# ACT Math – Complex Numbers

### I. The Basics

You’ve probably learned in algebra class that you can’t take the square root of a negative number because the result is not a real number. Instead, the result is what’s called an imaginary number.

The square root of negative 1 () is represented by the imaginary number i.

Here are some multiples of i that are commonly seen in problems involving complex numbers:

…and so on.

A complex number is a number that has both a real part and an imaginary part, and is often written in the form a+bi, where a and b are real numbers (for example, 3+4i and 5-6i are complex numbers).

### II. Applying Your Knowledge

Just like real numbers, complex numbers can be added, subtracted, multiplied, and divided. Here is an example of each.

Example 1 (Addition):

Combine like terms:

Example 2 (Subtraction):

Distribute the minus sign through the second term:

Combine like terms:

Example 3 (Multiplication):

Multiply the terms:

Remember

Combine like terms:

Example 4 (Division):

Multiply the numerator and denominator by the complex conjugate of the denominator: 2+3i. The complex conjugate is the same complex number but with the opposite sign in front of the imaginary number (Ex: 2-3i and 2+3i, 5+6i and 5-6i, etc.).

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Multiply the terms in the numerator and the terms in the denominator:

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Simplify and remember :

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