What Kinds of Numbers?
Before we count, before we measure, before we compute: what exists?
Three questions. Try to answer each.
How many electrons in a helium atom?
Answer: 2
What kind of number is 2? It counts discrete objects. It's a positive integer.
What is the bond length of H₂?
Answer: 0.74 Å
What kind of number is 0.74? It measures a continuous quantity. It falls between integers. It's a rational number (74/100).
What is the diagonal of a square with side length 1?
Answer: √2 ≈ 1.41421356...
What kind of number is √2? It's not a ratio of integers. The decimal never terminates, never repeats. It's irrational.
What are the solutions to x² + 1 = 0?
Answer: ...
There is no real number whose square is -1. Every real number squared is non-negative. Do we stop here? Or do we expand what "number" means?
Different questions demand different number systems.
The integers suffice for counting atoms. They fail for measuring lengths.
The rationals suffice for most measurements. They fail for √2.
The reals suffice for geometry. They fail for x² + 1 = 0.
Each limitation drives an expansion. Each expansion reveals structure that was always there.
We build from the ground up.
What they're for: Counting discrete objects.
The problem: 3 - 5 = ? Subtraction sometimes has no answer in ℕ.
What they add: Zero and negative numbers.
Why we need them: Charges (proton +1, electron -1), oxidation states (Fe²⁺, Fe³⁺, Fe⁰).
The problem: 3 ÷ 5 = ? Division sometimes has no answer in ℤ.
What they add: Fractions — ratios of integers.
Why we need them: Measurement (0.74 Å), concentrations (0.1 M), stoichiometry (half a mole).
Representation: Every rational has a decimal that terminates or repeats: 1/4 = 0.25, 1/3 = 0.333..., 1/7 = 0.142857...
The problem: The rationals have gaps.
Proof: Suppose √2 = p/q where p, q are integers with no common factor.
Then 2 = p²/q², so p² = 2q².
This means p² is even, so p is even. Write p = 2k.
Then 4k² = 2q², so q² = 2k², so q² is even, so q is even.
But then p and q share factor 2, contradicting our assumption.
Therefore √2 cannot be written as p/q. ∎
Irrational numbers fill the gaps: √2, √3, π = 3.14159..., e = 2.71828...
The real numbers ℝ comprise all rationals and irrationals together.
Geometric picture: The real line. Every point corresponds to exactly one real number. No gaps.
The problem: What is √(-1)?
Zoom in anywhere. Rationals (green) and irrationals (red) are densely packed at every scale.
Consider the equation x² + 1 = 0. This has no solution in ℝ. For any real x, x² ≥ 0, so x² + 1 ≥ 1 > 0.
We have two choices:
Mathematicians chose option 2 — and discovered that the expansion reveals deep structure.
We introduce a new number, denoted i, defined by the property:
This is not "imaginary" in the sense of "fake." The term is a historical accident. The number i is exactly as real as -1 or √2. All numbers are abstract entities.
Definition: A complex number is an expression of the form:
where a, b ∈ ℝ. a is the real part (Re(z) = a), b is the imaginary part (Im(z) = b).
| Operation | Formula | Example |
|---|---|---|
| Addition | (a+bi) + (c+di) = (a+c) + (b+d)i | (3+2i) + (1-5i) = 4-3i |
| Multiplication | (a+bi)(c+di) = (ac-bd) + (ad+bc)i | (3+2i)(1-5i) = 13-13i |
| Conjugate | z̄ = a - bi | (3+2i)̄ = 3-2i |
| Modulus | |z| = √(a² + b²) | |3+4i| = 5 |
Key property: z · z̄ = a² + b² = |z|². The product of a complex number with its conjugate is always a non-negative real.
The powers of i cycle with period 4:
| Power | Value | Pattern |
|---|---|---|
| i⁰ | 1 | n mod 4 = 0 |
| i¹ | i | n mod 4 = 1 |
| i² | -1 | n mod 4 = 2 |
| i³ | -i | n mod 4 = 3 |
| i⁴ | 1 | Cycle repeats |
Any nonzero complex number can be written:
where r = |z| and θ = arg(z).
This uses Euler's formula:
The most beautiful equation in mathematics
Setting θ = π gives:
Euler's identity — connecting e, i, π, 1, and 0
The nth roots of 1 are the solutions to zⁿ = 1.
They are e^(2πik/n) for k = 0, 1, 2, ..., n-1. They form a regular n-gon on the unit circle.
When you multiply by a complex number z = re^(iθ):
Theorem: Every polynomial equation of degree n ≥ 1 with complex coefficients has exactly n roots in ℂ (counting multiplicity).
This is why we stop at ℂ:
The complex numbers are the algebraic closure of the reals. There is no need to go further.
The wavefunction ψ(x, t) is complex-valued:
The Schrödinger equation has i built into its structure:
You cannot do quantum mechanics with real numbers alone.
Complex exponentials simplify calculations. Compare:
Fourier transforms produce complex results. The magnitude gives amplitude; the phase gives timing.
Spherical harmonics Y_l^m are complex. The real orbitals p_x, p_y are linear combinations of complex ones.
The phase oscillates, but |ψ|² = 1 everywhere (uniform probability).
| System | Symbol | Contains | Closed under |
|---|---|---|---|
| Natural | ℕ | 1, 2, 3, ... | +, × |
| Integer | ℤ | ..., -1, 0, 1, ... | +, -, × |
| Rational | ℚ | p/q (q ≠ 0) | +, -, ×, ÷ |
| Real | ℝ | All limits of rationals | +, -, ×, ÷, √(≥0) |
| Complex | ℂ | a + bi | All algebra, all √ |
| Operation | Formula |
|---|---|
| Addition | (a+bi) + (c+di) = (a+c) + (b+d)i |
| Multiplication | (a+bi)(c+di) = (ac-bd) + (ad+bc)i |
| Conjugate | a+bi → a-bi |
| Modulus | |a+bi| = √(a² + b²) |
| Polar form | z = r(cos θ + i sin θ) = re^(iθ) |
| Euler | e^(iθ) = cos θ + i sin θ |
Compute, expressing answers in the form a + bi:
(a) (2 + 3i) + (4 - i) (b) (2 + 3i)(4 - i) (c) (2 + 3i)/(4 - i) (d) i³ (e) i¹⁰⁰
For z = 3 - 4i:
(a) Find z̄ (b) Find |z| (c) Verify that z · z̄ = |z|² (d) Find z + z̄ and z - z̄
(a) Convert z = 1 + i to polar form (b) Convert z = 2e^(iπ/3) to Cartesian form (c) What is e^(2πi)?
(a) Find both square roots of i (b) Find all cube roots of 1 (c) Find all fourth roots of -1
(a) For ψ = e^(ikx), show that |ψ|² = 1 everywhere.
(b) For ψ = A e^(i(kx - ωt)), find ψ* and |ψ|².
(c) In NMR, signals are S(t) = S₀ e^(iωt) e^(-t/T₂). What is |S(t)|²?
Prove that √3 is irrational. (Follow the structure of the √2 proof.)