1. Numbers: From Counting to Complex
Physics uses several number systems, each extending the previous one to solve new problems.
| System | Symbol | Examples | New Capability |
|---|---|---|---|
| Natural | ℕ | 1, 2, 3, ... | Counting |
| Integers | ℤ | ..., -2, -1, 0, 1, 2, ... | Subtraction (debts) |
| Rationals | ℚ | 1/2, -3/4, 0.333... | Division (fractions) |
| Reals | ℝ | π, √2, e | Limits, continuity |
| Complex | ℂ | 3 + 4i, eiπ | √(-1), rotations |
Complex numbers are essential for quantum mechanics. The imaginary unit i = √(-1) allows us to describe wave functions, interference, and rotations naturally.
2. Complex Numbers
The Imaginary Unit
Define i = √(-1), so i² = -1.
A complex number has a real part and an imaginary part:
z = x + iy
🎮 Interactive: Complex Plane
3
4i
z =
3 + 4i
|z| =
5.00
θ =
53.1°
Operations
Addition: (a + bi) + (c + di) = (a+c) + (b+d)i
Multiplication: (a + bi)(c + di) = (ac - bd) + (ad + bc)i
Conjugate: z* = x - iy (flip the imaginary part)
Magnitude: |z| = √(x² + y²) = √(z · z*)
Z²
|z|² = z · z*
This pattern appears throughout quantum mechanics: probability = |ψ|²
3. Euler's Formula
The most beautiful equation in mathematics connects exponentials, trigonometry, and complex numbers:
eiθ = cos θ + i sin θ
Euler's Formula
🎮 Interactive: Euler's Formula on the Unit Circle
cos θ
0.707
sin θ
0.707
e^(iθ)
0.71 + 0.71i
eiθ = cos θ + i sin θ
At θ = π: eiπ = -1 → eiπ + 1 = 0
Special Cases
ei·0 = 1
θ = 0
eiπ/2 = i
θ = 90°
eiπ = -1
Euler's Identity!
Euler's identity: eiπ + 1 = 0
This connects the five most important constants: e, i, π, 1, and 0.
This connects the five most important constants: e, i, π, 1, and 0.
4. Calculus Essentials
The Derivative
The derivative measures the instantaneous rate of change:
f'(x) = limh→0 [f(x+h) - f(x)] / h
🎮 Interactive: Derivative as Limit of Secants
Secant slope (approximation)
[f(x+h) - f(x)] / h = 3.000
Tangent slope (exact)
f'(x) = 2x = 2.000
As h → 0, the secant line approaches the tangent line.
Error: 1.0000
Error: 1.0000
Key Derivatives
| f(x) | f'(x) | Example Use |
|---|---|---|
| xⁿ | nxⁿ⁻¹ | Polynomial potentials |
| eˣ | eˣ | Exponential decay |
| sin x | cos x | Wave functions |
| ln x | 1/x | Entropy |
The Integral
The integral is the "reverse" of the derivative — it accumulates area under a curve:
∫ f(x) dx = F(x) + C where F'(x) = f(x)
Fundamental Theorem of Calculus: Differentiation and integration are inverse operations.
∫ₐᵇ f(x) dx = F(b) - F(a)
5. Connection to Z² Framework
These prerequisites appear throughout the framework:
Z²
e^(iθ)
Complex exponentials describe quantum phases and wave functions
Z²
∫ d³x
Integrals over 3-space appear in the action principle
Z²
Z² = 32π/3
π appears because we're integrating over spheres!
Why These Matter
- • Complex numbers: Quantum mechanics lives in ℂ
- • Euler's formula: Rotations, symmetries, Fourier analysis
- • Calculus: Everything is described by differential equations
- • π: Appears in Z² = 32π/3 because spheres have curved surfaces
Exercises
- Calculate |3 + 4i| and verify it equals 5.
- Using Euler's formula, find ei(π/4) in the form a + bi.
- Differentiate f(x) = sin(2x) using the chain rule.
- Evaluate ∫₀^π sin(x) dx geometrically and algebraically.
- Show that i⁴ = 1. What does this tell you about powers of i?