During their math education most people are told that negative
numbers don't have square roots. Like a lot of other things you are
told firmly in school, this isn't true but it also isn't completely
false. A better way to explain what's going on is to say that the
square root of a negative number is a type of number that is not a
distance nor the negative of a distance. So why bother? Well, it
turns out that these strange numbers actually describe things that
exist. Many electronic circuits and quite a few biological population
models end up involving the square roots of negative numbers. Once
you know about complex numbers it is also possible to derive a lot of
trigonometric identites rather than memorize them.
The (perhaps unfortunate) name for square roots of negative
numbers is imaginary numbers. The square root of 1 is called
i and all other imaginary numbers are multiples of i.
The square roots of 4, for example, are +/2i. Notice that
i is also a square root of minus one. The "normal"
numbers, the ones that are either distances or the negative of
distances, are called real numbers. Complex numbers
have real and imaginary parts. So 3+2i is a complex number
with real part 3 and imaginary part 2. We sometimes write
re(z) for the real part of an imaginary number z and
im(z) for the imaginary part.
As in the example 3+2i, a complex number has a real and an
imaginary part which are written added together. The complex numbers
have arithmetic just like the real numbers using the fact that
i^{2} = 1. See if you can use that fact to prove the
following complex arithmetic rules for addition, subtraction,
multiplication, and division.
The rules for addition and multiplication just use the associative
law. The rule for multiplication requires that you FOIL out the two
terms and then use i^{2} = 1. The rule for division
requires that you first rationalize the denominator (i is a
square root, do it the usual way) and then apply the rule for
multiplication to the top of the fraction.
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