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 13. Complex Numbers

In the discussion on the quadratic formula, we saw how there is no real solution to problems when the discriminant  involves the square root of a negative number.  A solution is, however, possible with what is known as imaginary numbers. The imaginary number, i, is defined as the square root of negative 1.

                                                                                   (13-1)

The imaginary number is a difficult concept. But remember that there is a day in every student's life when they're introduced to the concept of negative numbers. At first, negative numbers seem like a very abstract idea. There was even a time when negative numbers were completely unknown.  But now bankers, accountants, and even football players could not do their work without negative numbers.  In the same way, some engineers and scientists simply could not do their work without imaginary numbers. Now that we're familiar with vibrations, AC circuits, and vectors, we can get a basic understanding of how i is used in science and engineering.

While the imaginary number is very difficult to understand, it is rather simple to use. The following equations for i are derived from Equation 13-1.

                                                                                      (13-2)

                                                                              (13-3)

                                                                                    (13-4)

The set of imaginary numbers is the product of i times the real numbers.  For example, -3i, 4i, 5.7i, and  are imaginary numbers.  When we add a real number and an imaginary number, we get a complex number such as 5 + 8i.  Complex numbers are added like other polynomials.  For example:

Complex numbers are also multiplied much like polynomials using the FOIL process. When you multiply an imaginary number and a real number, you get another imaginary number.  When two imaginary numbers are multiplied together, they become a real number.  For example:

When two complex numbers can be multiplied and eliminate the imaginary numbers, they're said to be conjugate numbers.  When you multiply two conjugate numbers, you get a real number. For example:

In general, the conjugate numbers have the form a + bi and a - bi.  When you get imaginary numbers from the quadratic formula, the two roots will be conjugates. The answers in engineering problems that involve imaginary numbers usually come in conjugates.

13.1 The Complex Plane

We know approximately where irrational numbers like p and  are on the number line.  But where is  on the number line?  The answer is that it isn't on the number line of real numbers but rather “off to the side”. In math we use a complex plane with real numbers on a horizontal axis and imaginary numbers on a vertical axis.

It is sometimes helpful to look at i as an operator like the minus sign or the square root sign. To see how, let’s look at what the minus sign means graphically. One could say that when you add a minus sign to a number (or multiply it by –1) you rotate its value 180˚ on the number line.  For example, if you multiply 3 by –1, you get –3 which is 180˚ from 3 on the number line.

When you multiply a number by , you rotate the value half of 180˚ or 90˚. If you multiply 2 times i, you get 2i. As you can see in Fig 13.3, this is like rotating the value of 2 by 90˚. 

This also works for complex numbers.  If you multiply the number –3 + 1i by i, you get the following result:

 As you can see from Figure 13.3, these numbers are also 90˚ apart. 

Generally, you could say that when you multiply a number by –1^x, you rotate it xp radians.  From this we can start to see how i is related to trigonometry and exponents.

Imaginary numbers occur in several areas of science and engineering. Many mathematicians regret that they were given the name “imaginary”.  Since imaginary numbers are “off to the side”, a better term would have been lateral numbers.