What is really the purpose of $i$? We started talking about imaginary numbers again this year and asked this question in class, but nobody could really give a straight answer. So if anyone could tell me the real reason we have imaginary numbers that would be great! :)

The real purpose is be able to carry out mathematical operations in an unrestricted manner. 
Why have negative numbers ?
So you can subtract: $2-3=-1$
Why have fractional  numbers ?
So you can divide: $2\div 3=\frac{2}{2}$
Why have imaginary numbers ?
So you can take square roots: $\sqrt{-1}=i$.
It turns out to be a minor miracle that with imaginary numbers ALL algebraic equations, eg, $x^7+4x^6+3x^5+76x^4+14x^3+12x^2+89x+100=0$ have (possibly imaginary) solutions.
The the range of applicability of the operations is greatly extended.
