Re: complex division

In article <1178856533.097725.154000@xxxxxxxxxxxxxxxxxxxxxxxxxxx> vsgdp <cloud00769@xxxxxxxxx> writes:
My book says to compute the quotient of two complex numbers, we
rationalize the denominator.

(a+bi)/(c+di) = (a+bi)(c-di) / [(c+di)(c-di)]

However, this bothers me because how do we even know (c-di)/(c-di) = 1
when we haven't even defined division yet. Now I admit, they don't
take this process as a definition, but sort of use it to justify the
way division is defined.

Thoughts?


As you say, it it done to justify the definition of division.
If x, y, z are real numbers, then we have properties such as

x/y = (x*z)/(y*z) (y, z nonzero)

We would like to preserve this property for complex division,
as we attempt to get the complex numbers to obey
familiar rules of algebra. Apply it with x = a + bi, y = c + di,
z = c - di. If we have already decided how to define complex
multiplication, then we get ((a*c + b*d) + (b*c - a*d)*i) / (c^2 + d^2).
Other familar algebra rules suggest this should be the same as

(a*c + b*d)/(c^2 + d^2) + ((b*c - a*d) / (c^2 + d^2)) * i.

This motivates the definition of complex division.
Now we check that all of the familiar rules are satisfied,
as we desired. It turns out that most laws remain valid
but some like sqrt(z1 * z2) <> sqrt(z1) * sqrt(z2) fail,
even though this holds for real numbers with real square roots.
We also have to give up the notions of positive and negative for
complex numbers.
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