25. Preparation for Algebra.—The calculation of the values of simple algebraical expressions for particular values of letters involved is a useful exercise, but its tediousness is apt to make the subject repulsive.
What is more important is to verify particular examples of general formulae. These formulae are of two kinds:—(a) the general properties, such as m(a+b) = ma+mb, on which algebra is based, and (b) particular formulae such as (x-a)(x+a) = x²-a². Such verifications are of value for two reasons. In the first place, they lead to an understanding of what is meant by the use of brackets and by such a statement as 3(7+2) = 3•7+3•2. This does not mean (cf. § 23) that the algebraic result of performing the operation 3(7+2) is 3•7+3•2; it means that if we convert 7+2 into the single number 9 and then multiply by 3 we get the same result as if we converted 3•7 and 3•2 into 21 and 6 respectively and added the results. In the second place, particular cases lay the foundation for the general formula.
Exercises in the collection of coefficients of various letters occurring in a complicated expression are usually performed mechanically, and are probably of very little value.
26. General Arithmetical Theorems.
- (i.) The fundamental laws of arithmetic (q.v.) should be constantly borne in mind, though not necessarily stated. The following are some special points.
- (a) The commutative law and the associative law are closely related, and it is best to establish each law for the case of two numbers before proceeding to the general case. In the case of addition, for instance, suppose that we are satisfied that in a+b+c+d+e we may take any two, as b and c, together (association) and interchange them (commutation). Then we have a+b+c+d+e = a+c+b+d+e. Thus any pair of adjoining numbers can be interchanged, so that the numbers can be arranged in any order.
- (b) The important form of the distributive law is m(A+B) = mA+mB. The form (m+n)A = mA+nA follows at once from the fact that A is the unit with which we are dealing.
- (c) The fundamental properties of subtraction and of division are that A-B+B = A and m× of A = A, since in each case the second operation restores the original quantity with which we started.
- (ii.) The elements of the theory of numbers belong to arithmetic. In particular, the theorem that if n is a factor of a and of b it is also a factor of pa±qb, where p and q are any integers, is important in reference to the determination of greatest common divisor and to the elementary treatment of continued fractions. Graphic methods are useful here (§ 34 (iv.)). The law of relation of successive convergents to a continued fraction involves more advanced methods (see § 42 (iii.) and Continued Fraction).
- (iii.) There are important theorems as to the relative value of fractions; e.g.
- (a) If = then each = .
- (b) is nearer to 1 than is; and, generally, if ≠ , then lies between the two. (All the numbers are, of course, supposed to be positive.)
27. Negative Quantities and Fractional Numbers.—
- (i.) What are usually called "negative numbers" in arithmetic are in reality not negative numbers but negative quantities. If a person has to receive 7s. and pay 5s., with a net result of +2s., the order of the operations is immaterial. If he pays first, he then has -5s. This is sometimes treated as a debt of 5s.; an alternative method is to recognize that our zero is really arbitrary, and that in fact we shift it with every operation of addition or subtraction. But when we say "-5s." we mean "-(5s.)," not "(-5)s."; the idea of (-5) as a number with which we can perform such operations as multiplication comes later (§ 49).
- (ii.) On the other hand, the conception of a fractional number follows directly from the use of fractions, involving the subdivision of a unit. We find that fractions follow certain laws corresponding exactly with those of integral multipliers, and we are therefore able to deal with the fractional numbers as if they were integers.
28. Miscellaneous Developments in Arithmetic.—The following are matters which really belong to arithmetic; they are usually placed under algebra, since the general formulae involve the use of letters.
- (i.) Arithmetical Progressions such as 2, 5, 8, . . .—The formula for the rth term is easily obtained. The problem of finding the sum of r terms is aided by graphic representation, which shows that the terms may be taken in pairs, working from the outside to the middle; the two cases of an odd number of terms and an even number of terms may be treated separately at first, and then combined by the ordinary method, viz. writing the series backwards.
- In this, as in almost all other cases, particular examples should be worked before obtaining a general formula.
- (ii.) The law of indices (positive integral indices only) follows at once from the definition of a2, a3, a4, . . . as abbreviations of a•a, a•a•a, a•a•a•a, ..., or (by analogy with the definitions of 2, 3, 4, . . . themselves) of a•a, a•a2, a•a3, . . . successively. The treatment of roots and of logarithms (all being positive integers) belongs to this subject; a = = being the inverses of n = ap (cf. §§ 15, 16). The theory may be extended to the cases of p = 1 and p = 0; so that a3 means a•a•a•1, a2 means a•a•1, a1 means a•1, and a0 means 1 (there being then none of the multipliers a).
- The terminology is sometimes confused. In n = ap, a is the root or base, p is the index or logarithm, and n is the power or antilogarithm. Thus a, a2, a3, . . . are the first, second, third, . . . powers of a. But ap is sometimes incorrectly described as "a to the power p"; the power being thus confused with the index or logarithm.
- (iii.) Scales of Notation lead, by considering, e.g., how to express in the scale of 10 a number whose expression in the scale of 8 is 2222222, to
- (iv.) Geometrical Progressions.—It should be observed that the radix of the scale is exactly the same thing as the root mentioned under (ii.) above; and it is better to use the term "root" throughout. Denoting the root by a, and the number 2222222 in this scale by N, we have
N = 2222222.
aN = 22222220.
- Thus by adding 2 to aN we can subtract N from aN+2, obtaining 20000000, which is = 2 • a7; and from this we easily pass to the general formula for the sum of a geometrical progression having a given number of terms.
- (v) Permutations and Combinations may be regarded as arithmetical recreations; they become important algebraically in reference to the binomial theorem (§§ 41, 44).
- (vi.) Surds and Approximate Logarithms.—From the arithmetical point of view, surds present a greater difficulty than negative quantities and fractional numbers. We cannot solve the equation 7s.+X = 4s.; but we are accustomed to transactions of lending and borrowing, and we can therefore invent a negative quantity -3s. such that -3s.+3s. = 0. We cannot solve the equation 7X = 4s.; but we are accustomed to subdivision of units, and we can therefore give a meaning to X by inventing a unit s = 1s. such that 7×s = 1s., and can thence pass to the idea of fractional numbers. When, however, we come to the equation x² = 5, where we are dealing with numbers, not with quantities, we have no concrete facts to assist us. We can, however, find a number whose square shall be as nearly equal to 5 as we please, and it is this number that we treat arithmetically as √5. We may take it to (say) 4 places of decimals; or we may suppose it to be taken to 1000 places. In actual practice, surds mainly arise out of mensuration; and we can then give an exact definition by graphical methods.
- When, by practice with logarithms, we become familiar with the correspondence between additions of length on the logarithmic scale (on a slide-rule) and multiplication of numbers in the natural scale (including fractional numbers), √5 acquires