Number Systems

Building the Number Line

Mathematics builds its numbers in stages, each extending the last to solve problems the previous system could not. Understanding this hierarchy is the foundation of arithmetic.

• Natural numbers $\mathbb{N} = \{0, 1, 2, 3, \dots\}$ — for counting. Closed under addition and multiplication, but you cannot always subtract (e.g. $3 - 5$).
• Integers $\mathbb{Z} = \{\dots, -2, -1, 0, 1, 2, \dots\}$ — add negatives so subtraction always works. But you cannot always divide ($3 \div 2$ is not an integer).
• Rational numbers $\mathbb{Q}$ — all fractions $\frac{a}{b}$ with integers $a, b$ and $b \neq 0$. Now division (except by zero) always works. Rationals are exactly the numbers with terminating or repeating decimal expansions.
• Real numbers $\mathbb{R}$ — fill in the gaps with irrationals like $\sqrt{2}$ and $\pi$, whose decimals neither terminate nor repeat. The reals form a continuous line with no holes.

These sets are nested: $\mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R}$. Each is a subset of the next, so every natural number is an integer, every integer is rational, and every rational is real — but not conversely.

Each extension is driven by closure: we enlarge the number system precisely so that an operation always has an answer. Naturals to integers makes subtraction total; integers to rationals makes division (by nonzero) total; rationals to reals makes limits and roots available. 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 the systems are separate rather than nested. Numbers like $\sqrt{2}$ and $\pi$ are real but not rational — their decimals never terminate or repeat. And the systems build on each other: every integer is a rational (e.g. $5 = \frac{5}{1}$), so they are not disjoint categories but a chain of ever-larger sets.