I just finished reading the "Reality Bits" in a recent copy of NewScientist. It discusses attempts to purge mathematics of the need for complex numbers. Started me thinking(danger, danger) of not how to get rid of the square root of negative one, but, more easily, simply find out where it enters the mathematical picture. The Pythagoreans were probably the first who wondered about this; ##a^2 + b^2 = c^2## We sure know how to represent the square root of ##c^2##, it's ##c##. But can we write an expression for the square root of say, ##1 + x^2##? Try as we must, it doesn't seem to exist. But wait, if we supplement our real number system with an imaginary number system, it is at least factorable into, ##(1 + xi)(1 - xi) = 1 + x^2## That's where it enters the mathematical picture; by the simple act of trying to factor a sum. Anybody wish to add to this? Or even to point out other places it enters the mathematical picture, other than the factoring of sums? Have any ideas of your own, on how to "purge" complex numbers from your physics homework? I'm adding the words UBIT and U-BIT because I cant figure out how edit my tags.