Number Systems
The hierarchy of numbers — naturals, integers, rationals, reals, and primes — and what each layer adds.
The Hierarchy
Numbers aren’t one thing. They were invented in layers, each extending the previous to solve problems the earlier set couldn’t handle.
Natural numbers ⊂ Integers ⊂ Rationals ⊂ Reals
Each set strictly contains the one before it.
Natural Numbers — ℕ
{1, 2, 3, 4, …}
The counting numbers. Some definitions include 0; some don’t. What you use to count objects, index positions, measure whole quantities.
Closed under: addition, multiplication. Not closed under: subtraction (3 − 5 has no answer here), division.
Integers — ℤ
{…, −3, −2, −1, 0, 1, 2, 3, …}
Extend the naturals by adding zero and negative numbers. Now subtraction always works.
Closed under: addition, subtraction, multiplication. Not closed under: division (3 ÷ 2 has no integer answer).
The letter Z comes from Zahlen — German for “numbers.”
Rational Numbers — ℚ
All numbers expressible as p/q where p and q are integers and q ≠ 0.
Examples: 1/2, −3/4, 7, 0.333…, 2.75
Closed under: addition, subtraction, multiplication, division (except ÷ 0).
Key property: every rational number has a decimal expansion that either terminates (1/4 = 0.25) or repeats (1/3 = 0.333…, 1/7 = 0.142857142857…). This is both necessary and sufficient — if a decimal terminates or repeats, it’s rational.
Irrational Numbers
Numbers that cannot be expressed as p/q. Their decimal expansions never terminate and never repeat.
- √2 = 1.41421356… — proved irrational by the ancient Greeks. Assume √2 = p/q in lowest terms; then 2q² = p², so p² is even, so p is even, so p = 2k, so 2q² = 4k², so q² = 2k², so q is even — contradicting lowest terms.
- π = 3.14159265… — ratio of circumference to diameter, proved irrational in 1761
- e = 2.71828… — base of natural logarithm, proved irrational in 1737
- φ = 1.61803… — golden ratio, (1 + √5)/2
Real Numbers — ℝ
Rationals ∪ Irrationals. Every point on the number line. No gaps.
Real numbers fill in all the holes the rationals leave. Between any two rationals there is an irrational (and between any two irrationals, a rational). The reals are dense — and also uncountably infinite, meaning there are strictly more real numbers than integers, even though both are infinite. Cantor’s diagonal argument proves this.
Primes
A prime is a natural number greater than 1 with no positive divisors other than 1 and itself.
{2, 3, 5, 7, 11, 13, 17, 19, 23, 29, …}
Fundamental Theorem of Arithmetic: every integer greater than 1 is either prime or can be written as a unique product of primes.
360 = 2³ × 3² × 5
This factorization is unique — there’s only one way to do it.
Why 1 is not prime: if it were, prime factorization wouldn’t be unique (you could multiply by 1 infinitely).
Infinitely many primes: suppose there were finitely many — p₁, p₂, …, pₙ. Construct N = (p₁ × p₂ × … × pₙ) + 1. N is either prime (contradiction — it’s not in the list) or has a prime factor not in the list (also a contradiction). So the list can’t be finite.
Useful primality facts
- 2 is the only even prime
- To test if n is prime, only need to check divisors up to √n
- Primes thin out as numbers grow — but never stop (Prime Number Theorem: the density of primes near n is approximately 1/ln(n))
Key Distinctions
| Property | ℕ | ℤ | ℚ | ℝ |
|---|---|---|---|---|
| Subtraction always works | ✗ | ✓ | ✓ | ✓ |
| Division always works | ✗ | ✗ | ✓ | ✓ |
| Every polynomial has a root | ✗ | ✗ | ✗ | partial |
| Complete (no gaps) | ✗ | ✗ | ✗ | ✓ |
Each layer is the minimal extension needed to close the gap the previous layer had.