Complex Arithmetic

This article introduces complex arithmetic. The word complex, in this context, means "involving the square root of minus one," rather than complicated or hard. This arithmetic is used in several other articles.

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 i2 = -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 i2 = -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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