
## Section2.3Division and Angle Measure

The division of the complex number $z$ by $w \neq 0\text{,}$ denoted $\frac{z}{w}\text{,}$ is the complex number $u$ that satisfies the equation $z = w \cdot u\text{.}$

For instance, $\frac{1}{i} = -i$ because $1 = i \cdot (-i)\text{.}$

In practice, division of complex numbers is not a guessing game, but can be done by multiplying the top and bottom of the quotient by the conjugate of the bottom expression.

###### Example2.3.1Division in Cartesian form

We convert the following quotient to Cartesian form:

\begin{align*} \frac{2+i}{3+2i} \amp = \frac{2+i}{3+2i}\cdot\frac{3-2i}{3-2i}\\ \amp = \frac{(6+2)+(-4+3)i}{9+4}\\ \amp = \frac{8-i}{13}\\ \amp = \frac{8}{13} - \frac{1}{13}i. \end{align*}
###### Example2.3.2Division in polar form

Suppose we wish to find $z/w$ where $z = re^{i\theta}$ and $w = se^{i\beta} \neq 0\text{.}$ The reader can check that

\begin{equation*} \frac{1}{w} = \frac{1}{s}e^{-i\beta}. \end{equation*}

Then we may apply Theorem 2.2.3 to obtain the following result:

\begin{align*} \frac{z}{w} \amp = z\cdot\frac{1}{w}\\ \amp = re^{i\theta}\cdot \frac{1}{s}e^{-i\beta}\\ \amp = \frac{r}{s}e^{i(\theta-\beta)}. \end{align*}

So,

\begin{equation*} \arg\bigg(\frac{z}{w}\bigg)=\arg(z)-\arg(w) \end{equation*}

where equality is taken modulo $2\pi\text{.}$

Thus, when dividing by complex numbers, we can first convert to polar form if it is convenient. For instance,

\begin{equation*} \frac{1+i}{-3 + 3i} =\frac{\sqrt{2}e^{i\pi/4}}{\sqrt{18}e^{i3\pi/4}} = \frac{1}{3}e^{-i\pi/2} = -\frac{1}{3} i. \end{equation*}
##### Angle Measure

Given two rays $L_1$ and $L_2$ having common initial point, we let $\angle(L_1,L_2)$ denote the angle between rays $L_1$ and $L_2$, measured from $L_1$ to $L_2\text{.}$ We may rotate ray $L_1$ onto ray $L_2$ in either a counterclockwise direction or a clockwise direction. We adopt the convention that angles measured counterclockwise are positive, and angles measured clockwise are negative, and admit that angles are only well-defined up to multiples of $2\pi\text{.}$ Notice that

\begin{equation*} \angle(L_1,L_2) = - \angle(L_2,L_1). \end{equation*}

To compute $\angle(L_1,L_2)$ where $z_0$ is the common initial point of the rays, let $z_1$ be any point on $L_1\text{,}$ and $z_2$ any point on $L_2\text{.}$ Then

\begin{align*} \angle(L_1,L_2) \amp = \arg\bigg(\frac{z_2-z_0}{z_1-z_0}\bigg) \notag\\ \amp = \arg(z_2-z_0)-\arg(z_1-z_0). \end{align*}
###### Example2.3.3The angle between two rays

Suppose $L_1$ and $L_2$ are rays emanating from $2+2i\text{.}$ Ray $L_1$ proceeds along the line $y=x$ and $L_2$ proceeds along $y = 3-x/2$ as pictured.

To compute the angle $\theta$ in the diagram, we choose $z_1 = 3+3i$ and $z_2 = 4+i\text{.}$ Then

\begin{equation*} \angle(L_1,L_2) = \arg(2-i)-\arg(1+i) = -\tan^{-1}(1/2) - \pi/4 \approx -71.6^\circ. \end{equation*}

That is, the angle from $L_1$ to $L_2$ is 71.6$^\circ$ in the clockwise direction.

###### The angle determined by three points

If $u,v,$ and $w$ are three complex numbers, let $\angle uvw$ denote the angle $\theta$ from ray $\overrightarrow{vu}$ to $\overrightarrow{vw}\text{.}$ In particular,

\begin{equation*} \angle uvw = \theta = \arg\bigg(\frac{w-v}{u-v}\bigg). \end{equation*}

For instance, if $u = 1$ on the positive real axis, $v= 0$ is the origin in $\mathbb{C}\text{,}$ and $z$ is any point in $\mathbb{C}\text{,}$ then $\angle uvz = \arg(z)\text{.}$

### SubsectionExercises

###### 1

Express $\frac{1}{x+yi}$ in the form $a + bi\text{.}$

###### 2

Express these fractions in Cartesian form or polar form, whichever seems more convenient.

\begin{equation*} \frac{1}{2i},~~ \frac{1}{1+i},~~ \frac{4+i}{1-2i},~~ \frac{2}{3+i}. \end{equation*}
###### 3

Prove that $\displaystyle|z/w| = |z|/|w|\text{,}$ and that $\displaystyle\overline{z/w} = \overline{z}/\overline{w}.$

###### 4

Suppose $z = re^{i\theta}$ and $w = se^{i\alpha}$ are as shown below. Let $u = z\cdot w\text{.}$ Prove that $\Delta 01z$ and $\Delta 0wu$ are similar triangles.

###### 5

Determine the angle $\angle uvw$ where $u = 2 + i\text{,}$ $v = 1 + 2i\text{,}$ and $w = -1 + i\text{.}$

###### 6

Suppose $z$ is a point with positive imaginary component on the unit circle shown below, $a = 1$ and $b = -1\text{.}$ Use the angle formula to prove that angle $\angle b z a = \pi/2.$