Imaginary and Complex Numbers
12.1 Logic
 
12.2 Sequences
 
12.3 Limits
 
12.4 Imaginary and Complex Numbers
 
 
12.5 Key Formulas
 
12.6 Review Questions
 
12.7 Explanations
 
Imaginary and Complex Numbers
Most of the Math IC test deals with real numbers. But every so often, a question will appear that does not involve 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 numbers 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 a + bi, where a and b are real numbers, and i = .
There are two 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
The Powers of i
The powers of i are easy to work with. For example:
Evaluate i2 i9.
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 i2i9 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).
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