Complex Numbers, An Introduction


Since -1 has no square root in the reals, boldly declare i2 = -1. Multiples of i are imaginary numbers. After all, we can only imagine the square root of -1. Numbers of the form a+bi, part real and part imaginary, are called complex numbers, probably because they are complicated.

Complex numbers are often graphed in the complex plane, which is really the xy plane with the x axis real and the y axis imaginary. Thus a+bi is the point with coordinates x=a and y=b.

Addition is performed per component and multiplication is determined by the distributive law. In other words, expand the product algebraically, replace i2 with -1, and simplify, giving this well known formula.

(a+bi) × (c+di) = (ac-bd) + (ad+bc)i

The norm of a complex number z is its distance from the origin, which is sqrt(a2+b2) by the pythagorean theorem. Let |z| be the norm squared, or a2+b2. This lets us define the inverse of z as follows. (Here z is a general number a+bi.)

a+bi × (a-bi)/|z| = 1

Expand the left hand side and get a2+b2 over a2+b2, or 1. Everything, other than 0, is invertible, so division is well defined. The complex numbers form a field.

A complex function f takes complex numbers to complex numbers. Portions of the complex plane are moved onto other portions of the complex plane. We can't really graph such a function, because we need four dimensions. But sometimes we can visualize regions of the complex plane moving about. For instance, f(z) = 2z pushes everything outward, away from the origin, until each point is twice as far from 0 as it was before. Similarly, ½z contracts the plane towards the origin. The function iz multiplies everything by i, and rotates the plane 90° counterclockwise.

Now for something nonlinear. To visualize z2, start with the unit circle. Break it at the point 1,0 and pull it around again, so that it wraps around twice. Then sew the broken ends back together. Thus 1 stays where it is, i is pulled around to -1, -1 is pulled around to 1, and so on. If the circle is larger, with radius r, break it at r,0 and wrap it around twice, as we did before, then increase the radius to r2. This will become clearer after the next theorem.