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.
10. Two other developments of algebra are of special importance. The 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.
11. 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.
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 ℔ + 5 ℔ − 7 ℔ + 2 ℔ means that 5 ℔ is to be added to 3 ℔, 7 ℔ subtracted from the result, and 2 ℔ added to the new result.
(ii.) The above operation is performed with 1 ℔ 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. − 7s. + 2s. Hence we can separate the numbers from the common unit, and replace 3 ℔ + 5 ℔ − 7 ℔ + 2 ℔ by (3 + 5 − 7 + 2) ℔, the additions and subtractions being then performed by means of an addition-table.
(iii.) Multiplications, represented by ×, are performed from right to left. Thus 5×3×7×1 ℔ means 5 times 3 times 7 times 1 ℔; i.e. it means that 1 ℔ 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 5×3×7 ℔, since 7 ℔ itself means 7×1 ℔, and the ℔ is the unit in each case. But it does not mean the same as 5×21 ℔, though the two are equal, i.e. give the same result (see § 23).
This rule as to the meaning of × 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 164.
(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 8×7+6, but (8×7)+6, or 8×(7+6). The sign × coming immediately before, or immediately after, a bracket may be omitted; e.g. 8×(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 “412+323 of 7+5” 412+(323 of 7)+5”; though it is not clear what “Find the value of 412+323 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+(7×6)+3 may be written 8+7.6+3, and 8+76+3 may be written 8+7/6+3. But 3 . 52 . 4 should be written (3.5)/(2.4), not 3.5/ 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. |
