Rotation Theorem in Complex Numbers

The Rotation Theorem is a fundamental concept in complex number theory that describes how multiplying a complex number by a unit complex number (a complex number with magnitude 1) results in a rotation of the original complex number in the complex plane.

Understanding Complex Numbers

Before diving into the Rotation Theorem, let's briefly review complex numbers. A complex number is of the form $z = a + bi$, where $a$ and $b$ are real numbers, and $i$ is the imaginary unit with the property that $i^2 = -1$. The number $a$ is called the real part, denoted as $\Re(z)$, and $b$ is called the imaginary part, denoted as $\Im(z)$.

Polar Form of Complex Numbers

A complex number can also be represented in polar form as $z = r(\cos \theta + i\sin \theta)$, where $r$ is the magnitude (or modulus) of the complex number and $\theta$ is the argument (or angle). The magnitude $r$ is given by $r = \sqrt{a^2 + b^2}$, and the argument $\theta$ is determined by $\tan \theta = \frac{b}{a}$.

Euler's Formula

Euler's formula provides a powerful way to represent complex numbers using exponentials: $e^{i\theta} = \cos \theta + i\sin \theta$. Using Euler's formula, a complex number in polar form can be written as $z = re^{i\theta}$.

Rotation Theorem

The Rotation Theorem states that if you multiply a complex number $z$ by a unit complex number $w = e^{i\alpha}$, where $\alpha$ is a real number, the result is a complex number that is a rotation of $z$ by an angle $\alpha$ in the complex plane.

Mathematically, if $z = re^{i\theta}$ and $w = e^{i\alpha}$, then the product $zw$ is given by:

$$ zw = re^{i\theta} \cdot e^{i\alpha} = re^{i(\theta + \alpha)} $$

This product has the same magnitude as $z$ but is rotated by an angle $\alpha$.

Important Points and Differences

Aspect	Description
Magnitude Preservation	The magnitude of the original complex number $z$ is preserved after the rotation.
Angle Addition	The angle of the rotated complex number is the sum of the original angle and the rotation angle.
Unit Complex Number	The multiplier must be a unit complex number, meaning its magnitude is 1.

Formulas

• Polar form: $z = r(\cos \theta + i\sin \theta)$ or $z = re^{i\theta}$
• Rotation: $zw = re^{i(\theta + \alpha)}$

Examples

Example 1: Basic Rotation

Let $z = 1 + i$, which in polar form is $z = \sqrt{2}e^{i\frac{\pi}{4}}$. Rotate $z$ by $\frac{\pi}{2}$ radians.

Let $w = e^{i\frac{\pi}{2}}$. Then,

$$ zw = \sqrt{2}e^{i\frac{\pi}{4}} \cdot e^{i\frac{\pi}{2}} = \sqrt{2}e^{i\frac{3\pi}{4}} $$

This is a rotation of $z$ by $\frac{\pi}{2}$ radians, resulting in a new complex number located at $\sqrt{2}e^{i\frac{3\pi}{4}}$ or $\sqrt{2}(\cos \frac{3\pi}{4} + i\sin \frac{3\pi}{4})$, which simplifies to $-1 + i$.

Example 2: Rotating by a Negative Angle

Let $z = 2e^{i\frac{\pi}{3}}$. Rotate $z$ by $-\frac{\pi}{6}$ radians.

Let $w = e^{-i\frac{\pi}{6}}$. Then,

$$ zw = 2e^{i\frac{\pi}{3}} \cdot e^{-i\frac{\pi}{6}} = 2e^{i\frac{\pi}{6}} $$

This is a rotation of $z$ by $-\frac{\pi}{6}$ radians, resulting in a new complex number located at $2e^{i\frac{\pi}{6}}$ or $2(\cos \frac{\pi}{6} + i\sin \frac{\pi}{6})$, which simplifies to $\sqrt{3} + i$.

The Rotation Theorem is a simple yet powerful tool in complex analysis and has applications in various fields such as engineering, physics, and computer graphics. Understanding this theorem is essential for anyone working with complex numbers and their geometric interpretations.