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