Page:EB1911 - Volume 01.djvu/641

Rh (ii.) In the above case the two different kinds of statement lead to arithmetical formulae of the same kind. In the case of division we get two kinds of arithmetical formula, which, however, may be regarded as requiring a single kind of numerical process in order to determine the final result.

(a) If 24d. is divided into 4 equal portions, how much will each portion be?

Let the answer be X; then $\textstyle {\rm 24d.=4 \times X, \ \therefore X= {1 \over 4} \ of \ 24d.}$

(b) Into how many equal portions of 6d. each may 24d. be divided?

Let the answer be x; then

$\rm 24d.=\it x \times \rm 6d., \ \therefore \it x=\rm24d. \div 6d.$

(iii.) Where the direct operation is evolution, for which there is no commutative law, the two inverse operations are different in kind.

(a) What would be the dimensions of a cubical vessel which would exactly hold 125 litres; a litre being a cubic decimetre?

Let the answer be X; then

$\rm 125 \ c.dm.=X^3,\ \therefore X = \sqrt[3]{125 \ \rm c.dm.} = \sqrt[3]{125} \ dm$

(b) To what power must 5 be raised to produce 125?

Let the answer be x; then

$125=5^x, \ \therefore \ x= \log_5 125.$

15. With regard to the above, the following points should be noted.

(1) When what we require to know is a quantity, it is simplest to deal with this quantity as a whole. In (i.), for instance, we want to find the amount by which 10s. exceeds 3s., not the number of shillings in this amount. It is true that we obtain this result by subtracting 3 from 10 by means of a subtraction-table (concrete or ideal); but this table merely gives the generalized results of a number of operations of addition or subtraction performed with concrete units. We must count with something; and the successive somethings obtained by the addition of successive units are in fact numerical quantities, not numbers.

Whether this principle may legitimately be extended to the notation adopted in (iii.) (a) of § 14 is a moot point. But the present tendency is to regard the early association of arithmetic with linear measurement as important; and it seems to follow that we may properly (at any rate at an early stage of the subject) multiply a length by a length, and the product again by another length, the practice being dropped when it becomes necessary to give a strict definition of multiplication.

(2) The results may be stated briefly as follows, the more usual form being adopted under (iii.) (a):—


 * &emsp;$$\rm (i.) \ If \ A=B+X, \ or=X+B, \ then \ X=A-B.$$


 * &ensp;$$(\rm ii.)\ (\it a \rm) \ If \ A=\it m \ \rm times \ X, \ then \ X = \textstyle \ of \ A.$$


 * &emsp;&emsp;&emsp;$$(\it b \rm) \ If \ A= \it x \ \rm times \ M, \ then \ \it x =\rm A \div M.$$


 * $$(\rm iii.)\ (\it a \rm) \ If \ \it n=x^p \rm, \ then \ \it x=\sqrt[p]{}n.$$


 * $$(\it b \rm) \ If \it \ n=a^x, \ \rm then \ \it x= \log_a n.$$

The important thing to notice is that where, in any of these five cases, one statement is followed by another, the second is not to be regarded as obtained from the first by logical reasoning involving such general axioms as that “if equals are taken from equals the remainders are equal”; the fact being that the two statements are merely different ways of expressing the same relation. To say, for instance, that X is equal to A − B, is the same thing as to say that X is a quantity such that X and B, when added, make up A; and the above five statements of necessary connexion between two statements of equality are in fact nothing more than definitions of the symbols $$ -, \ \textstyle \ \rm of, \ \div, \ \sqrt[p]{}$$, and $$\ \it \log_a$$.

An apparent difficulty is that we use a single symbol − to denote the result of the two different statements in (i.) (a) and (i.) (b) of § 14. This is due to the fact that there are really two kinds of subtraction, respectively involving counting forwards (complementary addition) and counting backwards (ordinary subtraction); and it suggests that it may be wise not to use the one symbol − to represent the result of both operations until the commutative law for addition has been fully grasped.

16. In the same way, a statement as to the result of an inverse operation is really, by the definition of the operation, a statement as to the result of a direct operation. If, for instance, we state that A＝X−B, this is really a statement that X＝A+B. Thus, corresponding to the results under § 15 (2), we have the following:—

(1) Where the inverse operation is performed on the unknown quantity or number:—


 * &emsp;$$\rm (i.) \ If \ A=X-B, \ then \ X=A+B.$$


 * &ensp;$$(\rm ii.)\ (\it a \rm) \ If \ M = \textstyle \ of \ X, \ then \ X=\it m \ \rm times \ M.$$


 * &emsp;&emsp;&emsp;$$(\it b \rm) \ If \ \it m \rm =X \div M,\ then \ X= \it m \ \rm times \ M.$$


 * $$(\rm iii.)\ (\it a \rm) \ If \it \ a=\sqrt[p]{}x \rm ,\ then \ \it x=a^p.$$


 * &emsp;&emsp;&emsp;$$(\it b \rm) \ If \it \ p= \log_a x \rm, \ then \it \ x=a^p.$$

(2) Where the inverse operation is performed with the unknown quantity or number:—


 * &emsp;$$\rm (i.) \ If \ B=A-X, \ then \ A=B+X.$$


 * &emsp;&emsp;&emsp;$$(\it a \rm) \ If \ \it m \rm =A \div X,\ then \ A= \it m \ \rm times \ X.$$


 * &ensp;$$(\rm ii.)\ (\it b \rm) \ If \ M = \textstyle \ of \ A, \ then \ A=\it x \ \rm times \ M.$$


 * $$(\rm iii.)\ (\it a \rm) \ If \it \ p= \log_x n \rm, \ then \it \ n=x^p.$$


 * &emsp;&emsp;&emsp;$$(\it b \rm) \ If \it \ a=\sqrt[x]{}n \rm ,\ then \ \it n=a^x.$$

In each of these cases, however, the reasoning which enables us to replace one statement by another is of a different kind from the reasoning in the corresponding cases of § 15. There we proceeded from the direct to the inverse operations; i.e. so far as the nature of arithmetical operations is concerned, we launched out on the unknown. In the present section, however, we return from the inverse operation to the direct; i.e. we rearrange our statement in its simplest form. The statement, for instance, that 32−x＝25, is really a statement that 32 is the sum of x and 25.

17. The five equalities which stand first in the five pairs of equalities in §15(2) may therefore be taken as the main types of a simple statement of equality. When we are familiar with the treatment of quantities by equations, we may ignore the units and deal solely with numbers; and (ii.) (a) and (ii.) (b) may then, by the commutative law for multiplication, be regarded as identical. The five processes of deduction then reduce to four, which may be described as (i.) subtraction, (ii.) division, (iii.) (a) taking a root, (iii.) (b) taking logarithms. It will be found that these (and particularly the first three) cover practically all the processes legitimately adopted in the elementary theory of the solution of equations; other processes being sometimes liable to introduce roots which do not satisfy the original equation.

18. It should be noticed that we are still dealing with the elementary processes of arithmetic, and that all the numbers contemplated in §§ 14-17 are supposed to be positive integers. If, for instance, we are told that 15＝ of (x−2), what is meant is that (1) there is a number u such that x=u+2, (2) there is a number v such that u＝4 times v, and, (3) 15＝3 times v. From these statements, working backwards, we find successively that v＝5, u＝20, x＝22. The deductions follow directly from the definitions, and such mechanical processes as “clearing of fractions” find no place (§ 21 (ii.)). The extension of the methods to fractional numbers is part of the establishment of the laws governing these numbers (§ 27 (ii.)).

19. Expressed Equations.—The simplest forms of arithmetical equation arise out of abbreviated solutions of particular problems. In accordance with § 15, it is desirable that our statements should be statements of equality of quantities rather than of numbers; and it is convenient in the early stages to have a distinctive notation, e.g. to represent the former by capital letters and the latter by small letters.

As an example, take the following. I buy 2 ℔ of tea, and have 6s. 8d. left out of 10s.; how much per ℔ did tea cost?

(1) In ordinary language we should say: Since 6s. 8d. was left, the amount spent was 10s. − 6s. 8d., i.e. was 3s. 4d. Therefore 1 ℔ of tea cost 1s. 8d.

(2) The first step towards arithmetical reasoning in such a case is the introduction of the sign of equality. Thus we say:—

Cost of 2 ℔ tea+6s. 8d.=10s. ∴&thinsp;Cost of 2 ℔ tea=10s.−6s. 8d.=3s. 4d. ∴&thinsp;Cost of 1 ℔ tea=1s. 8d