Edit made on February 17, 2013 by ColinWright at 16:46:11
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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 ~really work
when imaginary numbers are combined with the "normal"
numbers. This numbers, the
so-called "real numbers.".
This combination of real number with imaginary number gives what are
called the complex numbers.
The imaginary number EQN:i has the property EQN:i^2=-1