It might be enlightening/helpful to first define complex numbers. It then becomes easy to define i and see why it behaves the way it does.
You are probably already familiar with real numbers and their operators +, -, \*, and /. Most students are taught real numbers and their operators fairly early in their education.
Now let's define complex numbers and their operators:
* Define a complex number to be the following: (a,b) where a and b are real numbers. So, by our definition, the following are all examples of complex numbers: (1,2), (-100,0), (0,pi), (1/2,0.001). Often a complex number is plotted by plotting the first number a on a horizontal axis and the second number b on a vertical axis.
* Define (a,0) to be equal to the real number a.
* Define the addition of two complex numbers to be the following: (a,b) + (c,d) = (a+c,b+d).
* Define the multiplication of two complex numbers to be the following: (a,b) \* (c,d) = (a\*c-b\*d, a\*d+b\*c).
* Define the symbol i to be equal to the complex number (0,1). So, for example, if b is a real number, we can do the following: i\*b = (0,1)\*(b,0) = (0,b). The quantity, i\*b, is called an imaginary number. Also, people often write the complex number (a,b) as a+i\*b. This is acceptable because a+i\*b = (a,0)+(0,b) = (a,b).
* Finally, define e^i*b to equal cos(b) + i\*sin(b). This may seem like a strange definition for e^i*b. If anybody is interested, I can write a few paragraphs explaining *why* this is a good definition.
Using the above definitions, you can now verify that the symbol i agrees with everything you have been taught about imaginary numbers.
For example, i^2 = i\*i = (0,1) \* (0,1) = (-1,0) = -1.
For another example, e^i*pi = cos(pi) + i\*sin(pi) = -1 + i\*0 = -1 + 0 = -1.
As a final example, i^i = ( e^i\*pi/2 )^i = e^i\*i\*pi/2 = e^-pi/2 = ~0.2079.
If you come across anything in complex mathematics that seems mysterious or strange, you can trace it back to these definitions and the mysteriousness will go away.
