Complex Numbers |
Contents: This page corresponds to § 2.3 (p.191) of the text.
Suggested Problems from Text:
p. 198 #7, 8, 11, 19, 22, 23, 31, 32, 33, 37, 43, 45, 53, 55, 56, 61
Why Complex Numbers
The Number i
The Complex Plane
Complex Arithmetic
Why do we need new numbers?
The hardest thing about working with complex numbers is understanding why you might want to. Before introducing complex numbers, let's backup and look at simpler examples of the need to deal with new numbers.
If you are like most people, initially number meant whole number, 0,1,2,3,... Whole numbers make sense. They provide a way to answer questions of the form "How many ... ?" You also learned about the operations of addition and subtraction, and you found that while subtraction is a perfectly good operation, some subtraction problems, like 3 - 5, don't have answers if we only work with whole numbers. Then you find that if you are willing to work with integers, ...,-2, -1, 0, 1, 2, ..., then all subtraction problems do have answers! Furthermore, by considering examples such as temperature scales, you see that negative numbers often make sense.
Now we have fixed subtraction we will deal with division. Some, in fact most, division problems do not have answers that are integers. For example, 3 ÷ 2 is not an integer. We need new numbers! Now we have rational numbers (fractions).
There is more to this story. There are problems with square roots and other operations, but we will not get into that here. The point is that you have had to expand your idea of number on several occasions, and now we are going to do that again.
The "problem" that leads to complex numbers concerns solutions of equations.
Equation 1: x2 - 1 = 0.
Equation 1 has two solutions, x = -1 and x = 1. We know that solving an equation in x is equivalent to finding the x-intercepts of a graph; and, the graph of y = x2 - 1 crosses the x-axis at (-1,0) and (1,0).
Equation 2: x2 + 1 = 0
Equation 2 has no solutions, and we can see this by looking at the graph of y = x2 + 1.
Since the graph has no x-intercepts, the equation has no solutions. When we define complex numbers, equation 2 will have two solutions.
Consider Equations 1 and 2 again.
| Equation 1 |
Equation 2 |
| x2 - 1 = 0. |
x2 + 1 = 0. |
| x2 = 1. |
x2 = -1. |
Equation 1 has solutions because the number 1 has two square roots, 1 and -1. Equation 2 has no solutions because -1 does not have a square root. In other words, there is no number such that if we multiply it by itself we get -1. If Equation 2 is to be given solutions, then we must create a square root of -1.
Definition: The imaginary unit i is defined by
The definition of i tells us that i2 = -1. We can use this fact to find other powers of i.
Example 1.
i3 = i2 * i = -1*i = -i.
i4 = i2 * i2 = (-1) * (-1) = 1.
Simplify i8 and i11. Answer.
We treat i like other numbers in that we can multiply it by numbers, we can add it to other numbers, etc. The difference is that many of these quantities cannot be simplified to a pure real number.
For example, 3i just means 3 times i, but we cannot rewrite this product in a simpler form, because it is not a real number. The quantity 5 + 3i also cannot be simplified to a real number.
However, (-i)2 can be simplified. (-i)2 = (-1*i)2 = (-1)2 * i2 = 1 * (-1) = -1.
Because i2 and (-i)2 are both equal to -1, they are both solutions for Equation 2 above.
Definition: A complex number is one of the form a + bi, where a and b are real numbers. a is called the real part of the complex number, and b is called the imaginary part.
Two complex numbers are equal if and only if their real parts are equal and their imaginary parts are equal. I.e., a+bi = c+di if and only if a = c, and b = d.
Example 2.
2 - 5i.
6 + 4i.
0 + 2i = 2i.
4 + 0i = 4.
The last example above illustrates the fact that every real number is a complex number (with imaginary part 0). Another example: the real number -3.87 is equal to the complex number -3.87 + 0i.
It is often useful to think of real numbers as points on a number line. For example, you can define the order relation c < d, where c and d are real numbers, by saying that it means c is to the left of d on the number line.
We can visualize complex numbers by associating them with points in the plane. We do this by letting the number a + bi correspond to the point (a,b).
When a number system is extended the arithmetic operations must be defined for the new numbers, and the important properties of the operations should still hold. For example, addition of whole numbers is commutative. This means that we can change the order in which two whole numbers are added and the sum is the same: 3 + 5 = 8 and 5 + 3 = 8.
We need to define the four arithmetic operations on complex numbers.
To add or subtract two complex numbers, you add or subtract the real parts and the imaginary parts.
(a + bi) + (c + di) = (a + c) + (b + d)i. (a + bi) - (c + di) = (a - c) + (b - d)i.
Example 3.
(3 - 5i) + (6 + 7i) = (3 + 6) + (-5 + 7)i = 9 + 2i.
(3 - 5i) - (6 + 7i) = (3 - 6) + (-5 - 7)i = -3 - 12i.
Note
These operations are the same as combining similar terms in expressions that have a variable. For example, if we were to simplify the expression (3 - 5x) + (6 + 7x) by combining similar terms, then the constants 3 and 6 would be combined, and the terms -5x and 7x would be combined to yield 9 + 2x.
The Complex Arithmetic applet below demonstrates complex addition in the plane. You can also select the other arithmetic operations from the pull down list. The applet displays two complex numbers U and V, and shows their sum. You can drag either U or V to see the result of adding other complex numbers. As with other graphs in these pages, dragging a point other than U or V changes the viewing rectangle.
Complex Arithmetic
The formula for multiplying two complex numbers is
(a + bi) * (c + di) = (ac - bd) + (ad + bc)i.
You do not have to memorize this formula, because you can arrive at the same result by treating the complex numbers like expressions with a variable, multiply them as usual, then simplify. The only difference is that powers of i do simplify, while powers of x do not.
Example 4.
(2 + 3i)(4 + 7i) = 2*4 + 2*7i + 4*3i + 3*7*i2 = 8 + 14i + 12i + 21*(-1) = (8 - 21) + (14 + 12)i = -13 + 26i. Notice that in the second line of the example, the i2 has been replaced by -1.
Using the formula for multiplication, we would have gone directly to the third line.
Perform the following operations.
(a) (-3 + 4i) + (2 - 5i).
(b) 3i - (2 - 4i).
(c) (2 - 7i)(3 + 4i).
(d) (1 + i)(2 - 3i).
Definition: The conjugate (or complex conjugate) of the complex number a + bi is a - bi.
Conjugates are important because of the fact that a complex number times its conjugate is real; i.e., its imaginary part is zero.
(a + bi)(a - bi) = (a2 + b2) + 0i = a2 + b2.
Example 5
Number Conjugate Product 2 + 3i 2 - 3i 4 + 9 = 13 3 - 5i 3 + 5i 9 + 25 = 34 4i -4i 16
Suppose we want to do the division problem (3 + 2i) ÷ (2 + 5i). First, we want to rewrite
this as a fractional expression 
.
Even though we have not defined division, it must satisfy the properties of ordinary division. So, a number divided by itself will be 1, where 1 is the multiplicative identity; i.e., 1 times any number is that number.
So, when we multiply by 
, we are multiplying by 1 and the number is not changed.
Notice that the quotient on the right consists of the conjugate of the denominator over itself. This choice was made so that when we multiply the two denominators, the result is a real number. Here is the complete division problem, with the result written in standard form.
Write (2 - i) ÷ (3 + 2i) in standard form. Answer
We began this section by claiming that we were defining complex numbers so that some equations would have solutions. So far we have shown only one equation that has no real solutions but two complex solutions. In the next section we will see that complex numbers provide solutions for many equations. In fact, all polynomial equations have solutions in the set of complex numbers. This is an important fact that is used in many mathematical applications. Unfortunately, most of these applications are beyond the scope of this course. See your text (p. 195) for a discussion of the use of complex numbers in fractal geometry.