# 2.3: Division and Angle Measure

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

For instance, $$\dfrac{1}{i} = -i$$ 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.

##### Example $$\PageIndex{1}$$: Division in Cartesian Form

We convert the following quotient to Cartesian form:

$$\begin{array} z \dfrac{2+i}{3+2i} &= \dfrac{2+i}{3+2i}\cdot\dfrac{3-2i}{3-2i}\\ &= \dfrac{(6+2)+(-4+3)i}{9+4}\\ &= \dfrac{8-i}{13}\\ &= \dfrac{8}{13} - \dfrac{1}{13}i\text{.} \end{array}$$

##### Example $$\PageIndex{2}$$: Division 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

$\dfrac{1}{w} = \dfrac{1}{s}e^{-i\beta}\text{.}$

Then we may apply Theorem 2.2.1 to obtain the following result:

\begin{array} z \dfrac{z}{w} &= z\cdot\dfrac{1}{w}\\ &= re^{i\theta}\cdot \dfrac{1}{s}e^{-i\beta}\\ &= \dfrac{r}{s}e^{i(\theta-\beta)}\text{.} \end{array}

So,

$\arg\bigg(\frac{z}{w}\bigg)=\arg(z)-\arg(w)$

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,

$\dfrac{1+i}{-3 + 3i} =\dfrac{\sqrt{2}e^{i\pi/4}}{\sqrt{18}e^{i3\pi/4}} = \dfrac{1}{3}e^{-i\pi/2} = -\dfrac{1}{3} i\text{.}$

## 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

$\angle(L_1,L_2) = - \angle(L_2,L_1)\text{.}$

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

$$\begin{array} \angle(L_1,L_2) & = \arg\bigg(\dfrac{z_2-z_0}{z_1-z_0}\bigg) \notag\\ &= \arg(z_2-z_0)-\arg(z_1-z_0)\text{.} \end{array}$$

##### Example $$\PageIndex{3}$$: The 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

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

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

##### Note: 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,

$\angle uvw = \theta = \arg\bigg(\dfrac{w-v}{u-v}\bigg)\text{.}$ 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 $$\angle uvz = \arg(z)\text{.}$$

## Exercises

##### Exercise $$\PageIndex{1}$$

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

##### Exercise $$\PageIndex{2}$$

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

$\dfrac{1}{2i},\;\; \dfrac{1}{1+i},\;\; \dfrac{4+i}{1-2i},\;\; \dfrac{2}{3+i}\text{.}$

##### Exercise $$\PageIndex{3}$$

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

##### Exercise $$\PageIndex{4}$$

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. ##### Exercise $$\PageIndex{5}$$

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

##### Exercise $$\PageIndex{6}$$

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 $$\angle b z a = \dfrac{\pi}{2} \text{.}$$ This page titled 2.3: Division and Angle Measure is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Michael P. Hitchman via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.