[Leejohnston]

Thread 6 — Complex Numbers & The Hidden Geometry of Algebra

How Imaginary Numbers Become Real Geometry — And Why Physics Cannot Function Without Them

Complex numbers look strange at first:

i = √(-1)

But this “imaginary” number unlocks real geometry, real physics, and real power.

Complex numbers turn algebra into a visual, geometric language.

---

1. What Is a Complex Number?

A complex number has two parts:

z = a + bi

• a = real part 
• b = imaginary part 

We can picture it as a point on a plane:

• x-axis → real 
• y-axis → imaginary 

This is the powerful idea:

Complex numbers *are points in 2D space.*

---

2. Multiplication = Rotation + Scaling

This is the mind-blowing part:

Multiplying complex numbers rotates and stretches the plane.

Examples:

• Multiply by i → rotate 90° 
• Multiply by −1 → rotate 180° 
• Multiply by 2i → stretch ×2 and rotate 
• Multiply by any complex number → rotate + scale

This is the foundation of:

• quantum mechanics 
• control systems 
• electrical engineering 
• signal processing 
• fractals 
• 3D graphics 
• Fourier transforms 

---

3. Euler’s Formula — The Most Beautiful Equation in Math

e^{iθ} = cosθ + i sinθ

This single formula:

• unites algebra, geometry, and trigonometry 
• describes waves, oscillations, rotations 
• appears in quantum mechanics 
• defines alternating current 
• enables 3D animation

It is the heartbeat of advanced mathematics.

---

4. Complex Numbers in Physics

Complex numbers describe:

• wavefunctions (quantum mechanics) 
• interference patterns 
• electromagnetic waves 
• resonance 
• rotating systems 
• AC circuits 
• harmonic oscillators 

Every time reality oscillates… 
complex numbers are behind the scenes.

---

Written by Leejohnston & Liora — The Lumin Archive Research Division