Re: complex division
- From: Virgil <virgil@xxxxxxxxxxx>
- Date: Thu, 10 May 2007 22:59:42 -0600
In article <1178856533.097725.154000@xxxxxxxxxxxxxxxxxxxxxxxxxxx>,
vsgdp <cloud00769@xxxxxxxxx> wrote:
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?
In any definition of division in which x/x is defined at all, one can be
reasonably sure that x/x will equal 1, because 1*x = x.
But that is not the reason one "rationalizes" the denominator.
(a + b*i)(c - d*i) = a*c + b*c*i - a*d*i - i*b*d
= (a*c + b*d) + (-a*d - b*c)*i
(c + d*i)(c - d*i) = c^2 + c*d*i - c*d*i - d^2*i^2
= c^2 + d^2, which is real.
So (a + b*i)/(c + d*i)
= (a*c + b*d)/(c^2 + d^2) + [(-a*d - b*c)/(c^2 + d^2)]*i
So this result is again in standard complex form of real part plus or
minus imaginary part.
The point is to get the result into this standard form.
.
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