2.5 Complex Numbers
Many people find complex numbers disturbing or unintuitive. Before delving into their mathematics, let’s consider why we might be interested in these constructs.
Complex numbers act much like a bridge between two villages that are located on opposite sides of a river. If the nearest ford is 10 miles upstream, a bridge may provide a direct path between the two villages. In traveling between the two villages, we might take the ford or the bridge. Either way, our destination is the same.
Similarly, we may have a mathematical problem that is expressed entirely with real numbers and has a solution that depends only on real numbers. However, using complex numbers to reach that solution may provide a convenient shortcut compared to techniques that only involve real numbers. We use the complex numbers to bridge the gap between the problem and its solution. Doing so does not change the solution. It merely provides a convenient means—a bridge—for obtaining the solution.
2.5.1 The Number i
The real numbers contain no solution to the equation x 2 = −1 that satisfies
As with any number, we can add, multiply, take roots and perform other operations with this new number i. Multiplying i by 5 results in the number 5i. Adding 3 to this yields 3 + 5i. Squaring this yields 9 + 30i + 25i 2.
At this point, imaginary numbers may be starting to seem like a Pandora’s box. By adding a single number i to , we have actually added many numbers, and the expressions for these numbers seem to be getting more and more complicated. What would happen now if we were to divide our number 9 + 30i + 25i 2 into 7?
In fact, such concerns are unfounded. Although the addition of i to does add many numbers to , expressions for these numbers always simplify to the form
where a and b are real. For example, using [2.52] we can simplify our number 9 + 30i + 25i 2 as follows:
which has the form [2.53].
We call the set of numbers of the form [2.53] the complex numbers and denote this set . Given a complex number z = a + bi, we call the real number a the real part of z. We call the real number b the imaginary part of z. This motivates the Re and Im functions that map a complex number z = a + bi to its real and imaginary parts a and b, respectively:
2.5.2 Complex Operations
Operations on complex numbers are extensions of the familiar operations on real numbers. Indeed, we have already performed complex addition and multiplication. We now formally define the operations of complex addition, subtraction, multiplication, division, and the taking of square roots. Let a + bi and c + di be complex numbers where a, b, c, d . Then
With the exception of 0, every number has two square roots. For example, the square roots of 4 are 2 and −2. The square roots of -1 are i and −i .
In formulas [2.57] through [2.61], we observe several things. First, the formulas reduce to the corresponding operations for real numbers if they are applied to real numbers. Also, the right side of each formula is always defined, and it corresponds to a complex number of the form [2.53] The only exception is division by zero, which is undefined with regard to real as well as complex numbers.4
Recall that we were motivated to introduce complex numbers by the equation x 2 = −1, which has no solution in . Is it possible that there is an equation x 2 = a + bi that has no solution in ? If this were the case, we might feel compelled to extend the complex numbers through the addition of still another “imaginary” number to solve this new equation. Because the right side of [2.61] is always defined, this will never happen.
We can make a more sweeping statement. Consider a polynomial equation of the form
where the wi are constants. Every such equation has exactly n solutions z , including repeated solutions.5 This is an important result. It is called the fundamental theorem of algebra.
2.5.3 Complex Functions
We extend the exponential function to with
This is the famous Euler’s formula that links the exponential function with the sine and cosine functions. We extend the sine and cosine functions to with
Express the following in a + bi form:
- (5 + 3i)(2 − i),
- 5/(2 + i),
- e i.