Complex Numbers

Definition

A complex number is:

$z = x + iy$

where:

• $x \in \mathbb{R}$ is the real part: $\text{Re}(z) = x$
• $y \in \mathbb{R}$ is the imaginary part: $\text{Im}(z) = y$
• $i$ is the imaginary unit: $i^2 = -1$

The set of all complex numbers is $\mathbb{C}$.

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Complex Conjugate

The complex conjugate of $z = x + iy$ is:

$\bar{z} = z^* = x - iy$

Properties:

• $z\bar{z} = |z|^2 = x^2 + y^2$
• $z + \bar{z} = 2\text{Re}(z)$
• $z - \bar{z} = 2i\text{Im}(z)$
• $\overline{z_1 z_2} = \bar{z}_1 \bar{z}_2$
• $\overline{z_1 + z_2} = \bar{z}_1 + \bar{z}_2$

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The Argand Diagram

Complex numbers correspond to points in the complex plane (Argand diagram):

• Horizontal axis: real part
• Vertical axis: imaginary part

This geometric interpretation makes complex numbers useful for 2D rotations and oscillations.

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Modulus and Argument

Modulus (Magnitude)

$|z| = r = \sqrt{x^2 + y^2} = \sqrt{z\bar{z}}$

The distance from origin in the Argand diagram.

Argument (Angle)

$\arg(z) = \theta = \tan^{-1}\frac{y}{x}$

Warning: Must consider the signs of $x$ and $y$ to get the correct quadrant. The principal argument satisfies $-\pi < \theta \leq \pi$.

Use:

• $\theta = \tan^{-1}(y/x)$ if $x > 0$
• $\theta = \tan^{-1}(y/x) + \pi$ if $x < 0$ and $y \geq 0$
• $\theta = \tan^{-1}(y/x) - \pi$ if $x < 0$ and $y < 0$
• $\theta = \pi/2$ if $x = 0$ and $y > 0$
• $\theta = -\pi/2$ if $x = 0$ and $y < 0$

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Polar Form

Using modulus and argument:

$z = r(\cos\theta + i\sin\theta) = re^{i\theta}$

This is the polar form (also called modulus-argument form).

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Euler’s Formula

$e^{i\theta} = \cos\theta + i\sin\theta$

Derivation sketch: Expand $e^{i\theta}$, $\cos\theta$, $\sin\theta$ as power series and compare.

Euler’s identity (when $\theta = \pi$): $e^{i\pi} + 1 = 0$

This connects five fundamental constants: $e$, $i$, $\pi$, $1$, $0$.

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De Moivre’s Theorem

$(\cos\theta + i\sin\theta)^n = \cos(n\theta) + i\sin(n\theta)$

Or equivalently: $(e^{i\theta})^n = e^{in\theta}$

Applications

Trigonometric identities: $\cos n\theta = \text{Re}[(\cos\theta + i\sin\theta)^n]$ $\sin n\theta = \text{Im}[(\cos\theta + i\sin\theta)^n]$

Binomial expansion method: $\cos 3\theta = \text{Re}[(\cos\theta + i\sin\theta)^3] = \cos^3\theta - 3\cos\theta\sin^2\theta = 4\cos^3\theta - 3\cos\theta$

Expressing trig in terms of exponentials: $\cos\theta = \frac{e^{i\theta} + e^{-i\theta}}{2}, \quad \sin\theta = \frac{e^{i\theta} - e^{-i\theta}}{2i}$

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Roots of Unity

The $n$th roots of unity are solutions to $z^n = 1$:

$z_k = e^{2\pi i k/n} = \cos\frac{2\pi k}{n} + i\sin\frac{2\pi k}{n}$

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

Geometric interpretation: The $n$ roots are evenly spaced around the unit circle, forming a regular $n$-gon.

Example (cube roots): $z^3 = 1 \Rightarrow z_0 = 1, \quad z_1 = e^{2\pi i/3}, \quad z_2 = e^{4\pi i/3}$

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Roots of Complex Numbers

To solve $z^n = w$ where $w = Re^{i\phi}$:

$z_k = \sqrt[n]{R} \exp\left(i\frac{\phi + 2\pi k}{n}\right)$

for $k = 0, 1, \ldots, n-1$.

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Why Complex Numbers Matter

Complex numbers unify and simplify many areas:

• Differential equations: exponential solutions
• AC circuits: impedance extends resistance
• Quantum mechanics: wave functions are complex-valued
• Signal processing: Fourier transforms use complex exponentials
• 2D rotations: multiplication by $e^{i\theta}$ rotates by angle $\theta$

The algebra of complex numbers captures oscillatory and wave behaviour elegantly, replacing sines and cosines with exponentials.