GCT Division and Angle Measure

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Section 2.3 Division and Angle Measure

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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

$z = w \cdot u\text{.}$

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For instance,

$\frac{1}{i} = -i$ because

$1 = i \cdot (-i)\text{.}$

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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.

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Example 2.3.1. Division in Cartesian form.

We convert the following quotient to Cartesian form:

$\begin{align*} \frac{2+i}{3+2i} \amp = \frac{2+i}{3+2i}\cdot\frac{3-2i}{3-2i}\\ \amp = \frac{(6+2)+(-4+3)i}{9+4}\\ \amp = \frac{8-i}{13}\\ \amp = \frac{8}{13} - \frac{1}{13}i\text{.} \end{align*}$

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Example 2.3.2. 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

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

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

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

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,

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

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Angle Measure.

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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

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

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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

$\begin{align*} \angle(L_1,L_2) \amp = \arg\bigg(\frac{z_2-z_0}{z_1-z_0}\bigg) \notag\\ \amp = \arg(z_2-z_0)-\arg(z_1-z_0)\text{.} \end{align*}$

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Example 2.3.3. 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

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

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

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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,

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

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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

$\angle uvz = \arg(z)\text{.}$

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Exercises Exercises

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1.

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

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2.

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

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

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3.

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

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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.

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5.

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

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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 $\angle b z a = \pi/2\text{.}$