2.1. Complex Numbers
Real numbers are the numbers we are all familiar with, and it is difficult to imagine any other kind of numbers. However, there is a very important class of numbers known as complex numbers, which includes so-called imaginary numbers as well as real numbers. Despite the name imaginary, these numbers are incredibly useful in many areas of science, including electromagnetic theory, relativity and quantum mechanics.
What are imaginary numbers? Consider the quadratic equation x2 + 1 = 0, or more simply x2 = -1. Does this equation have any solutions? It obviously has no real solutions because the square of every real number is nonnegative. But what if we define a number i, known as the imaginary number, whose square is equal to -1? You might say this does not make sense, and in fact, imaginary numbers were dismissed for many years for this reason. But one must realize that in mathematics, we can define whatever concepts we like as long as they satisfy a set of logically consistent axioms, and remarkably enough, imaginary numbers do so. Not only that, but as we will see, they turn out to have plenty of beautiful properties.
Now that we have defined the number i, sometimes written as √-1, we may extend the real numbers to include it as well as a host of other imaginary numbers. The simplest way to do so is to define the set of complex numbers as the set of all numbers of the form z = x + yi, where x and y are arbitrary real numbers. The number x is known as the real part and y the imaginary part of the complex number z.
Complex numbers have a very nice geometric interpretation. Just as the real numbers may be viewed as a line, namely the real number line, the complex numbers may be viewed as a plane, known as the complex plane. We simply identify the complex number z = x + yi with the point (x, y) in the complex plane as shown in Figure 2.1.1.
Figure 2.1.1: The Complex Plane
There are two useful operations on complex numbers. One is known as complex conjugation. Given a complex number z = x + yi, we define the complex conjugate of z, written as z, as the complex number z = x - yi. In the complex plane, the point corresponding to z is (x, -y). Geometrically, this is the reflection of the point (x, y) corresponding to z about the x-axis, as shown in Figure 2.1.2.
Figure 2.1.2: Complex Conjugation
The other useful operation on complex numbers is the modulus, also known as the complex norm or absolute value. For a given complex number z = x + yi, its modulus, deonted |z|, is given by the formula
- (2.1.3) |z| = √x2 + y2.
Note that by the distance formula (1.3.1), this is equal to the distance d(O,P) from the origin O = (0, 0) to the point P = (x, y). Thus, the norm of a complex number is the length of the line segment going from the origin to the point in the complex plane corresponding to this number.