How do I find the quotient of two complex numbers in standard form?

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Answer 1 · 2018-06-26T19:35:01

Let $z_1 = a_1+b_1i$ and $z_2=a_2+b_2i$ . We want to find

$q=z_1/z_2=(a_1+b_1i)/(a_2+b_2i)$

Generally, we wish to write this in the form

$q=A+Bi$

Where $A$ and $B$ are real numbers. To do this, we must amplify the quotient by the conjugate of the denominator:

$q=z_1/z_2 * bar(z_2)/(bar(z_2))=(a_1+b_1i)/(a_2+b_2i)*(a_2-b_2i)/(a_2-b_2i)=((a_1a_2+b_1b_2)+(b_1a_2-b_2a_1)i)/(a_2^2+b_2^2)$

$q = (a_1a_2+b_1b_2)/(a_2^2+b_2^2) + (b_1a_2-b_2a_1)/(a_2^2+b_2^2) i$