The Argand diagram is a geometric representation of complex numbers, where:

• The horizontal axis (real axis) represents $\text{Re}(z)$
• The vertical axis (imaginary axis) represents $\text{Im}(z)$

A complex number $z = x + iy$ is represented by the point $P(x, y)$ on this diagram. The distance from the origin to point $P$ is the modulus, and the angle formed with the positive real axis is the argument.

For $z = x + iy$, the modulus (or absolute value) is: $|z| = \sqrt{x^2 + y^2}$

The modulus represents the distance from the origin to the point representing $z$ on the Argand diagram.

Properties of Modulus

1. $|z| = 0$ if and only if $z = 0$
2. $|z_1 z_2| = |z_1| |z_2|$
3. $\left|\frac{z_1}{z_2}\right| = \frac{|z_1|}{|z_2|}$ (for $z_2 \neq 0$)
4. $|z_1 + z_2| \leq |z_1| + |z_2|$ (triangle inequality)
5. $|z_1 - z_2|^2 + |z_1 + z_2|^2 = 2(|z_1|^2 + |z_2|^2)$

Example

Find the modulus of $z = -\sqrt{3} + i$:

Solution: $|z| = \sqrt{(-\sqrt{3})^2 + 1^2} = \sqrt{3 + 1} = \sqrt{4} = 2$

The argument of a complex number $z = x + iy$ is the angle $\theta$ that the line from the origin to the point $(x, y)$ makes with the positive real axis: $\arg(z) = \theta = \tan^{-1}\left(\frac{y}{x}\right)$

The principal argument lies in the range $-\pi < \theta \leq \pi$ (or $-180° < \theta \leq 180°$). The argument is undefined when $z = 0$.

Example

Find the argument of $z = 1 + i$:

Solution: $\theta = \tan^{-1}\left(\frac{1}{1}\right) = \tan^{-1}(1) = 45° = \frac{\pi}{4}$

For $z = 5 - 2i$, where $x = 5$ and $y = -2$: $\theta = \tan^{-1}\left(\frac{-2}{5}\right) = -21.8°$

Note: When the point lies in the second or third quadrant, you must add $\pi$ (or $180°$) to the calculated angle to get the correct argument.

A complex number $z = x + iy$ can be expressed in polar form as: $z = r(\cos\theta + i\sin\theta)$

where:

• $r = |z| = \sqrt{x^2 + y^2}$ is the modulus
• $\theta = \arg(z)$ is the argument

Product and Quotient in Polar Form

If $z_1 = r_1(\cos\theta_1 + i\sin\theta_1)$ and $z_2 = r_2(\cos\theta_2 + i\sin\theta_2)$, then:

1. $z_1 z_2 = r_1 r_2[\cos(\theta_1 + \theta_2) + i\sin(\theta_1 + \theta_2)]$
2. $\frac{z_1}{z_2} = \frac{r_1}{r_2}[\cos(\theta_1 - \theta_2) + i\sin(\theta_1 - \theta_2)]$

Also:

• $\arg(z_1 z_2) = \arg(z_1) + \arg(z_2)$
• $\arg\left(\frac{z_1}{z_2}\right) = \arg(z_1) - \arg(z_2)$

Example

Express $z = 1 + i$ in polar form:

Solution: $r = \sqrt{1^2 + 1^2} = \sqrt{2}$ $\theta = \tan^{-1}\left(\frac{1}{1}\right) = \frac{\pi}{4}$

Thus: $z = \sqrt{2}\left(\cos\frac{\pi}{4} + i\sin\frac{\pi}{4}\right)$

De Moivre's Theorem states that for any complex number in polar form $z = r(\cos\theta + i\sin\theta)$ and any integer $n$: $z^n = [r(\cos\theta + i\sin\theta)]^n = r^n(\cos n\theta + i\sin n\theta)$

This theorem is valid for positive, negative, and fractional values of $n$.

Finding nth Roots

The $n$ distinct $n$th roots of a complex number $z = r(\cos\theta + i\sin\theta)$ are given by: $z_k = \sqrt[n]{r}\left[\cos\left(\frac{\theta + 2\pi k}{n}\right) + i\sin\left(\frac{\theta + 2\pi k}{n}\right)\right]$

for $k = 0, 1, 2, ..., n-1$.

Example

Find the cube roots of $8i$.

Solution: First, express $8i$ in polar form: $r = 8, \quad \theta = \frac{\pi}{2}$

So $8i = 8\left(\cos\frac{\pi}{2} + i\sin\frac{\pi}{2}\right)$

For cube roots ($n = 3$): $z_k = 8^{1/3}\left[\cos\left(\frac{\pi/2 + 2\pi k}{3}\right) + i\sin\left(\frac{\pi/2 + 2\pi k}{3}\right)\right]$

For $k = 0$: $z_0 = 2\left(\cos\frac{\pi}{6} + i\sin\frac{\pi}{6}\right) = \sqrt{3} + i$

For $k = 1$: $z_1 = 2\left(\cos\frac{5\pi}{6} + i\sin\frac{5\pi}{6}\right) = -\sqrt{3} + i$

For $k = 2$: $z_2 = 2\left(\cos\frac{3\pi}{2} + i\sin\frac{3\pi}{2}\right) = -2i$

Example Using De Moivre's Theorem

Evaluate $(1 + i)^8$.

Solution: Express $1 + i$ in polar form: $r = \sqrt{2}, \quad \theta = \frac{\pi}{4}$

Using De Moivre's theorem: $(1 + i)^8 = (\sqrt{2})^8 \left(\cos 8 \cdot \frac{\pi}{4} + i\sin 8 \cdot \frac{\pi}{4}\right)$ $= 16(\cos 2\pi + i\sin 2\pi) = 16(1 + 0i) = 16$

Euler's formula establishes a fundamental relationship between complex numbers and exponential functions: $e^{i\theta} = \cos\theta + i\sin\theta$

Therefore, a complex number in polar form can be written in exponential form as: $z = r(\cos\theta + i\sin\theta) = re^{i\theta}$

The complex conjugate is: $\bar{z} = re^{-i\theta}$

Example

Express $z = 2e^{\frac{13\pi}{6}i}$ in the form $a + ib$.

Solution: $z = 2\left[\cos\left(\frac{13\pi}{6}\right) + i\sin\left(\frac{13\pi}{6}\right)\right]$ $= 2\left[\cos\left(\frac{\pi}{6}\right) + i\sin\left(\frac{\pi}{6}\right)\right]$ $= 2\left(\frac{\sqrt{3}}{2} + i \cdot \frac{1}{2}\right) = \sqrt{3} + i$

Relationship with Trigonometric Functions

From Euler's formula:

• $\cos\theta = \frac{e^{i\theta} + e^{-i\theta}}{2}$
• $\sin\theta = \frac{e^{i\theta} - e^{-i\theta}}{2i}$

These identities are useful for simplifying trigonometric expressions and proving identities.

Complex numbers are used in Tanzania's electrical engineering industry when analyzing alternating current (AC) circuits, which power homes and businesses. For example, when Tanesco engineers calculate power distribution in the national grid, they use the polar form of complex numbers to represent voltage and current, making it easier to determine power consumption. The modulus gives the magnitude (like the actual power in watts), while the argument represents the phase difference between voltage and current—essential information for ensuring that electrical equipment operates safely and efficiently across Tanzania.