Add, subtract, multiply, and divide complex numbers.

Simplify powers of .

Imaginary Numbers

We've been solving lots of equations, but not every equation involving real numbers is solvable with real numbers. For example, there is no real number, , with . To handle these kinds of equations, we must extend the real number system.

An imaginary number is a "number" of the form , where is a positive real number. The imaginary unit is .

has the property that . No real number has this property!

Every imaginary number can be written in terms of by using .

Examples

Write in terms of and simplify:

Write in terms of and simplify:

Write in terms of and simplify:

Complex Numbers

A complex number is an expression of the form , where and are real numbers.

In the complex number , is called the real part and is called the imaginary part. The and terms cannot be combined---they are not like terms. But every complex number can be simplified to the standard form.

Examples

Write as a complex number in standard form:

Write as a complex number in standard form:

Write as a complex number in standard form:

Write as a complex number in standard form:

Write as a complex number in standard form.

Conjugates and Dividing

The complex conjugate of the complex number is the complex number . Notice that in forming the complex conjugate, we negate the imaginary part, but not the real part.

An important property of conjugates is the following:

Example

Simplify:

In order to divide by a complex number, we really just use the conjugate (and multiplication) to make the denominator real: