Edit made on October 22, 2008 by GuestEditor at 23:27:33
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No ordinary, "real" number can give a negative result when squared.
Hence the equation EQN:x^2+4=0 has no solution.
This doesn't seem like such a problem, but the development of the
general solution to the cubic equation uses quantities of this
type, which then subsequently all cancel out leaving just real
solutions. Somehow these "imaginary" quantities seem to be useful.
Can they be added, subtracted, multiplied and divided like normal
numbers? Certainly they can be added and subtracted, but multiplying
two imaginary numbers gives a real number, so things only really work
when imaginary numbers are combined with the "normal" numbers. This
combination gives what are called the complex numbers.
The imaginary number EQN:i has the property EQN:i^2=-1