Complex numbers really sound very...well complex to the casual observer, but really, the situation is within the grasp of anyone who has completed a year of Algebra.

The Form of Complex Numbers

First, complex numbers can all be written in the form a+bi (commonly just represented in the form (a,b) for graphing purposes), where a and b are real numbers, and i is of course the imaginary part. i is representative of sqrt(-1), much in the same way that pi is representative of 3.1415926.... Both are set, and used often in specific types of problems.

Whats so great about i?

The first thing to realize is that i is not a real number. Sqrt(-1) cannot be represented on the number line, simply because the square root of a negative number cannot be done. No number can be squared to equal a negative number. This is why i is such a useful number. Any negative number can be expressed as (-1*x)=-x and so any negative square root can be expressed as sqrt(-x)=sqrt(-1*x)=i*sqrt(x). In this way, sqrt(-64)=sqrt(-1*64)=i*sqrt(64)=i*8=8i.

Operations With Complex Numbers

Complex numbers can be multiplied, divided, added, subtracted, squared, etc. just like regular numbers. When 1+2i, or (1,2), is added to (3,5), we add the real part (a) to the real part (a), and the imaginary part (b) to the imaginary part(b). This means we add 1 and 3 to get the new real part, and we add 2 and 5 to get the new imaginary part, resulting in the sum (4,7) which is 4+7i.

When multiplying complex numbers, it can get a little tricky. For anyone who has taken Algebra 1, use the FOIL method. FOIL stands for First, Outer, Inner, Last, meaning that when you multiply (1+2i)*(3+5i), you multiply in that order. First you multiply the First numbers in each set, 1 and 3, then you multiply the Outer numbers, 1 and 5i, then the Inner numbers, 2i and 3, followed by the Last numbers, 2i and 5i. Following this, just add them all up to equal your product. So, 1*3=3, 1*5i=5i, 2i*3=6i, and 2i*5i=10i²=10*(sqrt(-1)*sqrt(-1))=10*-1=-10 (remember that i² is always -1). 3+5i+6i-10=-7+11i=(-7,11)

Graphing Complex Numbers

Graphing complex numbers is a fairly simple business. Most poeple know about the Cartesian Coordinate System and how points can be graphed using x and y, but this does not accomodate complex numbers due to the fact that they have no real numerical value. Thus, the complex plane is born. It works in the same way as the Cartesian plane, but x and y are replaced with the axis a and b. So, a number 1+4i, written as (1,4) is graphed at the point (1,4) on the complex plane. This makes it visible to those of us looking at a graph. You could think of it as using the same plane as normal, but x=a and y=b. Not too difficult, eh?

When graphed, a point's magnitude is its distance from zero. On the complex plane, this would be the distance from the origin (0,0). Using the Pythagarean Therum, thats sqrt(a²+b²). This conveniently means that the magnitude of a complex number is represented as a rational number.

Expanding To Other Fields

Imaginary numbers are used every day by people in the fields of electricity and they are a key element in fractals like the Mandelbrot Set. And of course, its neat to see how they behave. They are so similar to the real set of numbers, yet so different.