Complex Numbers

The mathematicians Wessel, Argand, and Gauss, separately devised a graphical method of representing Figure 2. Illustration by Hans & Cassidy. Courtesy of Gale Group. complex numbers. Real numbers have only a real component and can be represented as points on a one-dimensional number line. Complex numbers, on the other hand, with their real and imaginary components, require a two-dimensional complex-number plane.

The complex number plane can also by represented with polar coordinates. The relation between these coordinates and rectangular coordinates are given by the equations

Thus the complex number x + iy can also be written in polar form r cos θ + ir sin θ or r(cos θ + i sin θ), abbreviated r cis θ. When written in polar form, r is called the "modulus" or "absolute value" of the number. When the point is plotted on a Gauss-Argand diagram, r represents the distance from the point to the origin.

The angle θ is called the "argument" or the "amplitude" of the complex number, and represents the angle shown in Figure 2. Because adding or subtracting 360° to θ will not change the position of the ray drawn to the point, r cis θ, r cis ( θ + 360°), r cis ( θ - 360°), and others all represent the same complex number. When θ is measured in radians, adding or subtracting a multiple of 2π will change the representation of the number, but not the number itself.

In polar form the five points shown in Figure 1 are A: √θ (cos 45° + I sin 45°) or √θ cis 45°. B: √+5 cis 153.4°. C: 2 cis 225°. D: 5 cis 306.9°. E: √+5 cis 333.4°. Except for A and C, the polar forms seem more awkward. Often, however, it is the other way about. 1 cis 72°, which represents the fifth root of 1 is simple in its polar form, but considerably less so in its rectangular form: .3090 +.9511 i, and even this is only an approximation.

The polar form of a complex number has two remarkable features which the rectangular form lacks. For one, multiplication is very easy. The product of r₁ cis θ₁ and r₂ cis θ₂ is simply r₁ r₂ cis ( θ₁ + θ₂). For example, (3 cis 30°) (6 cis 60°) is 18 cis 90°. The other feature is known as de Moivre's theorem: (r cis θ)ⁿ = rⁿ cis n θ, where n is any real number (actually any complex number). This is a powerful theorem. For example, if one wants to compute (1 + i)ⁿ, multiplying it out can take a lot of time. If one converts it to polar form, ≠l2 cis 45°, however, (≠2 cis 45°)⁵ ≠32 cis 225° or -4 -4i.

One can use de Moivre's theorem to compute roots. Since the n^th root of a real or complex number z is z^1/n, the n^th root of r cis θ is r^1/n cis θ/n.

It is interesting to apply this to the cube root of 1. Writing 1 as a complex number in polar form one has 1 cis 0°. Its cube root is 1^1/3 cis 0/3°, or simply 1. But 1 cis 0° is the same number as 1 cis 360° and 1 cis 720°. Applying de Moivre's theorem to these alternate forms yields 1 cis 120° and 1 cis 240°, which are not simply 1. In fact they are -1/2 + √+2 /2 i and -1/2 - √+3/2i in rectangular form.