A * complex number* is a number of the form
, where a and b are real numbers and (so ). For example, here are some complex numbers:

Notice that real numbers are special kinds of complex numbers --- namely, those that don't have an i-term.

Complex numbers are often called * imaginary
numbers*, though there is nothing "imaginary" about
them. It's unfortunate terminology, but it's very common.

For instance, in a complex number , a is called the * real part* and b is called the *
imaginary part*. Thus, in , the real part is 4 and the
imaginary part is 7.

* Example.* Express in the form
, where a and b are real numbers.

* Example.* Express in the form
, where a and b are real numbers.

* Example.* Express in the form
, where a and b are real numbers.

* Example.* (a) Express
in the form , where a and b are real numbers.

Since , I'm going to find the largest even number less than 37 and use that as the basis for breaking up the power.

(b) Express in the form , where a and b are real numbers.

(c) Express in the form , where a and b are real numbers.

* Example.* You can add or subtract complex
numbers by adding or subtracting the real parts and the imaginary
parts:

* Example.* Complex numbers can be represented
by points in the plane. Use the real part for the x-direction and the
imaginary part for the y-direction. Here are some examples:

* Example.* Multiply complex numbers by using
the distributive law:

Standard rules for exponents apply. For example,

I used the fact that -1 raised to an even power is 1 and -1 raised to an odd power is -1.

To divide one complex number by another, or to compute reciprocals of
complex numbers, use the technique of * multiplying
the top and bottom by the conjugate*.

What's the * conjugate* of a complex number? The
conjugate of is ; the conjugate of is . In others words, find the conjugate by
flipping the sign of the imaginary part.

* Example.* Express in the
form .

Multiply the top and bottom by :

* Example.* Express in
the form .

Multiply the top and bottom by :

Copyright 2008 by Bruce Ikenaga