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Section 2.3 Division 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.

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*}
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*}


\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*}
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)\text{.} \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)\text{.} \end{align*}
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.

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*}

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 Exercises


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


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*}

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


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.


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


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