COMPLEX NUMBERS

 

What on earth is a complex number you ask?  In order to understand the Mandelbrot Set, you need to know only a few basic concepts about complex numbers.  A complex number is made up of two parts: a real part and an imaginary part.  The real part is a real number and the imaginary part is a real number multiplied by i.  The letter i is the imaginary unit and is defined as the square root of -1, so we have the following equality: i2 = -1.  So when you square an imaginary number you get a negative number.  For example, (3*i)2 = -9.


Example: 5 + 10*i is a complex number.

We can associate the real part and the imaginary part of the complex plane by taking a Cartesian plane and taking the x-axis to be the real axis and y-axis to be the imaginary axis of the complex plane.  We note that a complex number has two parts and therefore we need two dimensions in order to graph them. Therefore we have a general point: C = x + i*y.  The picture below shows the correspondence:

 

Complex numbers have the operations that real numbers have, such as addition, subtraction, and multiplication.  Let Z1 = x1 + i*y1 and  Z2 = x2 + i*y2, so we have:

Addition: Z1 + Z2 = (x1 + x2) + i*(y1 + y2)
Subtraction: Z1 - Z2 = (x1 - x 2) + i*(y1 - y2)
Multiplication: Z1*Z2 = (x1*x2 - y1*y2) + i*(x1*y2 + y1*x2 )

Complex numbers have many applications in the physical sciences such as fluid flow, electrostatics, and especially in electricity and magnetism.  Click here to see some interesting applications of fractals.