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

$$x^2+R=0$$

for the variable $x$, we would naturally write

$$x=\pm\sqrt{-R},$$

but how exactly do we compute the square root of a negative number?  As long as we use ordinary 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

$$\sqrt{-R}=\sqrt{-1}\sqrt{R}$$

and then define

$$i^2\equiv -1$$

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

$$x=\pm i\sqrt R.$$

For example, solving

$$0=4+x^2$$

for $x$, we get

$$x=\pm 2i$$

Another problem arises if we try to solve the quadratic

$$ax^2+bx+c=0$$

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

$$\displaystyle x = \frac{-b\pm\sqrt{b^2 - 4ac}}{2a}.$$

If $b^2 \geq 4ac$, then $x$ is real, however, if  $b^2\leq 4ac$, then $x$ is a combination of real and imaginary numbers:

$$x = \alpha \pm i \beta$$

This combination, a real number plus an imaginary number, is called a complex number.  Complex numbers are normally written as

$$c=a+ ib$$

Notice that, since $i$ is defined as

$$i^2\equiv -1\;\;\Rightarrow\;\;i=\pm \sqrt{-1},$$

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

$$c=a+ib$$

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

$$c^{*}=a-ib$$

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

$$\displaystyle \Re\mbox{e}\left\{ c \right\} = a$$

and

$$\displaystyle \Im\mbox{m}\left\{ c \right\} = b$$

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

$$\displaystyle \Re\mbox{e}\left\{ c \right\} = \frac{c+c^{*}}{2}$$

and

$$\displaystyle \Im\mbox{m}\left\{ c \right\} = \frac{c-c^{*}}{2i}$$

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

$$|c|=\sqrt{c c^{*}}=\sqrt{a^2+b^2}$$

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:

$$|c|^2=c\cdot c^{*} = a^2 + b^2$$

whereas

$$c^2=c\cdot c = \left( a+ib\right)\left( a+ib\right) = a^2+2iab-b^2$$

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 $z$,

$$z=x+iy$$

and the standard notation for a complex function is

$$f(z)=u(x,y)+iv(x,y)$$

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

Next: Euler's formula