# How do I graphically divide complex numbers?

Mar 25, 2016

Please follow the following process for multiplication as well as division

#### Explanation:

Let us write the two complex numbers in polar coordinates and let them be

${z}_{1} = {r}_{1} \left(\cos \alpha + i \sin \alpha\right)$ and ${z}_{2} = {r}_{2} \left(\cos \beta + i \sin \beta\right)$

Their multiplication leads us to

${r}_{1} \cdot {r}_{2} \left\{\left(\cos \alpha \cos \beta - \sin \alpha \sin \beta\right) + \left(\sin \alpha \cos \beta + \cos \alpha \sin \beta\right)\right\}$ or

r_1*r_2{(cos(alpha+beta)+sin(alpha+beta))

Hence, multiplication of ${z}_{1}$ and ${z}_{2}$, will be given by

$\left({r}_{1} \cdot {r}_{2} , \left(\alpha + \beta\right)\right)$, so for multiplying complex numbers, take new angle as $\left(\alpha + \beta\right)$ and modulus the product of the modulus of two numbers.

Similarly $\frac{{r}_{1} \left(\cos \alpha + i \sin \alpha\right)}{{r}_{2} \left(\cos \beta + i \sin \beta\right)}$

To simplify let us multiply numerator and denominator by denominator's conjugate $\left({r}_{2} \left(\cos \beta - i \sin \beta\right)\right)$, then ${z}_{1} / {z}_{2}$ is given by

$\frac{{r}_{1} \left(\cos \alpha + i \sin \alpha\right) \cdot \left({r}_{2} \left(\cos \beta - i \sin \beta\right)\right)}{{r}_{2} \left(\cos \beta + i \sin \beta\right) \left({r}_{2} \left(\cos \beta - i \sin \beta\right)\right)}$ which when simplified becomes

$\frac{{r}_{1} \cdot {r}_{2} \left(\cos \alpha \cos \beta + \sin \alpha \sin \beta\right) + i \left(\sin \alpha \cos \beta - \cos \alpha \sin \beta\right)}{{r}_{2}^{2} \left({\cos}^{2} \beta + {\sin}^{2} \beta\right)}$ or

(r_1/r_2)*(cos(alpha-beta)+isin(alpha-beta) or

${z}_{1} / {z}_{2}$ is given by $\left({r}_{1} / {r}_{2} , \left(\alpha - \beta\right)\right)$, so for division complex number ${z}_{1}$ by ${z}_{2}$ , take new angle as $\left(\alpha - \beta\right)$ and modulus the ratio ${r}_{1} / {r}_{2}$ of the modulus of two numbers.