# Complex Numbers

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 :

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