mathematics itself, nor any branch or set of branches of mathematics, can be regarded as an isolated science. While, therefore, the logical development of algebraic reasoning must depend on certain fundamental relations, it is important that in the early study of the subject these relations should be introduced gradually, and not until there is some empirical acquaintance with the phenomena with which they are concerned.

8. The extension of the range of subjects to which mathematical methods can be applied, accompanied as it is by an extension of the range of study which is useful to the ordinary worker, has led in the latter part of the 19th century to an important reaction against the specialization mentioned in the preceding paragraph. This reaction has taken the form of a return to the alliance between algebra and geometry (§5), on which modern analytical geometry is based; the alliance, however, being concerned with the application of graphical methods to particular cases rather than to general expressions. These applications are sometimes treated under arithmetic, sometimes under algebra; but it is more convenient to regard graphics as a separate subject, closely allied to arithmetic, algebra, mensuration and analytical geometry.

9. The association of algebra with arithmetic on the one hand, and with geometry on the other, presents difficulties, in that geometrical measurement is based essentially on the idea of continuity; while arithmetical measurement is based essentially on the idea of discontinuity; both ideas being equally matters of intuition. The difficulty first arises in elementary mensuration, where it is partly met by associating arithmetical and geometrical measurement with the cardinal and the ordinal aspects of number respectively (see ARITHMETIC). Later, the difficulty recurs in an acute form in reference to the continuous variation of a function. Reference to a geometrical interpretation seems at first sight to throw light on the meaning of a differential coefficient; but closer analysis reveals new difficulties, due to the geometrical interpretation itself. One of the most recent developments of algebra is the algebraic theory of number, which is devised with the view of removing these difficulties. The harmony between arithmetical and geometrical measurement, which was disturbed by the Greek geometers on the discovery of irrational numbers, is restored by an unlimited supply of the causes of disturbance.

IO.

ance.

Two other developments of algebra are of special importThe theory of sequences and series is sometimes treated as a part of elementary algebra; but it is more convenient to regard the simpler cases as isolated examples, leading up to the general theory. The treatment of equations of the second and higher degrees introduces imaginary and complex numbers, the theory of which is a special subject.

II. One of the most difficult questions for the teacher of algebra is the stage at which, and the extent to which, the ideas of a negative number and of continuity may be introduced. On the one hand, the modern developments of algebra began with these ideas, and particularly with the idea of a negative number. On the other hand, the lateness of occurrence of any particular mathematical idea is usually closely correlated with its intrinsic difficulty. Moreover, the ideas which are usually formed on these points at an early stage are incomplete; and, if the incompleteness of an idea is not realized, operations in which it is implied are apt to be purely formal and mechanical. What are called negative numbers in arithmetic, for instance, are not really negative numbers but negative quantities (§ 27 (i.)); and the difficulties incident to the ideas of continuity have already been pointed out.

12. In the present article, therefore, the main portions of elementary algebra are treated in one section, without reference to these ideas, which are considered generally in two separate sections. These three sections may therefore be regarded as to a certain extent concurrent. They are preceded by two sections dealing with the introduction to algebra from the arithmetical and the graphical sides, and are followed by a section dealing briefly with the developments mentioned in §§ 9 and 10 above.

I. Arithmetical Introduction to Algebra.

13. Order of Arithmetical Operations.-It is important, before beginning the study of algebra, to have a clear idea as to the meanings of the symbols used to denote arithmetical operations. (i.) Additions and subtractions are performed from left to right. Thus 3 lb + 5 lb-7 lb + 2 lb means that 5 lb is to be added to 3 lb, 7 lb subtracted from the result, and 2 lb added to the new result.

(ii.) The above operation is performed with 1 lb as the unit of counting, and the process would be the same with any other unit; e.g. we should perform the same process to find 3s.+5s.-75.+2s. Hence we can separate the numbers from the common unit, and replace 3 lb+5 lb 7 lb+2 lb by (3+5-7+2) Ib, the additions and subtractions being then performed by means of an addition-table.

(iii.) Multiplications, represented by X, are performed from right to left. Thus 5X3X7X1 lb means 5 times 3 times 7 times 1 lb; i.e. it means that 1 lb is to be multiplied by 7, the result by 3, and the new result by 5. We may regard this as meaning the same as 5X3X7 lb, since 7 lb itself means 7X1 lb, and the lb is the unit in each case. But it does not mean the same as 5X21 lb, though the two are equal, i.e. give the same result (see § 23).

This rule as to the meaning of X is important. If it is intended that the first number is to be multiplied by the second, a special sign such as should be used.

(iv.) The sign means that the quantity or number preceding it is to be divided by the quantity or number following it. (v.) The use of the solidus / separating two numbers is for convenience of printing fractions or fractional numbers. Thus 16/4 does not mean 16÷4, but 5.

(vi.) Any compound operation not coming under the above descriptions is to have its meaning made clear by brackets, the use of a pair of brackets indicating that the expression between them is to be treated as a whole. Thus we should not write 8X7+6, but (8X7)+6, or 8X (7+6). The sign X coming immediately before, or immediately after, a bracket may be omitted; e.g. 8X(7+6) may be written 8(7+6).

This rule as to using brackets is not always observed, the convention sometimes adopted being that multiplications or divisions are to be performed before additions or subtractions. The convention is even pushed to such an extent as to make "4+3 of 7+5" mean "4+(3 of 7)+5"; though it is not clear what "Find the value of 4+3 times 7+5" would then mean. There are grave objections to an arbitrary rule of this kind, the chief being the useless waste of mental energy in remembering it.

(vii.) The only exception that may be made to the above rule is that an expression involving multiplication-dots only, or a simple fraction written with the solidus, may have the brackets omitted for additions or subtractions, provided the figures are so spaced as to prevent misunderstanding. Thus 8+ (7X6)+3 may be written 8+7.6+3, and 8+3+3 may be written 8+7/6+3. But 3.5 should be written (3.5)/(2.4), not 3.5/2.4.

2.4

14. Latent Equations.-The equation exists, without being shown as an equation, in all those elementary arithmetical processes which come under the head of inverse operations; i.e. processes which consist in obtaining an answer to the question 'Upon what has a given operation to be performed in order to produce a given result?" or to the question "What operation of a given kind has to be performed on a given quantity or number in order to produce a given result?"

(i.) In the case of subtraction the second of these two questions is perhaps the simpler. Suppose, for instance, that we wish to know how much will be left out of 10s. after spending 3s., or how much has been spent out of 10s. if 3s. is left. In either case we may put the question in two ways:-(a) What must be added to 3s. in order to produce 10s., or (b) To what must 3s. be added in order to produce 10s. If the answer to the question is X, we have either (a) 10s. =3s. +X, ..X=10s. -3s. (b) IOS. X+35., .. X=10s. -35.

or

=

125 c.dm. X3,.. X=√125 c.dm. = √125 dm.

(b) To what power must 5 be raised to produce 125? Let the answer be x; then

125=5*, .. x = log. 125.

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

(i.) If A=X-B, then X=A+B.

(ii.) (a) If M

I

=m

of X, then X=m times M.

(b) If m=X÷M, then X=m times M.

(iii.) (a) If a=x, then x = a.

(b) If p=log.x, then x=a”.

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

(i.) If BA-X, then A- B+X.

(ii.) (a) If m=A÷X, then A = m times X.

(b) If M = of A, then A=x times M.

x

(iii.) (a) Ifp=log2n, then n = XP.

(b) If an, then n=a*.

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 subtractiontable (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):—

(i.) If A=B+X, or=X+B, then X-A-B.
(ii.) (a) If A = m times X, then X = of A.
(b) If A=x times M, then x = A÷M.

Vn.

I m

(iii.) (a) If n = x2, then x = (b) If n=a*, then x = log. 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, I of,, V, and log..

m

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.

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 lb of tea, and have 6s. 8d. left out of 10s.; how much per ib did tea cost?

left, the amount spent was 10s. - 6s. 8d., i.e. was 3s. 4d. There(1) In ordinary language we should say: Since 6s. 8d. was fore 2 lb of tea cost 3s. 4d. Therefore 1 lb of tea cost is. 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 lb tea +6s. 8d. = 10s.

.. Cost of 2 lb tea 10s. 6s. 8d. =3s. 4d.
.. Cost of 1 lb tea Is. 8d

..2XX =10s.-6s. 8d. =3s. 4d. .. X= Is. 8d.

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 2X 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 X3

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.

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.) 4X3 lb 12 lb.

=

I

{x = 10.

By successive stages we obtain (§ 18) x=2, x=12; or we may write at once x=576 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+3x=10, then 3x+ 2x=60 (and similarly if 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 4(X-35.) 1045. From this we can deduce successively X-3s. = 26s., X=295. But, if we transform the equation into

4X-12s. = 104s.,

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)=25 to x+2=√25, is arithmetical; but the transition from x2+4x+4=25 to (x+2)=25 is algebraical, since it involves a change of the number we are thinking about.

IS.

IS.

I2d. X24d. =2s.

If the operator is omitted, the statement is really an equation, giving is. in terms of id. 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 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 and q of a particular unit, then pXQ=qXP; or the more abstract statement that pXq=qXp.

(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).