## 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:

###### 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

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

So,

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,

##### 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

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

###### 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

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,

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

###### 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.

###### 3.

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

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