Complex Numbers

This section begins with a review of the complex numbers. Complex numbers are useful in the solution of second order differential equations. The starting point is the imaginary number i, which is the square root of -1. The complex number system is an extension of the real number system that is formed by adding the number i and keeping the usual rules for sums and products. The set of complex numbers, or complex plane, is the set of all numbers of the form

z = x + iy

where x and y are real numbers. The number x is called the real part of z, and y is called the imaginary part of z. A complex number z can be represented by a point in the plane, with the real part drawn on the horizontal axis and the imaginary part on the vertical axis, as in Figure 14.5.1. The sum of two complex numbers is computed in the same way as the sum of two vectors,

(a + ib) + (c + id) = (a + c) + i(b + d).

Figure 14.5.1

The product of two complex numbers is computed using the basic rule i² = - 1 and the rules of algebra:

(a + ib) - (c + id) = ac + ibc + iad + i²bd = (ac - bd) + i(bc + ad).