say, eight objects, marked r to 8. Then, if we introduce another object anywhere in the series, all those coming. after it will be displaced so that each will have the mark formerly attached to the next following; and the last will therefore be q instead of 8. This is true, whatever the arrangement of the original objects may be, and wherever the new one is introduced; and therefore, if the theorem is true for 8, it is true for 9. But it is true for z; therefore it is true for 3; therefore for 4, and so on.
Ir. Numerical Quantities.— If the term number is confined to number in the abstract, then number in the concrete may be described as numerical quantity. Thus £3 denotes £r taken 3 times. The £I is termed the unit, A numerical quantity, therefore, represents a certain unit, taken a certain number of times. If we take £3 twice, we get £6; and if we take 3s. twice, we get 6s., i.e. 6 times Is. Thus arithmetical processes deal with numerical quantities by dealing with numbers, provided the unit is the same throughout. If we retain the unit, the arithmetic is, concrete; if we ignore it, the arithmetic is abstract. But in the latter case it must always be understood that there is some unit concerned, and the results have no meaning until the unit is reintroduced.
II. NOTATION, NUMERATION AND NUMBERIDEATION
12. Terms used.—The representation of numbers by spoken sounds is called numeration; their representation by written signs is called notation. The systems adopted for numeration and for notation do not always agree with one another; nor do they always correspond with the idea which the numbers subjectively present. This latter presentation may, in the absence of any accepted term, be called numberideation; this word covering not only the perception or recognition of particular numbers, but also the formation of a numberconcept.
13. Notation of Numbers.—The system which is now almost universally in use amongst civilized nations for representing cardinal numbers is the Hindu, sometimes incorrectly called the Arabic, system. The essential features which distinguish this from other systems are (I) the limitation of the number of different symbols, only ten being used, however large the number to be represented may be; (2) the use of the zero to indicate the absence of number; and (3) the principle of local value, by which a symbol in effect represents different numbers, according to its position. The symbols denoting a number are called its digits.
A brief account of the development of the system will be found under NUMERAL. Here we are concerned with the principle, the explanation of which is different according as we proceed on the grouping or the counting system.
(i) On the grouping system we may in the first instance consider that we have separate symbols for numbers from " one " to " nine," but that when we reach ten objects we put them in a group and denote this group by the symbol used for "one," but printed in a• different type or written of a different size or (in teaching) of a different colour. Similarly when we get to ten tens we denote them by a new representation of the figure denoting one. Thus we may have:
ones I 2 3 4 5 6 7 8 9
tens I 2 3 4 5 6 7 8 9
hundreds, 1 2 3 4 5 6 7 8 9
&c. &c. &c.
On this principle 24would represent twentyfour, 24 two hundred and forty, and 24 two hundred and four. To prevent confusion the zero or " nought " is introduced, so that the success sive figures, beginning from the right, may represent ones, tens, hundreds, . . We then have, e.g., 240 to denote two hundreds and four tens; and we may now adopt a uniform type for all the figures, writing this 240.
(ii) On the counting system we may consider that we have a
series of objects (represented in the adjoining diagram by dots), and that we attach to these objects in succession the
i • symbols 1, 2, 3, 4, 5, 6, 7, 8, 9, o, repeating this series
3 • indefinitely. There is as yet no distinction between the first object marked I and the second object marked 1. We can, however, attach to the o's the same sym
6 bols, 1, 2, . . . o in succession, in a separate column,
9 • repeating the series indefinitely; then do the same with
to • every o of this new series; and so on. Any particular
I • object is then defined completely by the combination
2 • of the symbols last written down in each series; and
3 • this combination of symbols can equally be used to denote the number of objects up to and including the last one (§ ro).
In writing down a number in excess of s000 it is (except where the number represents a particular year) usual in England and America to group the figures in sets of three, starting from the right, and to mark off the sets by commas. On the continent of Europe the figures are taken in sets of three, but are merely spaced, the comma being used at the end of a number to denote the commencement of a decimal.
The zero, called " nought," is of course a different thing from the letter 0 of the alphabet, but there may be a historical connexion between them (§ 79). It is perhaps interesting to note that the latterday telephone operator calls 1907 " nineteen O seven " instead of " nineteen nought seven."
14. Direction of the NumberSeries.—There is no settled convention as to the direction in which the series of symbols denoting the successive numbers one, two, three, . . . is to be written.
(i) If the numbers were written down in succession, they would naturally proceed from left to right, thus: 1, 2, 3, . . . This system, however, would require that in passing to " double figures " the figure denoting tens should be written either above or below the figure denoting ones, e.g.
I, 2, . . . , 8, 9, 0, I, 2, . . . or I, 2, . . . , 8, 9, 0, I, 2,
The placing of the tensfigure to the left of the onesfigure will not seem natural unless the numberseries runs either up or down.
(ii) In writing down any particular number, the successive powers of ten are written from right to left, e.g. 5,462,198 is
(6) (s) (4) (3) (2) (I) (o)
5 4 6 2 I 9 8
the small figures in brackets indicating the successive powers. On the other hand, in writing decimals, the sequence (of negative powers) is from left to right.
(iii) In making out lists, schedules, mathematical tables (e.g. a multiplicationtable), statistical tables, &c., the numbers are written vertically downwards. In the case of lists and schedules the numbers are only ordinals; but in the case of mathematical or statistical tables they are usually regarded as cardinals,though, when they represent values of a continuous quantity, they must be regarded as ordinals (§§ 26, 93).
(iv) In graphic representation measurements are usually made upwards; the adoption of this direction resting on certain deeply rooted ideas (§ 23).
This question of direction is of importance in reference to the
development of useful numberforms (§ 23); and the existence
of the two methods mentioned under (iii) and (iv) above produces
confusion in comparing numerical tabulation with graphical
representation. It is generally accepted that the horizontal
direction of increase, where a horizontal direction is necessary,
should be from left to right; but uniformity as regards vertical
direction could only be attained either by printing mathe
matical tables upwards or by taking " downwards," instead of
" upwards," as the " positive " direction for graphical purposes.
200 The downwards direction will be taken in this article as
5° the normal one for succession of numbers (e.g. in multipli
3 cation), and, where the arrangement is horizontal, it is to
253 be understood that this is for convenience of printing. It
should be noticed that, in writing the components of a
number 253 as 200, 5o and 3, each component beneath the next larger one, we are really adopting the downwards principle, since the figures which make up 253 will on this principle be successively 2, 5 and 3 (§ 13 (ii) ).
15. Roman Numerals.—Although the Roman numerals are no longer in use for representing cardinal numbers, except in certain special cases (e.g. clockfaces, milestones and chemists' prescriptions), they are still used for ordinals.
The system differs completely from the Hindu system. There are no single symbols for two, three, &c.; but numbers are represented by combinations of symbols for one, five, ten, fifty, one hundred, five hundred, &c., the numbers which have single symbols, viz. I, V, X, L, C, D, M, proceeding by multiples of five and two alternately. Thus 1878 is MDCCCLXXVIII, i.e. thousand fivehundred hundred hundred hundred fifty ten ten five one one one.
The system is therefore essentially a cardinal and grouping one, i.e. it represents a number as the sum of sets of other numbers. It is therefore remarkable that it should now only be used for ordinal purposes, while the Hindu system, which is ordinal in its nature, since a single series is constantly repeated, is used almost exclusively for cardinal numbers. This fact seems to illustrate the truth that the counting principle is the fundamental one, to which the interpretation of grouped numbers must ultimately be referred.
The normal process of writing the larger numbers on the left is in certain cases modified in the Roman system by writing a number in front of a larger one to denote subtraction. Thus four, originally written IIII, was later written IV. This may have been due to one or both of two causes; a primitive tendency to refer numbers, in numeration, to the nearest large number (§ 24 (iv) ), and the difficulty of perceiving the number of a group of objects beyond about three (§ 22). Similarly IX, XL and XC were written for nine, forty and ninety respectively. These, however, were later developments.
16. Scales of Notation.—In the Hindu system the numbering proceeds by tens, tens of tens, &c.; thus the figure in the fifth place, counting from the right, denotes the product of the corresponding number by four tens in succession. The notation is then said to be in the scale of which ten is the base, or in the denary scale. The Roman system, except for the use of symbols for five, fifty, &c., is also in the denary scale, though expressed in a different way. The introduction of these other symbols produces a compound scale, which may be called a quinarybinary, or, less correctly, a quinarydenary scale.
The figures used in the Hindu notation might be used to express numbers in any other scale than the denary, provided. new symbols were introduced if the base of the scale exceeded ten. Thus 1878 in the quinarybinary scale would be 1131213, and 1828 would be 1130213; the meaning of these is seen at once by comparison with MDCCCLXXVIII and MDCCCXXVIII. Similarly the number which in the denary scale is 215 would in the quaternary scale (base 4) be 3113, being equal to 3.4.4.4+ 1.4.4+1.4+3._
The use of the denary scale in notation is due to its use in, numeration (§ 18); this again being due (as exemplified by the use of the word digit) to the. primitive use of the fingers for counting. If mankind had had six fingers on each hand and six toes on each foot, we should be using a duodenary scale (base twelve), which would have been far more convenient.
17. Notation of Numerical Quantities.—Over a large part of the civilized world the introduction of the metric system (§ 118) has caused the notation of all numerical quantities to be in the denary scale. In Great Britain and her colonies, however, and in the United States, other systems of notation still survive, though there is none which is consistently in one scale, other than the denary. The method is to form quantities into groups, and these again into larger groups; but the number of groups making one of the next largest groups varies as we proceed along the scale. The successive groups or units thus formed are called denominations. Thus twelve pennies make a shilling, and twenty shillings a pound, while the penny is itself divided into four farthings (or
2 •
two halfpennies). There are, therefore, four denominations, the bases for conversion of one denomination into the next being successively four (or two), twelve and twenty. Within each denomination, however, the denary notation is employed exclusively, e.g. " twelve shillings " is denoted by 12F.
The diversity of scales appears to be due mainly to four causes: (i) the tendency to group into scores (§ 20); (ii) the tendency to subdivide into twelve; (iii) the tendency to subdivide into two or four, with repetitions, making subdivision into sixteen or sixtyfour; and (iv) the independent adoption of different units for measuring the same kind of magnitude.
Where there is a division into sixteen parts, abinary scale may be formed by dividing into groups of two, four or eight. Thus the weights ordinarily in use for measuring from a oz. up to 2 lb give the basis for a binary scale up to not more than eight figures, only o and t being used. The points of the compass might similarly be expressed by numbers in a binary scale; but the numbers would be ordinal, and the expressions would be analogous to those of decimals rather than to those of whole numbers.
In order to apply arithmetical processes to a quantity expressed in two or more denominations, we must first express it in terms of a single denomination by means of a varying scale of notation.
(20) (12)
Thus £254, 13s. 6d. may be written £254 « 13s. A 6d.; each of the numbers in brackets indicating the number of units in one denomination that go to form a unit in the next higher denomination. To express the quantity in terms of £, it ought
(20) (12) I ~_6S
to be written £254 v 13 , 6; this would mean £254 20 or 13 6
£(2J4+=Th I2)0 and therefore would involve a fractional 20 20'
number.
A quantity expressed in two or more denominations is usually called a compound number or compound quantity. The former term is obviously incorrect, since a quantity is not a number; and the latter is not very suggestive. For agreement with the terminology of fractional numbers (§ 62) we shall describe such a quantity as a mixed quantity. The letters or symbols descriptive of each denomination are usually placed after or (in actual calculations) above the figures denoting the numbers of the corresponding units; but in a few cases, e.g. in the case of £, the symbol is placed before the figures. There would be great convenience in a general adoption of this latter method; the combination of the two methods in such an expression as £I23, 16s. 4zd. is especially awkward.
18. Numeration.—The names of numbers are almost wholly based on the denary scale; thus eighteen means eight and ten, and twentyfour means twice ten and four. The words eleven and twelve have been supposed to suggest etymologically a denary basis (see, however, NUMERAL).
Two exceptions, however, may be noted.
(i) The use of dozen, gross (=dozen dozen), and great gross = dozen gross) indicates an attempt at a duodenary basis. But the system has never spread; and the word " dozen " itself is based on the denary scale.
(ii) The score (twenty) has been used as a basis, but to an even more limited extent. There is no essential difference, however, between this and the denary basis. As the latter is due to fingerreckoning, so the use of the fingers and the toes produced a vigesimal scale. Examples of this are given in § 20; it is worthy of notice that the vigesimal (or, rather, quinaryquaternary) system was used by the Mayas of Yucatan, and also, in a more perfect form, by the Nahuatl (Aztecs) of Mexico.
The number ten having been taken as the basis of numeration, there are various methods that might consistently be adopted for naming large numbers.
(i) We might merely name the figures contained in the number. This method is often adopted in practical life, even as regards mixed quantities; thus £57,593, 16s. 4d. would be read as five seven, five nine three, sixteen and four pence.
(ii) The word ten might be introduced, esg. 593 would be five ten ten ninety (= nine ten) and three.
(iii) Names might be given to the successive powers of ten, up to the point to which numeration of ones is likely to go. Partial applications of this method are found in many languages.
(iv) A compromise between the last two methods would be to have names for the series of numbers, beginning with ten, each of which is the "square" of the preceding one. This would in effect be analysing numbers into components of the form a. lob where a is less than to, and the index b is expressed in the binary scale, e.g. 7,000,000 would be 7.1o4.102, and 700,000 would be 7.104.10'.
The British method is a mixture of the last two, but with an indexscale which is partly ternary and partly binary. There are separate names for ten, ten times ten (= hundred), and ten times ten times ten (= thousand) ; but the next single name is million, representing a thousand times a thousand. The next name is billion, which in Great Britain properly means a million million, and in the United States (as in France) a thousand million.
19. Discrepancies between Numeration and Notation.—Although numeration and notation are both ostensibly on the denary system, they are not always exactly parallel. The following are a few of the discrepancies. 
(i) A set of written symbols is sometimes read in more than one way, while on the other hand two different sets of symbols (at any rate if denoting numerical quantities) may be read in the same way. Thus 1820 might be read as one thousand eight hundred and twenty if it represented a number of men, but it would be read as eighteen hundred and twenty if it represented a year of the Christian era; while Is. 6d. and 18d. might both be read as eighteen pence. As regards the first of these twoexamples, however, it would be more correct to write 1,82o for the former of the two meanings (cf. § 13).
(ii) The symbols II and 12 are read as eleven and twelve, not (except in elementary teaching) as tenone and tentwo.
(iii) The names of the numbers next following these, up to 19 inclusive, only faintly suggest a ten. This difficulty is not always recognized by teachers, who forget that they themselves had to be told that eighteen means eightandten. 
(iv) Even beyond twenty, up to a hundred, the word ten is not used in numeration, e.g. we say thirty four, not three ten four.
(v) The rule that the greater number comes first is not universally observed in numeration. It is not observed, for instance, in the names of numbers from 13 to tg; nor was it in the names from which eleven and twelve are derived. Beyond twenty it is usually, but not always, observed; we sometimes instead of twentyfour say four and twenty. (This latter is the universal system in German, up to too, and for any portion of too in numbers beyond too.)
20. Other Methods of Numeration and Notation.—It is only possible here to make a brief mention of systems other than those now ordinarily in use.
(i) Vigesimal Scale.—The system of counting by twenties instead of by tens has existed it1 many countries; and, though there is no corresponding notation, it still exhibits itself in the names of numbers. This is the case, for instance, in the Celtic languages; and the Breton or Gaulish names have affected the Latin system, so that the French names for some numbers are on the vigesimal system. This system also appears in the Danish numerals. In English the use of the word score to represent twenty—e.g. in " threescore and ten " for seventy—is superimposed on the denary system, and has never formed an essential part of the language. The word; like dozen and couple, is still in use, but rather in a vague than in a precise sense.
(ii) Roman System.—The Roman notation has been explained above (§ 15). Though convenient for exhibiting the composition of any particular number, it was inconvenient for purposes of calculation; and in fact calculation was entirely (or almost entirely) performed by means of the abacus (q.v.). The numeration was in the denary scale, so that it did not agree absolutely with the notation. The principle of subtraction from a higher number, which appeared in notation, also appeared in numeration, but not for exactly the same numbers or in exactly the same way; thus XVIII was twofromtwenty, and the next number was onefromtwenty, but it was written XIX, not IXX.
(iii) Other Systems of Antiquity.—The Egyptian notation was purely denary, the only separate signs being those for 1, to, too, &c. The ordinary notation of the Babylonians was denary, but they also used a sexagesimal scale, i.e. a scale whose base was 6o. The Hebrews had a notation containing separate signs (the letters of the alphabet) for numbers from i to to, then for multiplies of to up to too, and then for multiples of too up to 400, and later up to r000.
The earliest Greek system of notation was similar to the Roman, except that the symbols for 50, 500, &c., were more complicated. Later, a system similar to the Hebrew was adopted, and extended by reproducing the first nine symbols of the series, preceded by accents, to denote multiplication by r000.
On the island of Ceylon there still exists, or existed till recently, a system which combines some of the characteristics of the later Greek (or Semitic) and the modern European notation; and it is conjectured that this was the original Hindu system.
For a further account of the above systems see NUMERAL, and the authorities quoted at the end of the present article.
21. The NumberConcept.—It is probable that very few people have any definite mental presentation of individual numbers (i.e. numbers proceeding by differences of one) beyond too, or at any rate beyond 144. Larger numbers are grasped by forming numbers into groups or by treating some large number as a unit. A person would appreciate the difference between 93,000,000 M. and 94,000,000 M. as the distance of the centre of the sun from the centre of the earth at a particular moment; but he certainly would not appreciate the relative difference between 93,000,000 M. and 93,000 001 M. In order to get an idea of 93,000,000, he must take a million as his unit. Similarly, in the metric system he cannot mentally compare two units, one of which is r000 times the other. The metre and the kilometre, for instance, or the metre and the millimetre, are not directly comparable; but the metre can be conceived as containing too centimetres.
On the other hand, it would seem that, for most educated people, sixteen and seventeen or twentysix and twentyseven, and even eightysix and eightyseven, are single numbers, just as six and seven are, and are not made up of groups of tens and ones. In other words, the denary scale, though adopted in notation and in numeration, does not arise in the corresponding mental concept until we get beyond too.
Again, in the use of decimals, it is unusual to give less than two figures. Thus 3.142 or 3.14 would be quite intelligible; but 3.1 does not convey such a good idea to most people as either 31bor 3.10, i.e. as an expression denoting a fraction or a percentage.,
There appears therefore to be a tendency to use some larger number than ten as a basis for grouping into new units or for subdivision into parts. The Babylonians adopted 6o for both these purposes, thus giving us the sexagesimal division of angles and of time.
This view is supported, not only by the intelligibility of percentages to ordinary persons, but also by the tendency, noted above (§ 19), to group years into centuries, and to avoid the use of thousands. Thus 1876 is not t thousand, 8 hundred, 7 tens and 6, but 18 hundred and 76, each of the numbers 18 and 76 being named as if it were a single number. It is also in accordance with what is so far known about numberforms (§ 23).
If there is this tendency to adopt too as a basis instead of to, the teaching of decimals might sometimes be simplified by proceeding from percentages to percentages of percentages, i.e. by commencing with centesimals instead of with decimals.
22. Perception of Number.—In using material objects as a basis for developing the numberconcept, it must be remembered that it is only when there are a few objects that their number can be perceived without either counting or the performance of some arithmetical process such as addition. If four coins are laid on a table, close together, they can (by most adults) be seen to be four, without counting; but seven coins have to be separated mentally into two groups, the numbers of which are added, or one group has to be seen and the remaining objects counted, before the number is known to be seven.
The actual limit of the number that can be " seen "—i.e. seen without counting or adding—depends for any individual on the shape and arrangement of the objects, but under similar conditions it is not the same for all individuals. It has been suggested that as many as six objects can be seen at once; but this is probably only the case with few people, and with them only when the objects have a certain geometrical arrangement. The limit for most adults, under favourable conditions, is about four. Under certain conditions it is less; thus IIII, the old Roman notation for four, is difficult to distinguish from III, and this may have been the main reason for replacing it by.IV (§ 15).
In the case of young children the limit is probably two. That this was also the limit in the case of primitive races, and that the classification of things was into one, two and many, before any definite process of counting (e.g. by the fingers) came to be adopted, is clear from the use of the " dual number " in language, and from the way in which the names for three and four are often based on those for one and two. With the individual, as with the race, the limit of the number that can be seen gradually increases up to four or five.
The statement that a number of objects can be seen to be three or four is not to be taken as implying that there is a simultaneous perception of all the objects. The attention may be directed in succession to the .different objects, so that the perception is rhythmical; the distinctive rhythm thus aiding the perception of the particular number.
In consequence of this limitation of the power of perception of number, it is practically impossible to use a pure denary scale in elementary numberteaching. If a quinarybinary system (such as would naturally fit in with counting on the fingers) is not adopted, teachers unconsciously resort to a binaryquinary system. This is commonly done where cubes are used; thus seven is represented by three pairs of cubes, with a single cube at the top.
23. Visualization of the Series.—A striking fact, in reference to ideas of number, is the existence of numberforms, i.e. of definite arrangements, on an imagined plane or in space, of the mental representations of the successive numbers from 1 onwards. The proportion of persons in whom numberforms exist has been variously estimated; but there is reason to believe that the forms arise at a very early stage of childhood, and that they did at some time exist in many individuals who have afterwards forgotten them. Those persons who possess them are also apt to make spatial arrangements.of days of the week or the month, months of the year, the letters of the alphabet, &c.; and it is practically certain that only children would make such arrangements of letters of the alphabet. The forms seem to result from a general tendency to visualization as an aid to memory; the letterforms may in the first instance be quite as frequent as the numberforms, but they vanish in early childhood, being of no practical value, while the numberforms continue as an aid to arithmetical work.
The forms are varied, and have few points in common; but the following tendencies are indicated.
(i) In the majority of cases the numbers lie on a continuous (but possibly zigzag) line.
(ii) There is nearly always (at any rate in English cases) a break in direction at 12. From 1 to 12 the numbers sometimes lie in the circumference of a circle, an arrangement obviously suggested by a clockface; in these cases the series usually mounts upwards from 12. In a large number of cases, however, the direction is steadily upwards from t to 12, then changing. In some cases the initial direction is from right to left or from left to right; but there are very few in which it is downwards.
(iii) The multiples of to are usually strongly marked; but special stress is also laid on other important numbers, e.g. the multiples of 12.
(iv) The series sometimes goes up to very high numbers, but sometimes stops at too, or even earlier.. It is not stated, in most cases, whether all the numbers within the limits of the series have definite positions, or whether there are only certain numbers which form an essential part of the figure, while others only
exist potentially. Probably the latter is almost universally the case.
These forms are developed spontaneously, without suggestion from outside. The possibility of replacing them by a standard form, which could be utilized for performing arithmetical operations, is worthy of consideration; some of the difficulties in the way of standardization have already been indicated (§ 14). The general tendency to prefer an upward direction is important; and our current phraseology suggests that this is the direction which increase is naturally regarded as taking. Thus we speak of counting up to a certain number; and similarly mathematicians speak of high and ascending powers, while engineers speak of high pressure, high speed, high power, &c. This tendency is probably aided by the use of bricks or cubes in elementary numberteaching.
24. Primitive Ideas of Number.—The names of numbers give an idea of the way in which the idea of number has developed. Where civilization is at all advanced, there are .usually certain names, the origin of which cannot be traced; but, as we go farther back, these become fewer, and the names, are found to be composed on certain systems. The systems are varied, and it is impossible to lay down any absolute laws, but the following seem to be the main conclusions.
(i) Amongst some of the lowest tribes, as (with a few exceptions) amongst animals, the only differentiation is between one and many, or between one, two and many, or between one, two, three and many. As it becomes necessary to use higher but still small numbers, they are formed by combinations of one and two, or perhaps of three with one or two. Thus many of the Australasian and South American tribes use only one and two; seven, for instance, would be two two two one.
(ii) Beyond ten, and in many cases beyond five, the names have reference to the use of the fingers, and sometimes of the toes, for counting; and the scale may be quinary, denary or vigesimal, according as one hand, the pair of hands, or the hands and feet, are taken as the new unit. Five may be signified by the word for hand; and either ten or twenty by the word for man. Or the
words signifying these numbers may have reference to the conepletion of some act of counting. Between five and ten; or beyond
ten, the names may be due to combinations, e.g. 16 may be Io+ 5+ 1; or they may be the actual names of 'the fingers .last counted.
(iii) There are a few, but only a few, cases in which, the number 6 or 8 is named as twice 3 or twice 4; and there are also a few cases in which 7, 8 and 9 are named as 6+,, 6+2 and 6+3. In the large majority of cases the numbers 6, 7, 8 and 9 are 5+1, 5+2, 5+3 and 5+4, being named either directly from their composition in this way or as the fingers on the second hand.
(iv) There is a certain tendency to name 4, 9, 14 and 19 as being one short of 5, ro, 15 and 20 respectively; the principle being thus the same as that of the Roman IV, IX, &c. It is possible that at an early stage the number of the fingers on one hand or on the two hands together was only thought of vaguely as a large number in comparison with 2 or 3, and that the number did not attain definiteness until it was linked up with the .smaller by insertion of the intermediate ones; and the linking up might take place in both directions.
(v) In a few cases the names, of certain small numbers are the names of objects' which present these numbers in some conspicuous way. Thus the word used by the Abipones to denote 5 was the name of a certain hide of five colours. It has been suggested that names of this kind may have been the origin of the numeral words of different races; but it is improbable that direct visual perception would lead to a name for a number unless a name based on a process of counting had previously been given to it.
25. Growth of the NumberConcept.—The general principle that the development of the individual follows the development of the race holds good to a certain extent in the.case of the numberconcept, but it is modified by the existence of language dealing with concepts which are beyond the reach of the child, and also, of course, by the direct attempts at instruction. One result is the formation of a numberseries as a mere succession of nameswithout any corresponding ideas of number; the series not being necessarily correct.
When numbering begins, the names of the successive numbers are attached to the individual objects; thus the numbers are originally ordinal, not cardinal.
The conception of number as cardinal, i.e. as something belonging to a group of objects as a whole, is a comparatively late one, and does not arise until the idea of a whole consisting of its parts has been formed. This is the quantitative aspect of number.
The development from the nameseries to the quantitative conception is aided by the numbering of material objects and the performance of elementary processes of comparison, addition, &c., with them. It may also be aided, to a certain extent, by the tendency to find rhythms in sequences of sounds. This tendency is common in adults as well as in children; the strokes of a clock may, for instance, be grouped into fours, and thus eleven is represented as two fours and three. Fingercounting is of course natural to children, and leads to grouping into fives, and ultimately to an understanding of the denary system of notation.
26. Representation of Geometrical Magnitude by Number.—The application of arithmetical methods to geometrical measurement presents some difficulty. In reality there is a transition from a cardinal to an ordinal system, but to an ordinal system which does not agree with the original ordinal system from which the cardinal system was derived. To see this, we may represent ordinal numbers by the ordinary numerals 1, 2, 3, . . and cardinal numbers by the Roman I, II, III, . . . Then in the earliest stage each object counted is indivisible; either we are
counting it as a whole, or we are not
counting it at all. The symbols 1, 2,
3, . . . then refer to the individual
objects, as in fig. 1; this is the primary Figs. 2 and 3 represent the cardinal stage; fig. 2
•
n
III
showing how the I, II, III, . . denote the successively larger groups of objects, while fig. 3 shows how the name II of the whole is determined by the name 2 of the last one counted.
When now we pass to geometrical measurement, each " one " is a thing which is itself divisible, and it cannot be said that at any moment we are counting it; it is only when one is completed that we can count it. The names 1, 2, 3, ... for the individual objects cease to have an intelligible meaning, and measurement is effected by the cardinal numbers I, II, III, . . . , as in fig. 4.
These cardinal numbers have now, however, come to denote individual points in the line of measurement, i.e. the points of separation of the individual units of length. The point III in fig. 4 does not include the point II in the same way that the number III includes the number II in fig; 2, and the points must. therefore be denoted by the ordinal numbers 1, 2, 3, . . as in fig. 5, the zero o falling into its natural place immediately before the commencement of the first unit.
Thus, while arithmetical numbering refers to units, geometrical numbering does not refer to units but to the intervals between units.
(i.) Preliminary
27. Equality and Identity. =There is a certain difference between the use of words referring to equality and identity in
I 2
3 I 2 3
• I • II • III FIG. 3.
I
2
3
III
0 1 2 FIG. 5.
arithmetic and in algebra respectively; what' is an equality in the former becoming an identity in the latter. Thus the statement that 4 times 3 is equal to 3 times 4, or, in abbreviated form, 4X3 =3 X4(§ 28), is a statement not of identity but of equality; i.e. 4 X 3 and 3 X4 mean different things, but the operations which they denote produce the same result. But in algebra a X b= b X a is called an identity, in the sense that it is true whatever a and b may be; while nXX=A is called an equation, as being true, when n and A are given, for one value only of X. Similarly the numbers represented by and z are not identical, but are equal.
28. Symbols of Operation.— The failure to observe the distinction between an identity and an equality often leads to loose reasoning; and in order to prevent this it is important that definite meanings should be attached to all symbols of operation, and especially to those which represent elementary operations. The symbols — and = mean respectively that the first quantity mentioned is to be reduced or divided by the second; but there is some vagueness about + and X. In the present article a+b will mean that a is taken first, and b added to it; but a X b will mean that b is taken first, and is then multiplied by a. In the case of numbers the X may be replaced by a dot; thus 4.3 means 4 times 3. When it is necessary to write the multiplicand before the multiplier, the symbol >e will be used, so that bse a will mean the same as a X b.
29. Axioms.—There are certain statements that are sometimes regarded as axiomatic; e.g. that if equals are added to equals the results are equal, or that if A is greater than B then A+X is greater than B+X. Such statements, however, are capableof logical proof,and are generalizations of results obtained empirically at an elementary stage; they therefore belong more properly to the laws of arithmetic (§ 58).
(ii.) Sums and Differences.
30. Addition and Subtraction. — Addition is the process of expressing (in numeration or notation) a whole, the parts of which have already been expressed; while, if a whole has been expressed and also a part or parts, subtraction is the process of expressing the remainder.
Except with very small numbers, addition and subtraction, on the grouping system, involve analysis and rearrangement. Thus the sum of 8 and 7 cannot be expressed as ones; we can either form the whole, and regroup it as ro and 5, or we can split up the 7 into 2 and 5, and add the 2 to the 8 to form ro, thus getting 8+7 =8+ (2+5) = (8+2) +5=10+5= 15. For larger numbers the rearrangement is more extensive; thus 24+31= (20+4) + (30+ I) = (20+30) + (4+ I) = 50+5 = 55, the processbeing still more complicated when the ones together make more than ten. Similarly we cannot subtract 8 from 15, if 15 means 1 ten + 5 ones; we must either write 15—8=(I0+5)—8= (ro—8)+5=2+5=7, or else resolve the 15 into an inexpressible number of ones, and then subtract 8 of them, leaving 7.
Numerical quantities, to be added or subtracted, must be in the same denomination; we cannot, for instance, add 55 shillings and roo pence, any more than we can add 3 yards and 2 metres.
31. Relative Position in the Series.—The above method of dealing with addition and subtraction is synthetic, and is appropriate to the grouping method of dealing with number. We commence with processes, and see what they lead to; and thus get an idea of sums and differences. If we adopted the counting method, we should proceed in a different way, our method being analytic.
One number is less or greater than another, according as the symbol (or ordinal) of the former comes earlier or later than that of the latter in the numberseries. Thus (writing ordinals in light type, and cardinals in heavy type) 9 comes after 4, and therefore 9 is greater than 4. To find how much greater, we compare two series, in dne of which we go up to 9, while in the other we stop at 4 and then recommence our counting. The series are shown below, the numbers being placed horizontally for convenience of printing, instead of vertically (§ 14):
I 2 3 4 5 6 7 8 9
1 2 3 4 1 2 3 4 5This exhibits 9 as the sum of 4 and 5; it being understood that the sum of 4 and 5 means that we add 5 to 4. That this gives the same result as adding 4 to 5 may be seen by reckoning the series backwards.
It is convenient to introduce the zero; thus
o I 2 3 4 5 6 7 8 9
o I 2 3 4 5 indicates that after getting to 4 we make a fresh start from 4 as our zero.
To subtract, we may proceed in either of two ways. The subtraction of 4 from 9 may mean either " What has to be added to 4 in order to make up a total of 9," or " To what has 4 to be added in order to make up a total of 9." For the former meaning we count forwards, till we get to 4, and then make a new count, parallel with the continuation of the old series, and see at what number we arrive when we get to 9. This corresponds to the concrete method, in which we have 9 objects, take away 4 of them, and recount the remainder. The alternative method is to retrace the steps of addition, i.e. to count backwards, treating g of one (the standard) series as corresponding with 4 of the other, and finding which number of the former corresponds with o of the latter. This is a more advanced method, which leads easily to the idea of negative quantities, if the subtraction is such that we have to go behind the o of the standard series.
32. Mixed Quantities.—The application of the above principles, and of similar principles with regard to multiplication and division, to numerical quantities expressed in any of the diverse British denominations, presents no theoretical difficulty if the successive denominations are regarded as constituting a varying scale of notation (§17). Thus the expression 2 ft. 3 in. implies that in counting inches we use o to eleven instead of o to g as our first repeating series, so that we put down i for the next denomination when we get to twelve instead of when we get to ten. Similarly 3 yds. 2 ft. means
yds. o I 2 3
ft. O I 2 0 I 2 O I 2 0 I 2 The practical difficulty, of course, is that the addition of two numbers produces different results according to the scale in which we are for the moment proceeding; thus the sum of g and 8 is 17, 15, 13 or I I according as we are dealing with shillings, pence, pounds (avoirdupois) or ounces. The difficulty may be minimized by using the notation explained in § 17.
(iii.) Multiples, Submultiples and Quotients.
33. Multiplication and Division are the names given to certain numerical processes which have to be performed in order to find the result of certain arithmetical operations. Each process may arise out of either of two distinct operations; but the terminology is based on the processes, not on the operations to which they belong, and the latter are not always clearly understood.
34. Repetition and Subdivision.—Multiplication occurs when a certain number or numerical quantity is treated as a unit (§ II), and is taken a certain number of times. It therefore arises in one or other of two ways, according as the unit or the number exists first in consciousness. If pennies are arranged in groups of five, the total amounts arranged are successively once 5d., twice 5d., three times 5d., . .; which are written r X 5d., 2 X 5d., 3 X 5d., . .. (§ 28). This process is repetition, and the quantities 1 X 5d., 2 X 5d., 3 X 5d are the successive multiples of 5d. If, on the other hand, we have a sum of 5s., and treat a shilling as being equivalent to twelve pence, the 5s. is equivalent to 5 X 12d.; here the multiplication arises out of a subdivision of the original unit Is. into 12d.
Although multiplication may arise in either of these two ways, the actual process in each case is performed by commencing with the unit and taking it the necessary number of times. In the above case of subdivision, for instance, each of the 5 shillings is separately converted into pence, so that we do in fact find in
succession once 12d., twice 12d ; i.e. we find the multiples of 12d. up to 5 times.
The result of the multiplication is called the product of the unit by the number of times it is taken.
35. Diagram of Multiplication.—The process of multiplication reducing £3 to shillings, since each £ becomes 205., we find the
is performed in order to obtain such results as the following: value of 3.20.
If i boy receives 7 apples, A B
then 3 boys receive 21 apples;
or
If is. is equivalent to 12d.,
id.
then 5s. is equivalent to 6od. £3 4f.
The essential portions of these statements, from the arith t 20 is. . 12d.
metical point of view, may be exhibited in the form of the
diagrams A and B
A B
I2 £720 £t 20S.
i boy 7 apples is. I2d.
£2880 £3 6os. 720d. 288of.
3 boys 21 apples 5s. 6od.
or more briefly, as in C or C' and D or D':
C C' D
I 7 apples
3 21 apples 3
the general arrangement of the diagram being as shown in E or E'
E E'
Multiplication is therefore equivalent to completion of the diagram by entry of the product.
36. MultipleTables.—The diagram C or D of § 35 is part of a complete table giving the successive multiples of the particular unit. If we take several different units, and write down their successive multiples in parallel columns, preceded by the numberseries, we obtain a multipletable such as the following:
I I 2 9 Is. 5d. 3Yds.2ft. 17359 ••••.
2 2 4 18 2S. Iod. 7 yds. i ft. 34718 .... .
3 3 6 27 4s. 3d. I I yds. oft. 52077 .... .
4 4 8 36 5s. 8d. 14 yds. 2 ft. 69436 • .
5 5 10 45 7s. id. 18 yds. i ft. 86795 • .
It is to be considered that each column may extend downwards indefinitely.
37. Successive Multiplication.—In multiplication by repetition the unit is itself usually a multiple of some other unit, i.e. it is a product which is taken as a new unit. When this new unit has been multiplied by a number, we can again take the product as a unit for the purpose of another multiplication; and so on indefinitely. Similarly where multiplication has arisen out of the subdivision of a unit into smaller units, we can again subdivide these smaller units. Thus we get successive multiplication; but it represents quite different operations according as it is due to repetition, in the sense of § 34, or to subdivision, and these operations will be exhibited by different diagrams. Of the two diagrams below, A exhibits the successive multiplication of £3 by 20, 12 and 4, and B the successive reduction of £3 to shillings, pence and farthings. The principle on which the diagrams are constructed is obvious from § 35. It should be noticed that in multiplying 6 by 20 we find the value of 20.3, but that in
38. Submultiples.—The relation of a unit to its successive multiples as shown in a multipletable is expressed by saying that it is a submultiple of the multiples, the successive submultiples being onehalf, onethird, one fourth, . . . Thus, in the diagram of § 36, is. 5d. is onehalf of 2s. rod., onethird of 4S. 3d., onefourth of 5s. 8d ; these being written " of 2S. rod.," " a of 4s. 3d.," " 4 of 5s. 8d.,"
The relation of submultiple is the converse of that of multiple; thus if a is i of b, then b is 5 times a. The determination of a submultiple is therefore equivalent to completion of the diagram E or E' of § 35 by entry of the unit, when the number of times it is taken, and the product, are given. The operation is the converse or repetition; it is usually called partition, as representing division into a number of equal shares.
39. Quotients.—The converse of subdivision is the formation of units into groups, each constituting a larger unit; the number of the groups so formed out of a definite number of the original units is called a quotient. The determination of a quotient is equivalent to completion of the diagram by entry of the number when the unit and the product are given. There is no satisfactory name for the operation, as distinguished from partition; it is sometimes called measuring, but this implies an equality in the original units, which is not an essential feature of the operation.
40. Division.—From the commutative law for multiplication, which shows that 3 X 4d. = 4 X 3d. = 12d., it follows that the number of pence in onefourth of I2d. is equal to the quotient when 12 pence are formed into units of 4d.; each of these numbers being said to be obtained by dividing 12 by 4. The term division is therefore used in textbooks to describe the two processes described in §§ 38 'and 39; the product mentioned in § 34 is the dividend, the number or the unit, whichever is given, is called the divisor, and the unit or number which is to be found is called the quotient. The symbol = is used to denote both kinds of division; thus A = n denotes the unit, n of which make up A, and A= B denotes the number of times that B has to be taken to make up A. In the present article this confusion is avoided by writing the
former as n of A.
Methods of division are considered later (§§ 106108).
41. Diagrams of Division.—Since we write from left to right or downwards, it may be convenient for division to interchange the rows or the columns of the multiplicationdiagram. Thus the uncompleted diagram for partition is F or G, while for measuring it is usually H; the vacant compartment being for the unit in
F G H K
Number Product Product
F or G, and for the number in H. In some cases it may be convenient in measuring to show both the units, as in K.
. 42. Successive Division may be performed as the converse of successive multiplication. The diagrams A and B below are the converse (with a slight alteration) of the corresponding diagrams
D'
7 apples
12d.
12d.
21 apples
5
6od.
5
6od.
I
Unit
Unit
Number
Product
Number
Product
I
Number
Product
Unit
12d. is. 6od.
in § 37; A representing the determination of 6 of of ,x of 288o farthings, and B the conversion of 288o farthings into £.
A B
(iv.) Properties of Numbers.
(A) Properties not depending on the Scale of Notation.
43• Powers, Roots and Logarithms.—The standard series 1, 2, 3,
.. is obtained by successive additions of x to the number last
found. If instead of commencing with x and making successive
additions of r we commence with any number such as 3 and make
successive multiplications by 3, we get a series 3, 9, 27, . . . as
shown below the line in the margin. The first memo I = 3° n° ber of the series is 3; the second is the product of I 3=3' n' two numbers, each equal to 3; the third is the pro
2 9=32 n2 duct of three numbers, each equal to 3; and so on.
3 27=3' n4 These are written 31 (or 3), 32, :3', 34, . . where
4 SI 3 n nP denotes the product of p numbers, each equal to
: n. If we write nP = x, then, if any two of the three
numbers n, p, u are known, the third is determinate. If we know n and p, p is called the index, and n, n2, . .. nP are called the first power, second power, . . . pth power of n, the series itself being called the powerseries. The second power and third power are usually called the square and cube respectively. If we know p and N, is is called the pth root of N,•so that is is the second (or square) root of n2, the third (or cube) root of n3, the fourth root of n4, . . . If we know is and N, then p is the logarithm of N to base is.
The calculation of powers (i.e. of N when is and p are given) is involution; the calculation of roots (i.e. of is when p and N are given) is evolution; the calculation of logarithms (i.e. of p when n and N are given) has no special name.
Involution is a direct process, consisting of successive multiplications; the other two are inverse processes. The calculation of a logarithm can be performed by successive divisions; evolution requires special methods.
The above definitions of logarithms, &c., relate to cases in which n and p are whole numbers, and are generalized later.
44• Law of Indices.—If we multiply nP by 19, we multiply the product of p n's by the product of q n's, and the result is therefore nPl4. Similarly, if we divide nP by n4, where q is less than p, the result is nPQ. Thus multiplication and division in the powerseries correspond to addition and subtraction in the indexseries, and vice versa.
If we divide nP by nP, the quotient is of course 1. This should be written n°. Thus we may make the powerseries commence with r, if we make the indexseries commence with o. The added terms are shown above the line in the diagram in § 43.
45• Factors, Primes and Prime Factors.—If we take the suc
cessive multiples of 2, 3, . . .
as in § 36, and place each 2 2
multiple opposite the same 3 3
number in the original series, 4 4 4
we get an arrangement as 6 6
in the adjoining diagram. If 7 8 8
any number N occurs in the 8
vertical series commencing 9 9
with a number is (other than Io IO 12 12 IO
I) then n is said to be a factor 12 12 I2
of N. Thus 2, 3 and 6 are .
factors of 6; and 2, 3, 4, 6
and 12 are factors of 12.
A number (other than r) which has no factor except itself iscalled a prime number, or, more briefly, a prime. Thus 2, 3, 5, 7 and r r are primes, for each of these occurs twice only in the table. A number (other than i) which is not a prime number is called a composite number.
If a number is a factor of another number, it is a factor of any multiple of that number. Hence, if a number has factors, one at least of these must be a prime. Thus 12 has 6 for a factor; but 6 is not a prime, one of its factors being 2; and therefore 2 must also be a factor of 12. Dividing 12 by 2, we get a submultiple 6, which again has a prime 2 as a factor. Thus any number which is not itself a prime is the product of several factors, each of which is a prime, e.g. 12 is the product of 2, 2 and 3. These are called prime factors.
The following are the most important properties of numbers in reference to factors:
(i) If a number is a factor of another number, it is a factor of any multiple of that number.
(ii) If a number is a factor of two numbers, it is a factor of their sum or (if they are unequal) of their difference. (The words in brackets are inserted to avoid the difficulty, at this stage, of saying that every number is a factor of o, though it is of course true that o. n=o, whatever is may be.)
(iii) A number can be resolved into prime factors in one way only, no account being taken of their relative order. Thus 12=2X2X3=2X3X2=3X2X2, but this is regarded as one way only. If any prime occurs more than once, it is usual to write the number of times of occurrence as an index; thus 144=2X2X2X2X3X3=24 32.
The number r is usually included amongst the primes; but, if this is done, the last paragraph requires modification, since 144 could be expressed as I. 24. 32, or as I2. 24. 32, or as V. z4. 32, where p might be anything.
If two numbers have no factor in common (except x) each is said to be prime to the other.
The multiples of 2 (including 1.2) are called even numbers; other numbers are odd numbers.
46. Greatest Common Divisor.—If we resolve two numbers into their prime factors, we can find their Greatest Common Divisor or Highest Common Factor (written G.C.D. or G.C.F. or H.C.F.), i.e. the greatest number which is a factor of both. Thus 144=24. 32, and 756=22 33 7, and therefore the G.C.D. of 144 and 756 is 22. 32=36. If we require the G.C.D. of two numbers, and cannot resolve them into their prime factors, we use a process described in the textbooks. The process depends on (ii) of § 45, in the extended form that, if x is a factor of a and b, it is a factor of paqb, where p and q are any integers.
The G.C.D. of three or more numbers is found in the same way.
47. Least Common 'Multiple.—The Least Common Multiple, or L.C.M., of tWo numbers, is the least number of which they are both factors. Thus, since 144 = 24. 32, and 756 = 22. 33 7, the L.C.M. of 144 and 756 is 24 33 7. It is clear, from comparison with the last paragraph, that the product of the G.C.D. and the L.C.M. of two numbers is equal to the product of the numbers themselves. This gives a rule for finding the L.C.M. of two numbers. But we cannot apply it to finding the L.C.M. of three or more number; if we cannot resolve the numbers into their prime factors, we must find the L.C.M. of the first two, then the L.C.M. of this and the next number, and so on.
(B) Properties depending on the Scale of Notation.
48. Tests of Divisibility.The following are the principal rules for testing whether particular numbers are factors of a given number. The number is divisible
(i) by xo if it ends in o;
(ii) by 5 if it ends in o or 5;
(iii) by 2 if the last digit is even;
(iv) by 4 if the number made up of the last two digits is divisible by 4;
(v) by 8 if the number made up of the last three digits is divisible by 8;
(vi) by 9 if the sum of the digits is divisible by 9;
(vii) by 3 if the sum of the digits is divisible by 3;
20S.
£I
Is.
4f.
12d.
Id.
3
6os.
288of.
720d.
288of.
4
I
I2
I
72of.
6of.
20
I
3f.
(viii) by 1 1 if the difference between the sum of the 1st, 3rd, sth, . .. digits and the sum of the 2nd, 4th, 6th, . is zero or divisible by Ir.
(ix) To find whether a number is divisible by 7, 1r or 13, arrange the number in groups of three figures, beginning from the end, treat each group as a separate number,; and then find the difference between the sum of the 1st, 3rd, ... of these numbers and the sum of the 2nd, 4th, . . Then, if this difference is zero or is divisible by 7, II or 13, the original number is also so divisible; and conversely. For example, 31521 gives 52f 31 ..490, and therefore is divisible by7,. but not by.II or 13.
49. Casting out Nines is a process based on (vi) of the last paragraph. The remainder when a number is divided by g is equal to the remainder when the sum of its digits is divided by 9. Also, if the remainders when two numbers are divided by 9 are respectively a and b, the remainder when their product is divided by 9 is the same as the remainder when a.b is divided by g. This gives a rule for testing multiplication, which is found in most textbooks. It is doubtful, however, whether such a rule, giving a test which is necessarily incomplete, is of much educational value.
(v.) Relative Magnitude. _
5o. Fractions.—A fraction of a quantity is a suhmultiple, or a multiple of a submultiple, of that quantity. Thus, since 3XIs. 5d.=4s. 3d., 1s: 5d. may be denoted by i of 4S. 3d.; and any multiple of Is. sd., denoted by nXrs. 5d., may also be
denoted by of 4S. 3d. We therefore use "a of A" to mean that
we find a quantity X such that aXX=A, and then multiply X by n.
It must be noted (i) that this is a definition of "a of, " not a definition of "a,"and (ii) that it is not necessary that t should be
less than a.
51. Subdivision of Submultiple.—By r. of A we mean 5 times the unit, 7 times which is A. If we regard this unit as being 4 times a lesser unit, then A_is 7.4 times this lesser unit, and 4 of A is 5.4
times the lesser unit. Hence 4 of A is equal to 4 of A; and, conversely, 4 of A is equal to 4 of A. Similarly each of these is equal to 4 of A. Hence the value of a fraction is not
altered by substituting for the numerator and denominator the corresponding numbers in any other column of a multipletable
(§ 36). If we write 7+55_44In the form 44'.7 5 we may say that the
value of a fraction is not altered by multiplying or dividing the numerator and denominator by any number.
52. Fraction of a Fraction.—To find ~; of . 4 of A we must convert ? of A into 4 times some unit. This is done by the pre
ceding paragraph. For 4 of A=5:1 of A= 4.1 of A; i.e. it is 7.4
4 times a unit which is itself 5 times another unit, 7.4 times which
is A. Hence, taking the former unit 11 times instead of 4 times,
loft,ofA=rl. of A.
7.4
A fraction of a fraction is sometimes called a compound fraction.
53. Comparison, Addition and Subtraction of Fractions.—The quantities ; of A and 4 of A are expressed in terms of different units. To compare them, or to add or subtract them, we must express them in terms of the same unit:' Thus, taking 21$ of A as the unit, we have (§ 51)
ofA=liofA; ofA=HHofA.
Hence the former is greater than the latter; their sum is;$ of A; and their difference is of A.
Thus the fractions must be reduced to a common denominator. This denominator must, if the fractions are in their lowest terms (§ 54), be a multiple of each of the denominators; it is usually most convenient that it should be their L.C.M. (§ 47).
54. Fraction in its Lowest Terms.—A fraction is said to be in its lowest terms when its numerator and denominator have no common
1
7d.
10
5S. Tod.
24
14s.
factor; or to be reduced to its lowest terms when it is replaced by such a fraction. Thusv of A is said to be reduced to its lowest terms when it is replaced by A of A. It is important always to bear in mind that  of A is not the same as  of A, though it is equal to it.
55. Diagram of Fractional Relation.—To find T of 14s. we have
to take lo of the units, 24 of which make up 14s. Hence the required amount will, in the multipletable of § 36, be opposite 10 in the column in which the amount opposite 24 is 14s.; the quantity at the head of this column, representing the unit, will be found to be 7d. The elements of the multipletable with which we are concerned are shown in the diagram in the margin. This diagram serves equally for the two statements that (i) 4 of 14s. is
5s. iod., (ii) ofss. iod: is 14s. The two statements are in fact
merely different aspects of a single relation, considered in the
next section.
56. Ratio.—If we omit the two upper compartments of the
diagram in the last section, we obtain the diagram A. This diagram exhibits a relation between the two
amounts 5s. to1 and 14S. on the one hand,
io 5 d and the numbers Io and 24 of the standard series on the other, which is expressed by saying that 5s. rod. is to 14S. in the ratio of io
24 to 24, or that 14s. is to 5s. rod. in the ratio of
24 to 10. If we had taken Is. 2d. instead of
B " 7d. as the unit for the second column, we should have obtained the diagram B. Thus we must regard the ratio of a to b as being the same as the ratio of c to d; if the fractions
b and a are equal. For this reason the
ratio of a to b is sometimes written but
the more correct method is to write it a:b.
If two quantities or numbers P and Q are to each other in the 'ratio of p to q, it is clear from the diagram that p times Q=
q times P, so that Q = p of P.
57. Proportion.—If from any two columns in the table of § 36 we remove the numbers or quantities in any two rows, we get
a diagram such as that here shown.
The pair of compartments on either
side may, as here, contain numerical
quantities, or may contain numbers.
But the two pairs of compartments
will correspond to a single pair of numbers, e.g. 2 and 6, in the standard series, so that, denoting them by M, N and P, Q respectively, M will be to N.in the same
ratio that P' is to Q. This is expressed by saying
M that M is to N' as P. to Q, the relation being written
M :N :: P : Q; the four quantities are ,then said
to be in proportion or to be proportionals.
N Q This' is the most general' expression of the
relative magnitude of two quantities; i.e. the relation expressed by proportion includes the relations expressed by multiple, submultiple, fraction and ratio.
If M and N are respectively m and n times a unit, and P and Q are respectively p and q times a unit, then the quantities are in proportion if mq=np; and conversely.
IV. LAWS OF ARITHMETIC
58. Laws of Arithmetic.—The arithmetical processes which we have considered in reference to positive integral numbers are subject to the following laws:
(i) Equalities and Inequalities.—The following are sometimes called Axioms (§ 29), but their truth should be proved, even if at an early stage it is assumed_ The symbols " > " and " < mean respectively " is greater than " and "is less than." The numbers represented by a, b, c, x and m are all supposed to be positive.
A
145.
S. IO .
12
5s: iod.
14S.,
2S. Iod.
8s. 6d.
7 yds. I ft. 22 ,yds..
P
(a) If a=b, and b=c, then a=c;
(b) If a=b, then a+x=b+x, and a—x=b—x;
(c) If a>b, then a+x>b+x, and a—x>b—x;
(d) If ab, then ma>mb, and am>b+m;
(g) If acc; and it states that, subject to this, successive operations of multiplication or division may be grouped in sets in any way; e.g. aabaecacdiee=f=a+(b=c)ac(daee=f).
(v) Commutative Law for Multiplications and Divisions, that multiplications and divisions may be performed in any order: e.g. a=ba 

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