Say we’re looking for u= √z, i.e., we want a number uthat, multiplied by itself, equals z. Multiplication multiplies the magnitudes, so the
magnitude of ucan be found by taking the square root of the magnitude of z. Since multiplication also adds the arguments of the numbers, squaring a number doubles its
argument. Therefore we can simply divide the argument of zby two to find the argument of u. This results in one of the square roots of z. There is another one, which
is −u, since 
is the same as 
. This may seem a little odd:
if u was chosen so that doubling its argument gave the argument of z, then how can the same be true for −u? Well for example, suppose the argument of zis 4 ◦. Then arg u= 2 ◦, and arg(−u) = 182 ◦. Doubling 182 gives 364, which is actually a synonym for 4 degrees.
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