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

Jun 26, 2018

Let ${z}_{1} = {a}_{1} + {b}_{1} i$ and ${z}_{2} = {a}_{2} + {b}_{2} i$. We want to find

$q = {z}_{1} / {z}_{2} = \frac{{a}_{1} + {b}_{1} i}{{a}_{2} + {b}_{2} i}$

Generally, we wish to write this in the form

$q = A + B i$

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

$q = {z}_{1} / {z}_{2} \cdot \frac{\overline{{z}_{2}}}{\overline{{z}_{2}}} = \frac{{a}_{1} + {b}_{1} i}{{a}_{2} + {b}_{2} i} \cdot \frac{{a}_{2} - {b}_{2} i}{{a}_{2} - {b}_{2} i} = \frac{\left({a}_{1} {a}_{2} + {b}_{1} {b}_{2}\right) + \left({b}_{1} {a}_{2} - {b}_{2} {a}_{1}\right) i}{{a}_{2}^{2} + {b}_{2}^{2}}$

$q = \frac{{a}_{1} {a}_{2} + {b}_{1} {b}_{2}}{{a}_{2}^{2} + {b}_{2}^{2}} + \frac{{b}_{1} {a}_{2} - {b}_{2} {a}_{1}}{{a}_{2}^{2} + {b}_{2}^{2}} i$