Imaginary and Complex Numbers
12.1 Logic
12.2 Sequences
12.3 Limits
12.4 Imaginary and Complex Numbers
12.5 Key Terms
12.6 Review Questions
12.7 Explanations
Imaginary and Complex Numbers
Most of the Math IIC test deals with real numbers. But every so often, a question will appear that takes place outside the set of real numbers. These questions deal with imaginary and complex numbers.
Imaginary Numbers
Imaginary numbers are used to represent the even roots of negative numbers. They use the quantity i, where i = . For example:
Square roots of negative numbers are called imaginary because they do not lie on the real number line.
Complex Numbers
A complex number is the sum of a real number and an imaginary number. A complex number is written in the form of a + bi, where a and b are real numbers, and i = .
There are three things you need to be able to do with complex numbers:
  1. Know the powers of i
  2. Know how to do operations like addition, subtraction, and multiplication on complex numbers
  3. Know how to graph a complex number in the complex plane and how to find its absolute value
The Powers of i
The powers of i are easy to work with. For example:
Evaluate i2i9.
The trick is to remember that the powers of i work in cycles of four:
  • i1 = i
  • i2 = = ()2 = –1
  • i3 = = ()2 = –i
  • i4 = = ()4 = 1
This way, the expression i2 i9 becomes (–1)(i) = –i. If you know these cycles, you can reduce any exponent of i to a much more manageable size. The expression i2 i9 becomes (–1)(i) = –i.
Operations on Complex Numbers
Algebraic manipulation of complex numbers is exactly like dealing with real numbers. See for yourself:
Simplify the expression (3x + i)(x – 2i).
The Complex Plane
The complex plane is a plane similar to the coordinate plane, except that instead of having an x-axis and a y-axis, it has axes that allow you to plot complex numbers as well as real numbers.
The horizontal axis of a complex plane is called the real axis, and the vertical axis is called the imaginary axis. The complex number a + bi is plotted in the complex plane exactly where the point (a, b) would be plotted in the standard coordinate plane. Take a look at the picture of the complex plane below, with various complex numbers plotted on it.
The magnitude, or absolute value, of a complex number is the distance from the origin to that number in the complex plane. You can use the Pythagorean theorem to calculate the magnitude of the complex number.
The magnitude of a + bi, which can also be written as |a + bi|, is equal to Try the following example:
What is the magnitude of the complex number 4 – 9i?
The complex number 4 – 9i would be graphed on the plane as the point (4, –9i). Using the Pythagorean theorem to calculate its magnitude: =
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