Division of Complex Numbers
Key Questions

Answer:
By definition if
#z=a+bi# is a complex number, then his conjugate is#barz=abi# Explanation:
A conjugate of a complex number is the other complex number with the same real part and opposite imaginary part

You can do it by multiplying the numerator and the denominator by the complex conjugate of the denominator.
#1/{a+bi}=1/{a+bi}cdot{abi}/{abi}={abi}/{a^2+b^2}#
I hope that this was helpful.

If complex numbers
#z_1# and#z_2# in polar form are#{(z_1=r_1(costheta_1+isin theta_1)),(z_2=r_2(cos theta_2+i sin theta_2)):}# ,then we can write in exponential form
#{(z_1=r_1e^{i theta_1}),(z_2=r_2 e^{i theta_2}):}# .So, the quotient
#z_1/z_2# can be written as#z_1/z_2={r_1e^{i theta_1}}/{r_2e^{i theta_2}}=r_1/r_2e^{i(theta_1theta_2)}# #=r_1/r_2[cos(theta_1theta_2)+isin(theta_1theta_2)]#
I hope that this was helpful.

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_2b_2i)/(a_2b_2i)=((a_1a_2+b_1b_2)+(b_1a_2b_2a_1)i)/(a_2^2+b_2^2)# #q = (a_1a_2+b_1b_2)/(a_2^2+b_2^2) + (b_1a_2b_2a_1)/(a_2^2+b_2^2) i#