- (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 a < b, then a + b < b + x, and a − x < b − x;
- (e) If a = b, then ma = mb, and a ÷ m = b ÷ m;
- (f) If a > b, then ma > mb, and a ÷ m > b ÷ m;
- (g) If a < b, then ma < mb, and a ÷ m < b ÷ m.
(ii) Associative Law for Additions and Subtractions.—This law includes the rule of signs, that a − (b − c) = a − b + c; and it states that, subject to this, successive operations of addition or subtraction may be grouped in sets in any way; e.g. a − b + c + d + e − f = a − (b − c) + (d + e − f).
(iii) Commutative Law for Additions and Subtractions, that additions and subtractions may be performed in any order; e.g. a − b + c + d = a + c − b + d = a − b + c − b.
(iv) Associative Law for Multiplications and Divisions.—This law includes a rule, similar to the rule of signs, to the effect that a ÷ (b ÷ c) = a ÷ b × c; and it states that, subject to this, successive operations of multiplication or division may be grouped in sets in any way; e.g. a ÷ b × c × d × e ÷ f = a ÷ (b ÷ c) × (d × e ÷ f).
(v) Commutative Law for Multiplications and Divisions, that multiplications and divisions may be performed in any order: e.g. a ÷ b × c × d = a × c ÷ b × d = a × d × c ÷ b.
(vi) Distributive Law, that multiplications and divisions may be distributed over additions and subtractions, e.g. that m(a + b − c) = m.a + m.b − m.c, or that (a + b − c) ÷ n = (a ÷ n) + (b ÷ n) + (c ÷ n).
In the case of (ii), (iii) and (vi), the letters a, b, c, ... may denote either numbers or numerical quantities, while m and n denote numbers; in the case of (iv) and (v) the letters denote numbers only.
59. Results of Inverse Operations.—Addition, multiplication and involution are direct processes; and, if we start with positive integers, we continue with positive integers throughout. But, in attempting the inverse processes of subtraction, division, and either evolution or determination of index, the data may be such that a process cannot be performed. We can, however, denote the result of the process by a symbol, and deal with this symbol according to the laws of arithmetic. In this way we arrive at (i) negative numbers, (ii) fractional numbers, (iii) surds, (iv) logarithms (in the ordinary sense of the word).
60. Simple Formulae.—The following are some simple formulae which follow from the laws stated in § 58.
(i) (a + b + c + ...)(p + q + r + ...) = (ap + aq + ar + ...) + (bp + bq + br + ...) + (cp + cq + cr + ...) + ...; i.e. the product of two or more numbers, each of which consists of two or more parts, is the sum of the products of each part of the one with each part of the other.
(ii) (a + b)(a − b) = a2 − b2; i.e. the product of the sum and the difference of two numbers is equal to the difference of their squares.
(iii) (a + b)2 = a2 + 2ab + b2 = a2 + (2a + b)b.
V. Negative Numbers
61. Negative Numbers may be regarded as resulting from the commutative law for addition and subtraction. According to this law, 10 + 3 + 6 − 7 = 10 + 3 − 7 + 6 = 3 + 6 − 7 + 10 = &c. But, if we write the expression as 3 − 7 + 6 + 10, this means that we must first subtract 7 from 3. This cannot be done; but the result of the subtraction, if it could be done, is something which, when 6 is added to it, becomes 3 − 7 + 6 = 3 + 6 − 7 = 2. The result of 3 − 7 is the same as that of 0 − 4; and we may write it “−4,” and call it a negative number, if by this we mean something possessing the property that −4 + 4 = 0.
This, of course, is unintelligible on the grouping system of treating number; on the counting system it merely means that we count backwards from 0, just as we might count inches backwards from a point marked 0 on a scale. It should be remembered that the counting is performed with something as unit. If this unit is A, then what we are really considering is −4A; and this means, not that A is multiplied by −4, but that A is multiplied by 4, and the product is taken negatively. It would therefore be better, in some ways, to retain the unit throughout, and to describe −4A as a negative quantity, in order to avoid confusion with the “negative numbers” with which operations are performed in formal algebra.
The positive quantity or number obtained from a negative quantity or number by omitting the “−” is called its numerical value.
VI. Fractional and Decimal Numbers
62. Fractional Numbers.—According to the definition in § 50 the quantity denoted by 36 of A is made up of a number, 3, and a unit, which is one-sixth of A. Similarly pn of A, qn of A, rn of A, ... mean quantities which are respectively p times, q times r times, ... the unit, n of which make up A. Thus any arithmetical processes which can be applied to the numbers p, q, r, ... can be applied to pn, qn, rn, ... , the denominator n remaining unaltered.
If we denote the unit 1/n of A by X, then A is n times X, and pn of n times X is p times X; i.e. pn of n times is p times.
Hence, so long as the denominator remains unaltered, we can deal with pn, qn, rn, ... exactly as if they were numbers, any operations being performed on the numerators. The expressions pn, qn, rn, ... are then fractional numbers, their relation to ordinary or integral numbers being that pn times n times is equal to p times.
This relation is of exactly the same kind as the relation of the successive digits in numbers expressed in a scale of notation whose base is n. Hence we can treat the fractional numbers which have any one denominator as constituting a number-series, as shown in the adjoining diagram. The result of taking 13 sixths of A is then seen to be the same as the result of taking twice A and one-sixth of A, so that we may regard 136 as being equal to 216. A fractional number is called a proper fraction or an improper fraction according as the numerator is or is not less than the denominator; and an expression such as 216 is called a mixed number. An improper fraction is therefore equal either to an integer or to a mixed number. It will be seen from § 17 that a mixed number corresponds with what is there called a mixed quantity. Thus £3, 17s. is a mixed quantity, being expressed in pounds and shillings; to express it in terms of pounds only we must write it £31720.
63. Fractional Numbers with different Denominators.—If we divided the unit into halves, and these new units into thirds, we should get sixths of the original unit, as shown in A; while, if we divided the unit into thirds, and these new units into halves, we should again get sixths, but as shown in B. The series of halves in the one case, and of thirds in the other, are entirely different series of fractional numbers, but we can compare them by putting each in its proper position in relation to the series of sixths. Thus 32 is equal to 96, and 53 is equal to 106, and conversely; in other words, any fractional number is equivalent to the fractional number obtained by multiplying or dividing the numerator and denominator by any integer. We can thus find fractional numbers equivalent to the sum or difference of any two fractional numbers. The process is the same as that of finding the sum or difference of 3 sixpences and 5 fourpences; we cannot subtract 3 sixpenny-bits from 5 fourpenny-bits, but we can express each as an equivalent number of
