Introduction to Complex Math
It turns out the complex form of the equations makes things a lot simpler and more elegant. As such,
everyone uses complex numbers, from physicists, to engineers, and mathematicians. So get used to it,
it is actually a very beautiful thing.
On this page we'll start by introducing complex numbers and some simple properties, useful in the
study of the Fourier Transform.
A complex number z can be written in standard form as:
The complex number z has a real part given by x and an imaginary part
given by y. The real part of z is written as:
The imaginary part of z is written as:
In equations [1,2,3], i is given as:
Example. Z = 4 + i5 ==> Then Re[Z]=4, Im[Z]=5.
Addition and subtraction are straightforward. The addition of two complex numbers (z1 and z2) are the sums
of their real and imaginary parts:
Subtraction can be performed in a similar manner to equation [5]. Multiplication of complex numbers follows
algebra-style rules:
The magnitude of a complex number z is given by:
Equation [6] can be derived by expanding the left side in a Taylor series (with variable theta). Then
expand the right side using the Taylor series expansions for cosine and sine and the results are identical.
Fourier Transforms are performed using complex numbers. Since Fourier Transforms are used to analyze
real-world signals, why is it useful to have complex (or imaginary) numbers involved at all? 
[1] [2]
[3]
[4]
Addition and Multiplication
[5]
[6]
Complex Conjugate and Magnitude
The complex conjugate of a complex number can be found by replacing the i in equation [1]
with -i. The complex conjugate operator is written as a *, as shown in equation [7]: [7]
[8]
Polar Form
The above notation for complex numbers, using real and imaginary parts, is often called standard or rectangular
form. Another common way to express complex numbers is polar form. Before, this is presented,
please recall Euler's identity (which is one of the greatest equations of all time and should be memorized): [6]
Next: The Gaussian