COMPLEX NUMBERS are such as consist each of (at least) two constituents, no part of either equalling any part of the other. To elucidate : Numbers seem implied, though dimly, as well-nigh primary features of psychic experience; in perception "objects" are posited singly (sun, moon) or in groups (stars, fin gers), and certain likenesses among groups are slowly conceived as numbers. Hence the class of integers, 1, 2, 3, 4, 5 (fingers), etc. ; much later the two hands yield ten as (an unfortunate) base of notation. Such counting, conceptualizing, once started, pro ceeds indefinitely ; the groups soon become no longer envisageable.
Negatives.—These "natural" or whole numbers are combined in two direct operations (addition, multiplication), which present certain uniformities called commutative, associative and dis tributive laws (q.v.) expressed thus: a+b=b+a, a+(b+c)= a+b+c, and ab=be, abc=a(bc), a(b+c) =ab+ac, each de claring equivalence between two ways of counting. To these the "Law of Cancellation" may be added: If a + b = a + b', then b = b'; if a.b = a.b', then b = b'. Both operations may proceed at will, without end; but not so their inverses, subtraction (a—b) and division (a=b, a/b). A special case requires notice: if from any group as 5, all (5) be subtracted, nothing is left as remainder. To symbolize this we invent a number zero (o), defined thus: a — a = o; also a -!- o = a, a X o = o. The operation a+b is possible for every b, but a—b only for b not more than a. The mind chafes at such limitation and would make such subtraction always possible by fashioning new numbers (a'), such that a+a'=o. Since a — a = o, adding a' is equivalent to subtracting a; hence the new numbers (called negative to distinguish them from the originals, called positive—compare north and south latitudes) are marked thus a' = (— a) . The domain of numbers thus doubled, all subtractions become possible, the foregoing laws still holding.
Fractions.—Similarly, among integer-groups most divisions (partitions into equal groups) are impossible (as of 7 by 2 or 3) ; and again, refusing to be hemmed • in activity, our minds frame new numbers by definition : Any fraction n/d shall be a number, which multiplied by d yields a product n; or (n/d) d = n (n and d being either -I- or — ). In this new realm of number all divisions also are possible, except division by o, which remains undefined and excluded.
The mind now proposes intricate inverse prob lems, asking, "What is the number on which given operations pro duce a given result?" i.e., "What is the x that satisfies a given equation?" For simple x, of 1st degree, the answer is ready : if ax+ b = c, then x= (c — b)/d, a number already created. Not so, perhaps, for x in the second degree, as if
2 ; no integer or fraction has 2 as its square. The question, then, remains un answerable till a new number is made, whose square is 2 ; so de fined, it is named second (or square) root of 2 and is written Similar definition is extended to all positives (by Theodorus of Cyrene, c. 490 B.C., up to 17, thence on by Theaetetus of Athens, c. 38o B.c.), to higher powers and roots, as the 3rd, 4th, .. nth. Such algebraic irrationals were long a myth in mathe matics; the Greek mind shrank from creating such monsters. But it was imperative : the diagonal of a unit-square showed a length requiring such an irrational as its symbol. The Greek, however, had no such symbol; for him the irrational existed, but as a length only, and for centuries still he persisted in regarding only integers as numbers proper.
Far more difficult the question, "What is x, if x2+1 = o, or x2= —12?" No answer is found among numbers thus far defined, since all their squares are +. So again the un escapable alternative: accept the impossible, or else make new numbers with negative squares. At last, with extreme reluctance, the mind ordains a new unit (i, so designated by Euler, 1748), defining : i.i= 1, /2+ =o. (The 2d root of a negative seems to present itself for the first time in the Stereometrica [I., 34] of Heron the Great Measurer [of Alexandria, c. A.D. so?-2oo?], where V-63 is taken as V63.) Though unfortunately called "imaginary" (if not by Albert Girard, 1629, by Descartes, 1637, who seems to regard such "roots" as non-existent, in his Geome trie), it is neither more nor less imaginary than the (so-called real) unit i, though of later birth from intellectual activity of higher refinement.
Thus coined to meet an emergency, the i-unit is subjected to the same laws as its elders, and to the fundamental postulate: If two factors form a product o (ab=o), then one of them is o (a = o or b=o). Compounds of these two disparate units (1 and i) are called "complex numbers" (Gauss) or "quan tities" and are written x+iy to make the composition explicit; otherwise, z may be put for x-Fiy. As x and y are quite inde pendent of each other, evidently the domain of complex numbers is twofold extended. The factors of x2-1-y2, x-Fiy and x—iy are conjugate.
Is a third unit needed? The central theorem of algebra answers that, however complex the conditions, imposed, if expressible through an algebraic equation of the nth degree in z, they are satisfied by n complex z—values, of the form a-Fib (Gauss, 1799) ; so a third unit is not needed. The complex "field" or domain is closed and self-consistent; there is no way to pass without it by operations within it. However, for n>4, the roots in general are no longer expressible through algebraic functions of the coefficients, as foreseen by Gauss (1777-1855), declared by Ruffini, and strictly proved first by Abel (18°2-29). Hence we must insert an endless class of transcendents, like 7r and e, none expressible precisely by algebraic operations in finite numbers, but all precise as cuts (Dedekind) and determinable to any degree of precision, wherewith the linear series (of so-called reals and hence also of "imaginaries") becomes "dense," "compact," with out gap. Their existence, also, is secured solely by definition, and some (as 7r and e) play significant rOles in other quarters, e.g., metric geometry.
But even if not necessary, may not new units be admissible? So thought, even at a sacrifice of the commutative law, Grass mann ("Ausdehnungslehre," 1844) and Hamilton (Quaternions, 1853, 1866). Their brilliant creations have won more wonder than imitation. As Schering iemarked, "All results attainable by Quaternions may be reached by shorter and smoother paths." (See QUATERNIONS.) Axes of Reals and Imaginaries—Graphic representation or depiction of number is of prime importance. As it is natural to objectify mental experience, nothing seems simpler than to pic ture integers on a scale or axis by lengths in units starting from a zero-point o, successive ends be ing marked 1, 2, 3, right and left respectively + and — (fig. I). Fractions and irrationals fall between the integers, and the whole line or axis depicts by its points (as distant from o) the class of so-called reals.
How to depict the other unit i and the universe of complex numbers was long a vexing puzzle. Two centuries after the cubic had slowly yielded (1506-76) to the Italians, Dal Ferro, Tar taglia, Cardano and others, the "imaginary" still hovered in the air. The first fixation was by John Wallis ("De Algebra Tracta tus," T685). His constructions, like Heinrich Kiihn's (1753), though ingenious, were too complicated to be satisfactory. Far superior was the essay of the surveyor Caspar Wessel ("Om Directionens analytische Betegning"), laid before the Danish Royal Academy 1797, printed in its Memoirs 1799, there for gotten till rediscovered and published in French 1897—a kind of vector-analysis akin to the developments of the Paris account ant Jean Robert Argand, 1806, to whom credit for the accepted representation is commonly assigned. The decisive step, uncon sciously in Wessel's tracks, was taken in Gauss's memoir, April 15,1831, contributed to the Royal Society of Gottingen.
The simple re flection seems to be this: Since i.i= 1, two multiplications by i equal one multiplication by —I, this is depicted by rotation (fig. 2) about 0 through 18o°, since —a= I) hence, con sistently, one multiplication by i may be depicted by rotation through 9o°, which leaves all pure i-numbers ranged on an axis through 0 perpendicular to the (so-called) real axis. Where, then, is the complex, as 3+4i? Answer: the part 3 is represented by OF on the "real" axis, the part 4i by FP vertical, the whole 3+4i by the broken line OFP, or by the point P (of co-ordinates 3, 4); and so for all sudi, which together fill the plane.
The length OP, imaging in size a number r=a+bi is Va2-1-b2, called the "norm" or "absolute value" (or "amount"). Denote it by / and the director-angle of OP by (15, then at once r=l (cos0+i sin0). That complex multiplication is pictured by rotation round 0 is vividly shown in Cotes-DeMoivre-Euler formulae (171o, 1730, 1743); eick =cos(k+i sinch, (eifl))° = (cos ck+i sinctOn =cosnck+i sinn4); also eiCeie = (cos0+isin4)) (cos 0+i sin0) = cos(0+0) +i sin(cA +0). Plainly eie, or cos° +i sin°, is a complex number of norm imaged by a point on the unit-circle about 0 at the end of the arc of the angle 0, and each multiplication by eick rotates the point through an arc ci) (fig. 3). Now to multiply a+ib (OF) by c+di (OF'), resolve each into norm and director, thus: a-l-bi=Va2-1-b2.
c+di = c2-4-d2, elf); the product is -‘1 a2+b2.1 e2 - F d2. ei (0+ ), i.e., norm of the prod uct is the product of the norms, and the directive angle is the sum of the directives of the factors (fig. 4). Multiplication is thus, graphically, simultaneous turn ing (rotation, rotor) and stretch ing (tension, tensor). Addition of complex numbers is easy: thus (a-Fib) + (c-1-id) = a+c-Fi= (b+d). The point S, standing for sum, is also reached by adding OP, OQ as vectors, i.e., by trans porting OQ to the parallel posi tion PS, or by drawing OS as a diagonal of the parallelogram de termined by OP, OQ (fig. 5). Such is the addition or composi tion of vectors (representing directed magnitudes), as in the Paral lelogram of Forces, etc.: Subtraction and division are like addition and multiplication, mutatis mutandis.
In analytic geometry the plane is fully possessed by x- and y-axes, each point depicting a pair of reals (x, y) ; any line an equation, as a circle
a right line as bx+ay = ab. Intersections are found by solving two equations as simultaneous; points common to two loci, as line and circle, picture pairs of values (of x and y) satisfying both equations. If both pairs be real, their corresponding points are in the plane (X, Y) ; if complex they are not—line and circle not meeting in the plane (fig. 2, under Coordinates). Con sistency bids them meet in complex points of a complex domain —but where? The problem is too profound for discussion here ; the final answer, complete and perfect, must be awaited. Yet in any case it will exemplify the fact that numbers and their graphs are creatures of Mind, and even though come of age they still obey their creator.
Any adequate treatment of the logical bases of the algebra of complex quantities and its "iso morphic" systems, in the sense of Peano and the Italian school (since 189o), with its 27 inde pendent postulates, or of Dede kind's arithmetic system of "cuts," and of "couples" with sums and products so defined as to introduce the "imaginary" incognito (Essays on Number, tr. by W. W. Beman, 19o9) without naming it, or of Riemann's representation of functions of complex argu ment, would lead too far afield and belong rather to algebra and logic, and to function theory in general, which the reader may find discussed under appropriate titles.
R. Argand, Essai sur une maniere de representer Bibliography.-J. R. Argand, Essai sur une maniere de representer ... imaginaires (1806-1874) ; J. L. Coolidge, Geometry of the Complex Domain (1924) ; G. Darboux, Sur une classe . . . et sur la theorie des imaginaires (1895) ; A. Frankel and A. Ostrovski, Zahl begriff u. Algebra bei Gauss (192o) ; F. Klein, Elementarmath. vom hoheren Standpunkte aus, I (14)
J. L. S. Hatton, The Theory of the Imaginary in Algebra, etc. (1920) ; H. Minkowski, Geometrie der Zahlen; A. Vergibrusson, Memoire sur les classes des nombres complexes (1912) ; H. Wieleitner, D. Begriff d. Zahlen in seiner log. u. historischen Entwicklung (1918). See also E. V. Huntington, "The Fundamental Propositions of Algebra" in Monographs on Modern Mathematics, by J. W. A. Young (191 I) ; E. Study, Theorie der gemeinen and hoheren complexen Grossen in Heft 2, I. Teil, I. Bd. Encyklopadie der mathematischen Wissenschaften (1899) ; and P. Bachmann, K. Th. Vahlen, D. Hilbert, H. Weber on "Theory of Numbers," etc., in Heft 5 and Heft 6, II. Teil of same vol. (190o) ; E. Landau, Einfuhrung in die elementare and analytische Theorie der algebraischen Zahlen and der Ideale (1927). (W. B. SAL)
