(3) The next step is to show more distinctly the unit we are dealing with (in addition to the money unit), viz. the cost of 1 ℔ tea. We write:—
(2 × cost of 1 ℔ tea)+6s. 8d.＝10s.
∴ 2 × cost of 1 ℔ tea＝10s.−6s. 8d.＝3s. 4d.
∴ Cost of 1 ℔ tea＝1s. 8d.
(4) The stage which is introductory to algebra consists merely in replacing the unit “cost of 1 ℔ tea” by a symbol, which may be a letter or a mark such as the mark of interrogation, the asterisk, &c. If we denote this unit by X, we have
20. Notation of Multiples.—The above is arithmetic. The only thing which it is necessary to import from algebra is the notation by which we write 2X instead of 2 × X or 2 . X. This is rendered possible by the fact that we can use a single letter to represent a single number or numerical quantity, however many digits are contained in the number.
It must be remembered that, if a is a number, 3a means 3 times a, not a times 3; the latter must be represented by a × 3 or a . 3.
The number by which an algebraical expression is to be multiplied is called its coefficient. Thus in 3a the coefficient of a is 3. But in 3 . 4a the coefficient of 4a is 3, while the coefficient of a is 3 . 4.
21. Equations with Fractional Coefficients.—As an example of a special form of equation we may take
(i.) There are two ways of proceeding.
(a) The statement is that (1) there is a number u such that x＝2u, (2) there is a number v such that x＝3v, and (3) u+v＝10. We may therefore conveniently take as our unit, in place of x, a number y such that
x＝6y.
We then have
3y+2y＝10,
whence
5y＝10, y＝2, x＝6y＝12.
(b) We can collect coefficients, i.e. combine the separate quantities or numbers expressed in terms of x as unit into a single quantity or number so expressed, obtaining
By successive stages we obtain (§ 18) x＝2, x＝12; or we may write at once x＝ of 10＝ of 10＝12. The latter is the more advanced process, implying some knowledge of the laws of fractional numbers, as well as an application of the associative law (§ 26 (i.)).
(ii.) Perhaps the worst thing we can do, from the point of view of intelligibility, is to “clear of fractions” by multiplying both sides by 6. It is no doubt true that, if +x+x＝10, then 3x+2x＝60 (and similarly x+x+x＝10, then 3x+2x+x＝60); but the fact, however interesting it may be, is of no importance for our present purpose. In the method (a) above there is indeed a multiplication by 6; but it is a multiplication arising out of subdivision, not out of repetition (see Arithmetic), so that the total (viz. 10) is unaltered.
22. Arithmetical and Algebraical Treatment of Equations.—The following will illustrate the passage from arithmetical to algebraical reasoning. “Coal costs 3s. a ton more this year than last year. If 4 tons last year cost 104s., how much does a ton cost this year?”
If we write X for the cost per ton this year, we have
From this we can deduce successively X−3s.＝26s., X＝29s. But, if we transform the equation into
we make an essential alteration. The original statement was with regard to X−3s. as the unit; and from this, by the application of the distributive law (§ 26 (i.)), we have passed to a statement with regard to X as the unit. This is an algebraical process.
In the same way, the transition from (x2+4x+4)−4＝21 to x2+4x+4＝25, or from (x+2)2＝25 to x+2＝√25, is arithmetical; but the transition from x2+4x+4＝25 to (x+2)2＝25 is algebraical, since it involves a change of the number we are thinking about.
Generally, we may say that algebraic reasoning in reference to equations consists in the alteration of the form of a statement rather than in the deduction of a new statement; i.e. it cannot be said that “If A＝B, then E＝F” is arithmetic, while “If C＝D, then E＝F” is algebra. Algebraic treatment consists in replacing either of the terms A or B by an expression which we know from the laws of arithmetic to be equivalent to it. The subsequent reasoning is arithmetical.
23. Sign of Equality.—The various meanings of the sign of equality (=) must be distinguished.
(i.) 4 × 3 ℔＝12 ℔.
This states that the result of the operation of multiplying 3 ℔ by 4 is 12 ℔.
(ii.) 4 × 3 ℔＝3 × 4 ℔.</math>
This states that the two operations give the same result; i.e. that they are equivalent.
(iii.) A’s share＝5s., or
3 × A’s share＝15s.
Either of these is a statement of fact with regard to a particular quantity; it is usually called an equation, but sometimes a conditional equation, the term “equation” being then extended to cover (i.) and (ii.).
This is a definition of x3; the sign＝is in such cases usually replaced by ≡.
This is usually regarded as being, like (ii.), a statement of equivalence. It is, however, only true if 1s. is equivalent to 12d., and the correct statement is then
If the operator is omitted, the statement is really an equation, giving 1s. in terms of 1d. or vice versa.
The following statements should be compared:—
X＝A’s share＝ of £10＝3×5£＝£15.
X ＝A’s share＝ of £10 = of £30＝£15.
In each case, the first sign of equality comes under (iv.) above, the second under (iii.), and the fourth under (i.); but the third sign comes under (i.) in the first case (the statement being that ½ of £10＝£5) and under (ii.) in the second.
It will be seen from § 22 that the application of algebra to equations consists in the interchange of equivalent expressions, and therefore comes under (i.) and (ii.). We replace 4(x−3), for instance, by 4x−4.3, because we know that, whatever the value of x may be, the result of subtracting 3 from it and multiplying the remainder by 4 is the same as the result of finding 4x and 4.3 separately and subtracting the latter from the former.
A statement such as (i.) or (ii.) is sometimes called an identity.
The two expressions whose equality is stated by an equation or an identity are its members.
24. Use of Letters in General Reasoning.—It may be assumed that the use of letters to denote quantities or numbers will first arise in dealing with equations, so that the letter used will in each case represent a definite quantity or number; such general statements as those of §§ 15 and 16 being deferred to a later stage. In addition to these, there are cases in which letters can usefully be employed for general arithmetical reasoning.
- (i.) There are statements, such as A+B＝B+A, which are particular cases of the laws of arithmetic, but need not be expressed as such. For multiplication, for instance, we have the statement that, if P and Q are two quantities, containing respectively p and q of a particular unit, then p×Q＝q×P; or the more abstract statement that p×q＝q×p.
- (ii.) The general theory of ratio and proportion requires the use of general symbols.
- (iii.) The general statement of the laws of operation of fractions is perhaps best deferred until we come to fractional numbers, when letters can be used to express the laws of multiplication and division of such numbers.
- (iv.) Variation is generally included in text-books on algebra, but apparently only because the reasoning is general. It is part of the general theory of quantitative relation, and in its elementary stages is a suitable subject for graphical treatment (§ 31).
