Number Systems

Building the Number Line

Mathematics builds numbers in stages to solve problems older systems could not. Each extension is driven by closure: enlarging a set so operations always yield an answer.

These sets are nested: $\mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R}$. Every natural is an integer, and every integer is a rational (e.g., $5 = \frac{5}{1}$).

Nested Systems & Pitfalls

Knowing which system a number lives in tells you which operations are guaranteed to stay inside it.

Common pitfall: Assuming every real number is rational, or that these systems are separate categories rather than a chain of ever-larger sets.

• Irrationals like $\sqrt{2}$ and $\pi$ are real but not rational; their decimal expansions neither terminate nor repeat.
• Closure rules: Naturals are closed under addition and multiplication, but not subtraction. Integers are closed under subtraction. Rationals are closed under division by nonzero values.

Each expansion unlocks roots, limits, and total operations. Always check your number system before assuming an operation has an internal solution.