Question

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

Answer 1

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`