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 continuous, so that continuity can only be achieved by an artificial development. The development is based on the necessity of being able to represent geometrical magnitude by arithmetical magnitude; and it may be regarded as consisting of three stages. Taking any number n to be represented by a point on a line at distance nL from a fixed point O, where L is a unit of length, we start with a series of points representing the integers 1, 2, 3, This series is of course discontinuous. The next step is to suppose that fractional numbers are represented in the same way. This extension produces a change of character in the series of numbers. In the original integral series each number had a definite number next to it, on each side, except 1, which began the series. But in the new series there is no first number, and no number can be said to be next to any other number, since, whatever two numbers we take, others can be inserted between them. On the other hand, this new series is not continuous; for we know that there are some points on the line which represent surds and other irrational numbers, and these numbers are not contained in our series. We therefore take a third step, and obtain theoretical continuity by considering that every point on the line, if it does not represent a rational number, represents something which may be called an irrational number.

This insertion of irrational numbers (with corresponding negative numbers) requires for its exact treatment certain special methods, which form part of the algebraic theory of number, and are dealt with under.

66. The development of the theory of equations leads to the amplification of real numbers, rational and irrational, positive and negative, by imaginary and complex numbers. The quadratic equation x2+b2 = 0, for instance, has no real root; but we may treat the roots as being +b√−1, and −b√−1, if √−1 is treated as something which obeys the laws of arithmetic and emerges into reality under the condition √−1.√−1 = −1. Expressions of the form b√−1 and a+b√−1, Where a and b are real numbers, are then described as imaginary and complex numbers respectively; the former being a particular case of the latter.

Complex numbers are conveniently treated in connexion not only with the theory of equations but also with analytical trigonometry, which suggests the graphic representation of a+b√−1 by a line of length (a2+b2)$1⁄2$ drawn in a direction different from that of the line along which real numbers are represented.

.—W. K. Clifford, The Common Sense of the Exact Sciences (1885), Chapters i. and iii., forms a good introduction to algebra. As to the teaching of algebra, see references under to works on the teaching of elementary mathematics. Among school-books may be mentioned those of W. M. Baker and A. A. Bourne, W. G. Borchardt, W. D. Eggar, F. Gorse, H. S. Hall and S. R. Knight, A. E. F. Layng, R. B. Morgan. G. Chrystal, Introduction to Algebra (1898); H. B. Fine, A College Algebra (1905); C. Smith, A Treatise on Algebra (1st ed. 1888, 3rd ed. 1892), are more suitable for revision purposes; the second of these deals rather fully with irrational numbers. For the algebraic theory of number, and the convergence of sequences and of series, see T. J. I’A. Bromwich, Introduction to the Theory of Infinite Series (1908); H. S. Carslaw, Introduction to the Theory of Fourier’s Series (1906); H. B. Fine, The Number-System of Algebra (1891); H. P. Manning, Irrational Numbers (1906); J. Pierpont, Lectures on the Theory of Functions of Real Variables (1905). For general reference, G. Chrystal, Text-Book of Algebra (pt. i. 5th ed. 1904. pt. ii. 2nd ed. 1900) is indispensable; unfortunately, like many of the works here mentioned, it lacks a proper index. Reference may also be made to the special articles mentioned at the commencement of the present article, as well as to the articles on ; ; ;. The following may also be consulted:—E. Borel and J. Drach, Introduction à l’étude de la théorie des nombres et de l’algèbre supérieure (1895); C. de Comberousse, Cours de mathématiques, vols. i. and iii. (1884–1887); H. Laurent, Traité d’analyse, vol. i. (1885); E. Netto, Vorlesungen über Algebra (vol. i. 1896, vol. ii. 1900); S. Pincherle, Algebra complementare (1893); G. Salmon, Lessons introductory to the Modern Higher Algebra (4th ed., 1885); J. A. Serret, Cours d’algèbre supérieure (4th ed., 2 vols., 1877); O. Stolz and J. A. Gmeiner, Theoretische Arithmetik (pt. i. 1900, pt. ii. 1902) and Einleitung in die Funktionen-theorie (pt. i. 1904, pt. ii. 1905)—these being developments from O. Stolz, Vorlesungen über allgemeine Arithmetic (pt. i. 1885, pt. ii. 1886); J. Tannery, Introduction à la théorie des fonctions d’une variable (1st ed. 1886, 2nd ed. 1904); H. Weber, Lehrbuch der Algebra, 2 vols. (1st ed. 1895–1896, 2nd ed. 1898–1899; vol. i. of 2nd ed. transl. by Griess as Traité d’algèbre supérieure, 1898). For a fuller bibliography, see ''Encyclopädie der math. Wissenschaften'' (vol. i., 1898). A list of early works on algebra is given in Encyclopedia Britannica, 9th ed., vol. i. p. 518.

1. A special algebra is one which differs from ordinary algebra in the laws of equivalence which its symbols obey. Theoretically, no limit can be assigned to the number of possible algebras; the varieties actually known use, for the most part, the same signs of operation, and differ among themselves principally by their rules of multiplication.

2. Ordinary algebra developed very gradually as a kind of shorthand, devised to abbreviate the discussion of arithmetical problems and the statement of arithmetical facts. Although the distinction is one which cannot be ultimately maintained, it is convenient to classify the signs of algebra into symbols of quantity (usually figures or letters), symbols of operation, such as +, √, and symbols of distinction, such as brackets. Even when the formal evolution of the science was fairly complete, it was taken for granted that its symbols of quantity invariably stood for numbers, and that its symbols of operation were restricted to their ordinary arithmetical meanings. It could not escape notice that one and the same symbol, such as √(a−b), or even (a−b), sometimes did and sometimes did not admit of arithmetical interpretation, according to the values attributed to the letters involved. This led to a prolonged controversy on the nature of negative and imaginary quantities, which was ultimately settled in a very curious way. The progress of analytical geometry led to a geometrical interpretation both of negative and also of imaginary quantities; and when a “meaning” or, more properly, an interpretation, had thus been found for the symbols in question, a reconsideration of the old algebraic problem became inevitable, and the true solution, now so obvious, was eventually obtained. It was at last realized that the laws of algebra do not depend for their validity upon any particular interpretation, whether arithmetical, geometrical or other; the only question is whether these laws do or do not involve any logical contradiction. When this fundamental truth had been fully grasped, mathematicians began to inquire whether algebras might not be discovered which obeyed laws different from those obtained by the generalization of arithmetic. The answer to this question has been so manifold as to be almost embarrassing. All that can be done here is to give a sketch of the more important and independent special algebras at present known to exist.

3. Although the results of ordinary algebra will be taken for granted, it is convenient to give the principal rules upon which it is based. They are

(a+b)+c = a+(b+c)&emsp;&emsp;(a×b)×c = a×(b×c)&emsp;(′) &emsp;a+b = b+a&emsp;&emsp;&emsp;&ensp;&ensp;&ensp;a×b = b×a&emsp;&emsp;&ensp;&emsp;&emsp;(′) a(b+c) = ab+ac&emsp;&emsp;&emsp; &emsp;(a−b)+b = a&emsp;&emsp;&emsp;&ensp;&ensp;(a÷b)×b = a(′)

These formulae express the associative and commutative laws of the operations + and ×, the distributive law of ×, and the definitions of the inverse symbols − and ÷, which are assumed to be unambiguous. The special symbols 0 and 1 are used to denote a−a and a÷a. They behave exactly like the corresponding symbols in arithmetic; and it follows from this that whatever “meaning” is attached to the symbols of quantity, ordinary algebra includes arithmetic, or at least an image of it. Every ordinary algebraic quantity may be regarded as of the form +√−1, where, are “real”; that is to say, every algebraic equivalence remains valid when its symbols of quantity are interpreted as complex numbers of the type +√−1 (cf. ). But the symbols of ordinary algebra do not necessarily denote numbers; they may, for instance, be interpreted as coplanar points or vectors. Evolution and involution are usually regarded as operations of ordinary algebra; this leads to a notation for powers and roots, and a theory of irrational algebraic quantities analogous to that of irrational numbers.