## Complex Numbers

Assume that represents a positive, real number. In order to solve equations of the form

for the variable , we would naturally write

but how exactly do we compute the square root of a negative number? As long as we use real numbers, there is no way to write the value of in this case. In order to remedy this, mathematicians invented (or discovered) a type of number called "imaginary numbers." The idea is simple; we can write the square root as

and then define

where is the imaginary unit, analogous to the number 1 in the real number system. This allows us to write the solution to the equation above as

For example, solving

for , we get

Another problem arises if we try to solve the quadratic

where , , and are arbitrary real numbers. To solve for , we use the quadratic formula,

If , then is real, however, if , then is a combination of real and imaginary numbers:

This combination, a real number plus an imaginary number, is called a

*complex number*. Complex numbers are normally written as

Notice that, since is defined as

complex numbers always come in pairs. These are known as

*conjugate*pairs. Suppose

Then the conjugate of , which we denote as , is

The quantity is called the

*real part of*and the quantity is called the

*imaginary part of*. These are denoted as

and

The real and imaginary parts of a complex number can be written in terms of the conjugate notation as

and

The absolute value (or magnitude) of a complex number is given by

This may remind you of the Pythagorean theorem, where This is not a coincidence. We'll examine this more when we discuss Euler's formula. Notice that and are not the same expression:

whereas

Until this point, we have limited the discussion to complex constants. Variables and functions can be complex as well. The standard notation for a complex

*variable*is ,

and the standard notation for a complex

*function*is

where , , , and are real.

Next: Euler's formula