Advanced Applications of Complex Numbers

Problem Statement

Complex numbers are not just for solving polynomials. They are a universal language used in higher-level mathematical proofs and real-world engineering.

For instance, the Triangle Inequality ($|z_1 + z_2| \le |z_1| + |z_2|$), which is a fundamental concept in metric spaces, is a direct generalisation of the real number case, visualised perfectly via vector addition on the Argand plane. Furthermore, De Moivre's theorem can be rigorously proven using Mathematical Induction.

Outside the classroom, complex numbers are used extensively in:

• Signal Processing: The Fast Fourier Transform (FFT), used in audio production, Wi-Fi, and JPEG compression, relies entirely on complex numbers to decompose signals into sine waves.
• Electrical Engineering: AC circuit analysis uses complex numbers (impedance) to model resistance and reactance simultaneously.
• Computer Graphics: Multiplying by a complex number is a highly efficient way to compute 2D rotations in game engines.

Let's look at how complex numbers are used to model a physical 2D rotation, an application heavily used in digital graphic design.

Consider a point in 2D space represented by the complex number $z = 3 + 4i$. We want to rotate this point counter-clockwise around the origin by $90^\circ$ ($\frac{\pi}{2}$ radians).

(a) Multiplying a complex number by $i$ rotates it. Perform the algebraic multiplication $i \times z$ to find the new coordinate $z'$.

(b) Prove that this multiplication represents a $90^\circ$ rotation by using the polar form (modulus-argument form) of $z$ and $i$.

(c) State the Triangle Inequality for complex numbers, and geometrically explain what it means using vectors on the Argand plane.

Hints

• Part (a): Distribute the $i$ into $3 + 4i$. Remember that $i^2 = -1$.
• Part (b): Express $i$ in polar form ($r \text{cis}\theta$). Then express $z$ in polar form (let $z = R \text{cis}\alpha$). Use the rule that when multiplying complex numbers, you multiply their moduli and add their arguments.
• Part (c): The Triangle Inequality is $|z_1 + z_2| \le |z_1| + |z_2|$. Think of $z_1$ and $z_2$ as vectors forming two sides of a triangle. The sum $z_1 + z_2$ is the third side. What does basic geometry say about the lengths of the sides of a triangle?

Solutions

Part (a): Algebraic Rotation

1. We have $z = 3 + 4i$.
2. Multiply by $i$: $z' = i \times (3 + 4i)$
3. Distribute: $z' = 3i + 4i^2$
4. Substitute $i^2 = -1$: $z' = 3i + 4(-1) = -4 + 3i$
5. The new point is $(-4, 3)$.

Part (b): Proving Rotation via Polar Form

1. Express $i$ in polar form. It lies on the positive imaginary axis, 1 unit from the origin. $i = 1 \text{cis} \left(\frac{\pi}{2}\right)$
2. Express $z$ in general polar form: $z = R \text{cis}(\alpha)$
3. Multiply them together using polar multiplication rules (multiply moduli, add arguments): $z' = i \times z = \left[ 1 \text{cis} \left(\frac{\pi}{2}\right) \right] \times \left[ R \text{cis}(\alpha) \right]$ $z' = (1 \times R) \text{cis} \left(\alpha + \frac{\pi}{2}\right)$ $z' = R \text{cis} \left(\alpha + \frac{\pi}{2}\right)$
4. The new complex number $z'$ has the exact same modulus ($R$) as the original number $z$, meaning its distance from the origin hasn't changed.
5. The new argument is exactly $\frac{\pi}{2}$ greater than the original argument.
6. Therefore, multiplying by $i$ physically rotates any complex number by exactly $90^\circ$ counter-clockwise without altering its scale.

Part (c): The Triangle Inequality

1. The Triangle Inequality states: $|z_1 + z_2| \le |z_1| + |z_2|$
2. Geometric Explanation: On the Argand plane, complex numbers add like vectors (head-to-tail).
   • $|z_1|$ is the length of the first vector.
   • $|z_2|$ is the length of the second vector.
   • $|z_1 + z_2|$ is the length of the resultant vector (the diagonal of the parallelogram they form).
3. These three vectors form a triangle. A fundamental theorem of geometry states that the length of any single side of a triangle must be less than or equal to the sum of the lengths of the other two sides.
4. Therefore, the "straight line" distance $|z_1 + z_2|$ must be shorter than (or equal to) the path taken by walking along $|z_1|$ and then along $|z_2|$. Equality only holds if $z_1$ and $z_2$ are parallel and point in the same direction.

Takeaways

• Multiplication is Rotation: While addition of complex numbers is a translation (sliding), multiplication is inherently a rotation and a scaling. This is why complex numbers are the premier math tool for computer graphics engines.
• Geometric Intuition for Algebra: Abstract algebraic inequalities like $|z_1 + z_2| \le |z_1| + |z_2|$ become painfully obvious once you translate them into visual geometry on the complex plane.
• The Deep Links: The HSC syllabus introduces complex numbers via quadratics, but their true power lies in Fourier analysis, abstract algebra, and quantum mechanics.

Further Readings

• HSC Complex Numbers: https://vumaths.com/booklets/hsc-complex-numbers/
• HSC Proofs: https://vumaths.com/booklets/hsc-proofs/
• HSC Vectors: https://vumaths.com/booklets/hsc-vectors/

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