A complex number can be thought of as an ordered pair: (real,imag). We could develop and memorize the various arithmetic operations that combine these in terms of ordered pair notation. As an example:

However, it is easier to introduce and use:

Therefore, a complex number can be written as:

This as an ordinary algebraic binomial expression with the i as an ordinary algebraic variable that can be combined using the rules of binomial algebra. The only difference is when we are done we can evaluate and substitute all powers of i as:

and then separate the result into the real part (that does not contain a factor of i) and the imaginary part (that does contain a factor of i). Note that we can apply this equality repeatedly if necessary to get rid of all powers of i higher than 1.

Addition, subtraction, and multiplication of two complex numbers therefore fits rules we have already learned. Division introduces an an additional wrinkle. Note that we always want to write the result as:

Before we can perform division, we must introduce a new concept: complex conjugate!

The complex conjugate of a complex number is formed by changing the sign of the imaginary part:

Then, the complex conjugate is:

We indicate a complex conjugate with a "bar" over the complex value.

Therefore if we have:

where α = a+bi and β = c+di, we have a slightly different scheme: we divide by multiplying the numerator and the denominator by the complex conjugate of the denominator, or

α	=	a+bi	=	(a+bi)(c−di)	=	(ac+bd) + (bc−ad)i	=	ac+bd	+	bc−ad	i
β		c+di		(c+di)(c−di)		c2 + d2		c2 + d2		c2 + d2

The result can be written a proper complex number with a real and an imaginary part.

Symbolically

α	=		αβ
β			ββ

and when we are finished "rationalizing" the denominator, we separate the result into explicit real and imaginary parts.

The reason this works is a property of the complex conjugate:

which is a purely real number.

There are other properties involving a complex number and its complex conjugate. As an example:

which is also a purely real number.

On the other hand:

which is also a purely imaginary number.

We also have:

_

α = α

That is, the complex conjugate of a complex conjugate is just the original value.

Perhaps the most useful properties is that any complex zeros of a polynomial with real coefficients always occur in complex conjugate pairs. Therefore, as an example, if 2−3i is a zero, then immediately 2+3i is known to be another zero!