Chapter C: C' D D'

+---+-----------+ | +---+------+ |
| 1 | 7 apples | | 7 apples | 1 | 12d. | | 12d.
+---+-----------+ ---+----------- +---+------+ ---+------
| 3 | 21 apples | 3 | 21 apples | 5 | 60d. | 5 | 60d.
+---+-----------+ | +---+------+ |

the general arrangement of the diagram being as shown in E or E':--

E E'
+--------+---------+ |
| 1 | Unit | | Unit
+--------+---------+ --------+---------
| Number | Product | Number | Product
+--------+---------+ |

Multiplication is therefore equivalent to completion of the diagram by entry of the product.

36. _Multiple-Tables._--The diagram C or D of S 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 number-series, we obtain a _multiple-table_ such as the following:--

+---+---+----+----+----------+---------------+-------+-----
| 1 | 1 | 2 | 9 | 1s. 5d. | 3 yds. 2 ft. | 17359 | ...
+---+---+----+----+----------+---------------+-------+-----
| 2 | 2 | 4 | 18 | 2s. 10d. | 7 yds. 1 ft. | 34718 | ...
+---+---+----+----+----------+---------------+-------+-----
| 3 | 3 | 6 | 27 | 4s. 3d. | 11 yds. 0 ft. | 52077 | ...
+---+---+----+----+----------+---------------+-------+-----
| 4 | 4 | 8 | 36 | 5s. 8d. | 14 yds. 2 ft. | 69436 | ...
+---+---+----+----+----------+---------------+-------+-----
| 5 | 5 | 10 | 45 | 7s. 1d. | 16 yds. 1 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 S 34, or to subdivision, and these operations will be exhibited by different diagrams. Of the two diagrams below, A exhibits the successive multiplication of L3 by 20, 12 and 4, and B the successive reduction of L3 to shillings, pence and farthings. The principle on which the diagrams are constructed is obvious from S 35. It should be noticed that in multiplying L3 by 20 we find the value of 20.3, but that in reducing L3 to shillings, since each L becomes 20s., we find the value of 3.20.

A B
+----+-------+ +-------+--------+
| 1 | L3 | | 1d. | 4f. |
+-----+----+-------+ +------+-------+--------+
| 1 | 20 | L60 | | 1s. | 12d. | |
+---+-----+----+-------+ +----+------+-------+--------+
| 1 | 12 | | L720 | | L1 | 20s. | | |
+---+-----+----+-------+ +----+------+-------+--------+
| 4 | | | L2880 | | L3 | 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 S 36, 1s. 5d. is one-half of 2s. 10d., one-third of 4s. 3d., one-fourth of 5s. 8d., ...; these being written "1/2 of 2s. 10d.," "1/3 of 4s. 3d.," "1/4 of 5s. 8d,"...

The relation of submultiple is the converse of that of multiple; thus if a is 1/5 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 S 35 by entry of the unit, when the number of times it is taken, and the product, are given. The operation is the converse of 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 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 SS 38 and 39; the product mentioned in S 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 1/n of A.

Methods of division are considered later (SS 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 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.

F G H K
+--------+---------+ +--------+---------+ +---------+---+ +------+-----+
| 1 | | | Number | Product | | Unit | 1 | | 12d. | 1s. |
+--------+---------+ +--------+---------+ +---------+---+ +------+-----+
| Number | Product | | 1 | | | Product | | | 60d. | |
+--------+---------+ +--------+---------+ +---------+---+ +------+-----+

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 in S 37; A representing the determination of 1/20 of 1/12 of 1/4 of 2880 farthings, and B the conversion of 2880 farthings into L.

A B
+---+--------+ +------+----+
| 4 | 2880f. | | 20s. | L1 |
+----+---+--------+ +-------+------+----+
| 12 | 1 | 720f. | | 12d. | 1s. | |
+----+----+---+--------+ +--------+-------+------+----+
| 20 | 1 | | 60f. | | 4f. | 1d. | | |
+----+----+---+--------+ +--------+-------+------+----+
| 1 | | | 3f. | | 2880f. | 720d. | 60s. | L3 |
+----+----+---+--------+ +--------+-------+------+----+

(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 1 to the number last found. If instead of commencing with 1 and making successive additions of 1 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 member 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 3^1 (or 3), 3^2, 3^3, 3^4, ... where n^p denotes the product of p numbers, each equal to n. If we write n^p = 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, n^2, ... n^p 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 p and N, n is called the _pth root_ of N, so that n is the _second_ (or _square_) _root_ of n^2, the _third_ (or _cube_) _root_ of n^3, the _fourth root_ of n^4, ... If we know n and N, then p is the _logarithm_ of N to _base_ n.

0 1 = 3^0 n^0
------------------
1 3 = 3^1 n^1
2 9 = 3^2 n^2
3 27 = 3^3 n^3
4 81 = 3^4 n^4
: : : :
: : : :

The calculation of powers (i.e. of N when n and p are given) is _involution_; the calculation of roots (i.e. of n 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 n^p by n^q, we multiply the product of p n's by the product of q n's, and the result is therefore n^(p + q). Similarly, if we divide n^p by n^q, where q is less than p, the result is n^(p - q). Thus multiplication and division in the power-series correspond to addition and subtraction in the index-series, and vice versa.

If we divide n^p by n^p, the quotient is of course 1. This should be written n^0. Thus we may make the power-series commence with 1, if we make the index-series commence with 0. The added terms are shown above the line in the diagram in S 43.

45. _Factors, Primes and Prime Factors._--If we take the successive multiples of 2, 3, ... as in S 36, and place each multiple opposite the same number in the original series, we get an arrangement as in the adjoining diagram. If any number N occurs in the vertical series commencing with a number n (other than 1) then n is said to be a _factor_ of N. Thus 2, 3 and 6 are factors of 6; and 2, 3, 4, 6 and 12 are factors of 12.

1 .. .. .. .. .. .. ..
2 2 .. .. .. .. .. ..
3 .. 3 .. .. .. .. ..
4 4 .. 4 .. .. .. ..
5 .. .. .. 5 .. .. ..
6 6 6 .. .. 6 .. ..
7 .. .. .. .. .. 7 ..
8 8 .. 8 .. .. .. 8
9 .. 9 .. .. .. .. ..
10 10 .. .. 10 .. .. ..
11 .. .. .. .. .. .. ..
12 12 12 12 .. 12 .. ..
: : : : : : : :
: : : : : : : :

A number (other than 1) which has no factor except itself is 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 0, though it is of course true that 0.n = 0, 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 = 2 X 2 X 3 = 2 X 3 X 2 = 3 X 2 X 2, 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 = 2 X 2 X 2 X 2 X 3 X 3 = 2^4 . 3^2.

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 . 2^4 . 3^2, or as 1^2 . 2^4 . 3^2, or as 1^p . 2^4 . 3^2, 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.), i.e. the greatest number which is a factor of both. Thus 144 = 2^4 . 3^2, and 756 = 2^2 . 3^3 . 7, and therefore the G.C.D. of 144 and 756 is 2^2 . 3^2 = 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 S 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 144 = 2^4 . 3^2, and 756 = 2^2 . 3^3 . 7, the L.C.M. of 144 and 756 is 2^4 . 3^3 . 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 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 0;

(ii) by 5 if it ends in 0 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 11.

(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 a.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 3 X 1s. 5d. = 4s. 3d., 1s. 5d. may be denoted by 1/3 of 4s. 3d.; and any multiple of 1s. 5d., denoted by n X 1s. 5d., may also be denoted by n/3 of 4s. 3d. We therefore use "n/a of A" to mean that we find a quantity X such that a X X = A, and then multiply X by n.

It must be noted (i) that this is a definition of "n/a of," not a definition of "n/a," and (ii) that it is not necessary that n should be less than a.

51. _Subdivision of Submultiple._--By 5/7 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 5/7 of A is 5.4 times the lesser unit. Hence 5/7 of A is equal to (5.4)/(7.4) of A; and, conversely, (5.4)/(7.4) of A is equal to 5/7 of A. Similarly each of these is equal to (5.3)/(7.3) 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 (S 36). If we write (5.4)/(7.4) in the form (4.5)/(4.7) 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 11/4 of 5/7 of A we must convert 5/7 of A into 4 times some unit. This is done by the preceding paragraph. For 5/7 of A = (5.4)/(7.4) of A = (4.5)/(7.4) 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,

11/4 of 5/7 of A = (11.5)/(7.4) of A

A fraction of a fraction is sometimes called a _compound fraction._

53. _Comparison, Addition and Subtraction of Fractions._--The quantities 3/4 of A and 5/7 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 1/28 of A as the unit, we have (S 51)

3/4 of A = 21/28 of A; 5/7 of A = 20/28 of A.

Hence the former is greater than the latter; their sum is 41/28 of A; and their difference is 1/28 of A.

Thus the fractions must be reduced to a _common denominator_. This denominator must, if the fractions are in their lowest terms (S 54), be a multiple of each of the denominators; it is usually most convenient that it should be their L.C.M. (S 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 factor; or to be reduced to its lowest terms when it is replaced by such a fraction. Thus 8/22 of A is said to be reduced to its lowest terms when it is replaced by 4/11 of A. It is important always to bear in mind that 4/11 of A is not the _same_ as 8/22 of A, though it is _equal_ to it.

+------+-----------+
| 1 | 7d. |
+------+-----------+
| 10 | 5s.10d. |
+------+-----------+
| 24 | 14s. |
+------+-----------+

55. _Diagram of Fractional Relation._--To find 10/24 of 14s. we have to take 10 of the units, 24 of which make up 14s. Hence the required amount will, in the multiple-table of S 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 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) 10/24 of 14s. is 5s. 10d., (ii) 24/10 of 5s. 10d. is 14s. The two statements are in fact merely different aspects of a single relation, considered in the next section.

A
+------+-----------+
| 10 | 5s. 10d. |
+------+-----------+
| 24 | 14s. |
+------+-----------+

B
+------+-----------+
| 5 | 5s. 10d. |
+------+-----------+
| 12 | 14s. |
+------+-----------+

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. 10d. and 14s. on the one hand, 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 1s. 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 a/b and c/d are equal. For this reason the ratio of a to b is sometimes written a/b, 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 = q/p of P.

57. _Proportion._--If from any two columns in the table of S 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.

+----------+---------------+
| 2s. 10d. | 7 yds. 1 ft. |
+----------+---------------+
| 8s. 6d. | 22 yds. |
+----------+---------------+

This is expressed by saying 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_.

+---+---+
| M | P |
+---+---+
| 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.