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

1 Answer
Jun 26, 2018

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#