## Complex Numbers

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 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
$$! \pm\sqrt{-R}=\pm\sqrt{-1}\sqrt{R}$$
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 $$b^2\lt 4ac$$, 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