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 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.

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:

The powers of i are easy to work with. For example:

The trick is to remember that the powers of i work in cycles of four:

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.

Algebraic manipulation of complex numbers is exactly like dealing with real numbers. See for yourself: