Editing DividingComplexNumbers
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Given two complex numbers, $a+bi$ and $c+di,$ we can ask about the quotient: * $\frac{a+bi}{c+di}$ There are two ways to proceed. One is to remember that when you multiply a complex number by its complex conjugate the result is a real number. So we get this: |>> $\frac{a+bi}{c+di}~=~\frac{a+bi}{c+di}\times\frac{c-di}{c-di}$ <<| The right-hand side multiples through, using the rules for multiplying rational numbers, to: |>> $\frac{a+bi}{c+di}\times\frac{c-di}{c-di}~=~\frac{(a+bi)(c-di)}{(c+di)(c-di)}$ <<| Using the rules of multiplying complex numbers we get: |>> $\frac{(a+bi)(c-di)}{(c+di)(c-di)}~=~\frac{(ac+bd)+(bc-ad)i}{c^2+d^2}$ <<| Alternatively, we can express the numbers using the polar representation of a complex number: |>> $z_1=re^{i\theta},\quad\quad~z_2=se^{i\phi}$ <<| Then the quotient is: |>> $\frac{z_1}{z_2}~=~\frac{r}{s}e^{i(\theta-\phi)}$ <<| This uses the index laws that $\frac{x^a}{x^b}~=~x^{a-b}.$