Complex Numbers

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

for the variable $x$, 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 $x$ 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 $i$ 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 $x$, we get

Another problem arises if we try to solve the quadratic

where $a$, $b$, and $c$ are arbitrary real numbers.  To solve for $x$, we use the quadratic formula,

If $b^2 \geq 4ac$, then $x$ is real, however, if  $b^2\lt 4ac$, then $x$ 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 $i$ is defined as

complex numbers always come in pairs. These are known as conjugate pairs.  Suppose

Then the conjugate of $c$, which we denote as $c^{*}$, is

The quantity $a$ is called the real part of $c$ and the quantity $b$ is called the imaginary part of $c$.  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 $|c|^2=a^2+b^2.$  This is not a coincidence.  We'll examine this more when we discuss Euler's formula. Notice that $c^2$ and $|c|^2$ 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 $z$,

and the standard notation for a complex function is

where $x$, $y$, $u$, and $v$ are real.

Next: Euler's formula