70 VARIABLE in succession 6 = and a = 0. Quantities thus expressed are subject to all the ordinary algebraic rules applicable to real quantities, and in addition to the relation i 2 = - 1 . The occurrence of this symbol i in an identity thus singles out two distinct identities : for instance, if the product of a + ib and a + ib be A + iB, we have A = ad - bb and B = ab + ba ; and in this sense Cauchy, as lately as 1844, still maintained that every imaginary equation was only the symbolic representation of two equations between real quantities. If either or both of the real quantities be variable, the quantity is called a complex variable ; adopt ing the usual notation, it is then written x + iy = z. In order that z may vanish, both x and y must do so, as it is not possible for the real part to cancel the imaginary. But either part becoming infinite renders z infinite, and a breach of continuity in the change of the variable z arises when either x or y is discontinuous ; but, as long as both change continuously, z is also a continuous complex variable. 2. As long as a variable is conceived to admit of real values only, the distances from a fixed point, measured along a right line, are sufficient to represent it. These may be taken positively or negatively, and in both direc tions, through all magnitudes, from a vanishing amount to values large beyond conception, i.e., infinite. But, in asmuch as a complex variable z = x + iy depends on two perfectly independent real variables, x and y, its geometri cal realization demands a field of two dimensions. When this is assumed to be a plane, to each value z = x + iy cor responds a point P of the plane whose rectangular coor dinates referred to an arbitrarily assumed pair of axes in the plane have the real values x and y. Thus to all real quantities in the old sense correspond the points of the axis of abscissae and to pure imaginaries those of the axis of ordinates. The coordinates of the point P may change quite independently, just as the variables x and y do. If the point P be given, the corre- spending value of z is known, and also for a given value of z the corresponding point P is known. The continuous varia tion of z is exhibited by the motion of P on a curve. The complex number may also be written in the following form, which is found as early as Euler. Joining the point P to the origin 0, the line OP = p is called the modulus, and the angle XOP = which this line forms with the positive axis of x, or the prime axis, is called the argument (or amplitude) of the complex x + iy. Thus p = + /# 2 + y 2 , cos 6 = xjp, sin = yip, and z = x + iy = p(cos 6 + i sin 0) = pe is . This, again, shows that the variable vanishes only when x = and y = 0, i.e., when p = 0, since no real angle makes cos + i sin 0=0. In this way the quantity is represented by rotating through the angle 6 from the prime axis a line whose length is p. The value of the radical v/# 2 + y 1 is always taken positively ; the difference of sign arises from a reversal of direction and is indicated by the angle in the factor e i9 . When P is given, p is unique and 6 is determined by cos and sin : that is to say, it admits only of a certain value or of those which differ from it by integer multiples of 2-n-. Thus two quantities are equal whose moduli are equal and their directions parallel. The modulus varies continuously with z, and so does the argument, except when z vanishes : that is, when the curve passes through the origin, the argument then undergoes generally a sudden change by TT. 3. Complex numbers form a system complete in them selves, and any process of calculation on complex numbers always reproduces a complex number. Thus, as any such quantity is represented by a point approached from a given Fig. 1. point, the result of calculation also has a point or points to represent it. This is illustrated as follows. The addition of the quantity z = x+iy to z = x + iy produces z l = x + x + i(y + y ) ; the point z 1 lias the coordinates x + x , y + y : that is, it is the extremity of the diagonal of the parallelogram included by the lines Oz and Oz . Y If a third quantity z" = x" + iy" were added on, the result would be represented by the diagonal of the parallelogram included by Oz 1 and Oz" ; and so on. Sub traction is the addition of a line with opposite sign. The modu lus of the sum or difference of two quantities is found by as- o suming p l cos 6^p cos 0/o cos 6 and p l sin 0j= p sin 0p sin 6 , whence pi 2 = p 2 + /> -2p/> cos (9 - ). Thus the modulus of the sum is never greater than the sum of the moduli and the modulus of the difference never less than their difference. The Hue zz l in fig. 2 is found by subtracting z from z lt and is thus seen to be equal to the parallel line Oz . The multiplication of z=pe i(> by z = p e iff gives Z=zz =pp e l ( l>+(t ") : that is, we have to measure along a line inclined to the prime axis at an angle equal to the sum of the arguments a length representing the product of the moduli ; in order to determine this, the unit of length must be introduced. If this be OU measured along the prime axis, Z is constructed by placing on Oz a triangle z OZ similar and similarly placed to UOz. From this we can proceed at once to division ; as an example, when ~, 1 _ j- 1 = < ,_ , the triangle formed by the three points represent-
- i ~ *i z z
ing 21, z, z" l is homothetic with that formed by z, z , z". In like manner we might proceed to powers of com plex quantities with real exponents : e.g., every complex number has n distinct nth roots included in the form %/z = %/p , gi(8+zkir)/n . then to powers with complex expon ents; to logarithms of complex numbers : for example, every number has infinitely many logarithms to the base e, which differ by multiples of 2iir, in accordance with the formula log z log p + i(d 2^-7r) ; and, finally, to powers with com plex base and complex exponent. All these in turn lead again to complex numbers, and to such only. 4. In dealing with real numbers any definite range of values is represented by the points of a finite right line ; but to realize a definite range of complex numbers we must have recourse in general to a " region " of two dimen sions of the plane bounded by some curve. For instance, all complex numbers whose modulus is less than p and greater than p occupy a region bounded by two concen tric circles with radii p and p round the origin. All those whose moduli are equal to p form a linear region, namely, the circumference of the circle round the origin with that radius. A region of two dimensions is called connected when we can pass from any point within it to any other point within it without crossing the boundary curve. A quantity is said to be unrestrictedly variable in a region when it can assume all numerical values in this region. It is said to be continuously variable when all values which it assumes always belong to a finite connected region. Thus a variable is said to be unrestrictedly continuous for a certain value when it can take all the values which belong to a finite region, however small, which includes this value. The variable is restrictedly continuous for this value when the values it takes near this one form a region on whose boundary the value itself occurs, or it may be a region of one dimension. It is discontinuous for this value when the point is isolated, and does not belong to any region. When we know two definite values of a real variable, we know all the intermediate values which it must assume in passing from one to the other ; but in the case of the complex variable there is an essential difference : it can pass continuously from one given value to another by infinitely numerous series of continuously consecutive values. In geometric terms,
with the real variable we can only travel along the axis