If I boy receives 7 apples, then 3 boys receive 21 apples;

or

If Is. is equivalent to 12d.,

then 5s. is equivalent to 60d.

The essential portions of these statements, from the arithmetical point of view, may be exhibited in the form of the diagrams A and B:

A

I

£3

IS.

3

21 apples

3

21 apples

£2880

£3 60s. 720d. 2880f.

38. Submultiples.-The relation of a unit to its successive multiples as shown in a multiple-table is expressed by saying that it is a submultiple of the multiples, the successive submultiples being one-half, one-third, one-fourth, . . . Thus, in the diagram of § 36, 1s. 5d. is one-half of 2s. 10d., one-third of 4s. 3d., one-fourth of 5s. 8d., ; these being written “ of 25. 10d.," "of 4s. 3d.,' of 5s. 8d.,"

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 £3 by 20 we find the value of 20.3, but that in

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 3X4d. =4X3d. = 12d., it follows that the number of pence in one-fourth of 12d. 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 text-books 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 of A.

Methods of division are considered later (§§ 106-108). 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 multiplication-diagram. 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

I 3=31 n1 2 9=32 m2 327=33 n3 481=34 n1

3f.

288of. 720d. 60s. £3

(iv.) Properties of Numbers.

of

(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 1 to the number last found. If instead of commencing with 1 and making successive additions of I 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 mem0 I=30 20 ber of the series is 3; the second is the product of two numbers, each equal to 3; the third is the product of three numbers, each equal to 3; and so on. These are written 3' (or 3), 32, 33, 3*, . . where no denotes the product of p numbers, each equal to n. If we write n2= N, then, if any two of the three : numbers n, p, N are known, the third is determinate. If we know n and p, p is called the index, and n, n2, . . . no are called the first power, second power, . . . pth power of n, the series itself being called the power-series. The second power and third power are usually called the square and cube respectively. If we know and N, n is called the pth root of N, so that n is the second (or square) root of n2, the third (or cube) root of n3, the fourth root of n', . . . If we know n and N, then p is the logarithm

: : : :

:

:

of N to base n.

called a prime number, or, more briefly, a prime. Thus 2, 3, 5, 7 and 11 are primes, for each of these occurs twice only in the table. A number (other than 1) 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 n 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=3X2 X2, 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=2X2 X2 X2 X3X3=21 32.

The number 1 is usually included amongst the primes; but, if this is done, the last paragraph requires modification, since 144 could be expressed as 1. 24. 32, or as 12. 24. 32, or as 1o. 24. 3a, where p might be anything.

If two numbers have no factor in common (except 1) 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.),

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 no by no, we multiply the product of pn's by the product of qn's, and the result is therefore +. Similarly, if we divide no by n, where q is less than p, the result is no-9. Thus multiplication and division in the power-series correspond to addition and subtraction in the index-series, and vice versa.

If we divide no by n”, the quotient is of course 1. This should be written no. Thus we may make the power-series commence with 1, if we make the index-series 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 successive multiples of 2, 3,

• I

as in 36, and place each multiple opposite the same 3 number in the original series, 4 we get an arrangement as 6 in the adjoining diagram. If 7 any number N occurs in the 8 8 vertical series commencing 9 with a number n (other than I) then n is said to be a factor 12 12 12 12

of N. Thus 2, 3 and 6 are

factors of 6; and 2, 3, 4, 6

and 12 are factors of 12.

10 ΙΟ II

9

A number (other than 1) which has no factor except itself is

1442 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 text-books. 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 pa-qb, 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 1442 32, and 756=22 33 7, the L.C.M. of 144 and 756 is 2 337. 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 numbers; 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 10 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;

(viii) by 11 if the difference between the sum of the 1st, 3rd, 5th, . . . digits and the sum of the 2nd, 4th, 6th, . . . is zero or divisible by II.

(ix) To find whether a number is divisible by 7, 11 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, 11 or 13, the original number is also so divisible; and conversely. For example, 31521 gives 521-31 490, and therefore is divisible by 7, but not by 11 or 13. 49. Casting out Nines is a process based on (vi) of the last paragraph. The remainder when a number is divided by 9 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 4.b is divided by 9. This gives a rule for testing multiplication, which is found in most text-books. It is doubtful, however, whether such a rule, giving a test which is necessarily incomplete, is of much educational value.

=

(v.) Relative Magnitude.

50. Fractions. A fraction of a quantity is a submultiple, or a multiple of a submultiple, of that quantity. Thus, since 3X1s. 5d. 4s. 3d., Is. 5d. may be denoted by of 4s. 3d.; and any multiple of is. 5d., denoted by nXIs. 5d., may also be แข denoted by of 4s. 3d. We therefore use of A" to mean that we find a quantity X such that aXX-A, and then multiply X by n.

n

a

7.4

7.4

51. Subdivision of Submultiple.-By 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 of A is 5.4 times the lesser unit. Hence of A is equal to 54 of A; and, conversely, 54 of A is equal to of A. Similarly each of these is equal to 3 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 multiple-table (836). If we write 4 in the form 45 we may say that the value of a fraction is not altered by multiplying or dividing the numerator and denominator by any number.

4.7

52. Fraction of a Fraction.-To find of of A. we must convert of A into 4 times some unit. This is done by the pre5.4 4.5 7.4 7.4

ceding paragraph. For of A= of A= of A; i.e. it is

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, of of A=11.5 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 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 of A as the unit, we have (§ 51)

of Aof A; of A-18 of A.

I

ΙΟ

7d.

55. Diagram of Fractional Relation.-To find of 145. we have to take 10 of the units, 24 of which make up 14s. Hence the required amount will, in the multiple-table 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 10 5s. 10d. unit, will be found to be 7d. The elements of the multiple-table with which we are concerned are shown in the diagram in the margin. This diagram serves equally for the two statements that (i) of 14s. is 5s. 10d., (ii) }† of 5s. 10d. 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 A amounts 5s. 1od. and 14s. on the one hand, 5s. Iod. and the numbers 10 and 24 of the standard series on the other, which is expressed by saying that 5s. 10d. is to 14s. in the ratio of 10 to 24, or that 14s. is to 5s. 10d. in the ratio of 24 to 10. If we had taken Is. 2d. instead of 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

5

12

and 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= times P, so that Q=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 that M is to N as P to Q, the relation being written MN: P:Q; the four quantities are then said to be in proportion or to be proportionals.

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

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.

pence, and then perform the subtraction. Generally, to find the sum or difference of two or more fractional numbers, we must replace them by other fractional numbers having the same denominator; it is usually most convenient to take as this denominator the L.C.M. of the original fractional numbers (cf. $53).

64. Complex Fractions.-A fraction (or fractional number), the numerator or denominator of which is a fractional number, is called a complex fraction (or fractional number), to distinguish it from a simple fraction, which is a fraction having integers for 53 numerator and denominator. Thus of A means that we take a unit X such that 11 times X is equal to A, and then take 53 times X. To simplify this, we take a new unit Y, which is of X. Then A is 34 times Y, and of A is 17 times Y, i.e. it is of A. 65. Multiplication of Fractional Numbers. To multiply by is to take times. It has already been explained (§ 62) that times is an operation such that times 7 times is equal to 5 times. Hence we must express, which itself means times, as being 7 times something. This is done by multiplying both numerator and denominator by 7; i.e. is equal to Z, which is the same which is the same thing as 7 times Hence times = times 7 times 7.35 8 5.8 times The rule for multiplying a fractional number 7.3 7.3' by a fractional number is therefore the same as the rule for finding a fraction of a fraction.

=

8

7.3'

7.8

7.3'

8

66. Division of Fractional Numbers.-To divide by is to find a number (i.e. a fractional number) x such that times x is equal to . But times & times x is, by the last section, equal to x. Hence x is equal to times. Thus to divide by a fractional number we must multiply by the number obtained by interchanging the numerator and the denominator, i.e. by the reciprocal of the original number.

If we divide 1 by we obtain, by this rule, . Thus the reciprocal of a number may be defined as the number obtained by dividing 1 by it. This definition applies whether the original number is integral or fractional.

By means of the present and the preceding sections the rule given in § 63 can be extended to the statement that a fractional number is equal to the number obtained by multiplying its numerator and its denominator by any fractional number.

67. Negative Fractional Numbers. We can obtain negative fractional numbers in the same way that we obtain negative integral numbers; thus - or - A means that or A is taken negatively.

68. Genesis of Fractional Numbers.-A fractional number may be regarded as the result of a measuring division (§ 39) which cannot be performed exactly. Thus we cannot divide 3 in. by II in. exactly, i.e. we cannot express 3 in. as an integral multiple of 11 in.; but, by extending the meaning of “times" as in § 62, we can say that 3 in. is times 11 in., and therefore call the quotient when 3 in. is divided by 11 in. Hence, if p and n are numbers, is sometimes regarded as denoting the result of dividing by n, whether and n are integral or fractional (mixed numbers being included in fractional).

n

The idea and properties of a fractional number having been explained, we may now call it, for brevity, a fraction. Thus " of A " no longer means two of the units, three of which make up A; it means that A is multiplied by the fraction, i.e. it means the same thing as "times A.”

69. Percentage. In order to deal, by way of comparison or addition or subtraction, with fractions which have different denominators, it is necessary to reduce them to a common denominator. To avoid this difficulty, in practical life, it is usual to confine our operations to fractions which have a certain standard denominator. Thus (§ 79) the Romans reckoned in twelfths, and the Babylonians in sixtieths; the former method supplied a basis for division by 2, 3, 4, 6 or 12, and the latter for division by 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, or 60. The modern method is to deal with fractions which have 100 as denominator;

70. Decimal Notation of Percentage.-An integral percentage, i.e. a simple fraction with 100 for denominator, can be expressed by writing the two figures of the numerator (or, if there is only one figure, this figure preceded by o) with a dot or "point" before them; thus 76 means 76 %, or 10%. If there is an integral number to be taken as well as a percentage, this number is written in front of the point; thus 23.76 × A means 23 times A, with 76% of A. We might therefore denote 76 % by 0·76.

76

If as our unit we take X=1 of A = 1% of A, the above quantity might equally be written 2376 X = 23.76 of A; i.e. 23.76XA is equal to 2376 % of A.

I 14

00

71. Approximate Expression by Percentage.-When a fraction cannot be expressed by an integral percentage, it can be so expressed approximately, by taking the nearest integer to the numerator of an equal fraction having 100 for its denominator. Thus =100, so that is approximately equal to 14%; and 7100, which is approximately equal to 29 %. The difference between this approximate percentage and the true value is less than%, i. e. is less than 200

2 284

7

=

100

If the numerator of the fraction consists of an integer and 3 37 1-e.8. in the case of -it is uncertain whether we should take the next lowest or the next highest integer. It is best in such cases to retain the; thus we can write =371 %=·374. 72. Addition and Subtraction of Percentages.-The sum or difference of two percentages is expressed by the sum or difference of the numbers expressing the two percentages.

37

73. Percentage of a Percentage.-Since 37 % of 1 is expressed by 0:37, 37. % of 1 % (i.e. of 0.01) might similarly be expressed by 0.00 37. The second point, however, is omitted, so that we write it 0.0037 or 0037, this expression meaning 10% of 100-10005. On the same principle, since 37 % of 45 % is equal to 10% of 1405% 10000 10%+(10% of T), we can express it by 1665; and 3% of 2 % can be expressed by 0006. Hence, to find a percentage of a percentage, we multiply the two numbers, put o's in front if necessary to make up four figures (not counting fractions). and prefix the point.

=

153

74. Decimal Fractions.-The percentage-notation can be extended to any fraction which has any power of 10 for its denominator. Thus 15300 100% can be written 153 and 100000 can be written 15300. These two fractions are equal to each other, and also to 1530. A fraction written in this way is called a decimal fraction; or we might define a decimal fraction as a fraction having a power of 10 for its denominator, there being a special notation for writing such fractions.

A mixed number, the fractional part of which is a decimal fraction, is expressed by writing the integral part in front of the point, which is called the decimal point. Thus 271000% can be written 27.1530. This number, expressed in terms of the fraction 1000 or 0001, would be 271530. Hence the successive figures after the decimal point have the same relation to each other and to the figures before the point as if the point did not exist. The point merely indicates the denomination in which the number is expressed: the above number, expressed in terms