The mathematicians Wessel, Argand, and Gauss, separately devised a graphical method of representing 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.
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 nth root of a real or complex number z is z1/n, the nth root of r cis θ is r1/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 11/3 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.