7  Number Systems

\(\mathbb{N} \to \mathbb{Z} \to \mathbb{Q} \to \mathbb{R} \to \mathbb{C}\) is usually presented as a list to memorize: naturals, then integers, then rationals, then reals, then complex numbers, each one “bigger.” That framing hides the actual pattern, which is more interesting and more useful: at each step, some equation the previous system couldn’t solve becomes solvable — and something else, some property the previous system had for free, is given up to make that happen. This chapter is that trade, made explicit five times.

7.1 Natural numbers: what you start with

\(\mathbb{N} = \{0, 1, 2, 3, \dots\}\) — counting numbers, with addition and multiplication. (Whether to include 0 is a genuine convention that varies by author; this site includes it.) Formally, \(\mathbb{N}\) can be built from nothing but sets — \(0 := \varnothing\), and each successor \(n+1 := n \cup \{n\}\) — but nothing downstream depends on knowing that construction; it’s mentioned once here as evidence that “numbers are built from sets” is more than a slogan, not as machinery you need to carry forward.

\(\mathbb{N}\)’s first limitation is subtraction: \(3 - 5\) has no answer in \(\mathbb{N}\).

7.2 Integers: solving \(x + a = b\)

\(\mathbb{Z} = \{\dots, -2, -1, 0, 1, 2, \dots\}\) closes that gap by adjoining additive inverses. One clean formal construction (rarely spelled out computationally, worth seeing once): define \(\mathbb{Z}\) as equivalence classes of pairs \((a, b) \in \mathbb{N} \times \mathbb{N}\) under \((a,b) \sim (c,d) \iff a + d = b + c\) — the class of \((a,b)\) represents “\(a - b\),” and the equivalence relation is exactly what makes \((5,3)\) and \((8,6)\) (both “meaning” \(2\)) the same integer. This is the equivalence-class machinery from Relations and Functions, now doing real work.

Gained: every equation \(x + a = b\) now has a solution. Lost: nothing structural yet — \(\mathbb{Z}\) keeps everything \(\mathbb{N}\) had. Historically, negative numbers were the slow step, not the fast one: they appear in Chinese and Indian mathematics over a thousand years before European mathematicians fully accepted them as legitimate numbers (well into the 17th–18th centuries, negative solutions to equations were routinely dismissed as “false”). Formal ease and historical ease are not the same thing.

7.3 Rationals: solving \(ax = b\)

\(\mathbb{Q}\) adjoins multiplicative inverses for everything but 0 — the same equivalence-class trick, now on pairs \((a, b) \in \mathbb{Z} \times (\mathbb{Z} \setminus \{0\})\) representing “\(a/b\).” Gained: \(ax = b\) solvable whenever \(a \neq 0\). \(\mathbb{Q}\) is also dense: between any two distinct rationals sits another one, so in a real sense there’s “no next rational” — a genuinely different flavor of infinite than \(\mathbb{N}\)’s.

Lost, in a sense that took the Pythagoreans by surprise: \(\mathbb{Q}\) still has gaps. \(x^2 = 2\) has no rational solution — \(\sqrt{2}\) is irrational, a fact with a short, classic proof by contradiction (assume \(\sqrt{2} = p/q\) in lowest terms, square both sides, derive that \(p\) and \(q\) are both even, contradicting “lowest terms”). Density is not the same as completeness; \(\mathbb{Q}\) is packed infinitely tight and still full of holes.

7.4 Reals: filling the gaps

\(\mathbb{R}\) fills them. There are two standard rigorous constructions — Dedekind cuts (a real number is a partition of \(\mathbb{Q}\) into “everything below it” and “everything at or above it”) and Cauchy sequences (a real number is an equivalence class of rational sequences that bunch arbitrarily close together) — both due, independently, to Cantor and Dedekind around 1872. Neither construction is needed to use \(\mathbb{R}\) computationally, and this site doesn’t develop either in detail; what matters is the property they’re built to guarantee: completeness — every bounded set of reals has a least upper bound in \(\mathbb{R}\) itself, which \(\mathbb{Q}\) conspicuously lacks (the set \(\{q \in \mathbb{Q} : q^2 < 2\}\) is bounded above but has no rational least upper bound). Completeness is the property that makes calculus work at all: limits, continuity, and the intermediate value theorem all lean on it directly, which is why real analysis — not calculus over \(\mathbb{Q}\) — is the theory that makes calculus rigorous. Gained: no more gaps. Lost: countability — \(\mathbb{R}\) is provably “bigger than” \(\mathbb{N}\), \(\mathbb{Z}\), or \(\mathbb{Q}\) (all countably infinite; \(\mathbb{R}\) is not), a fact Cantor proved with his diagonal argument in the same period.

7.5 Complex numbers: solving \(x^2 = -1\)

\(\mathbb{C} = \{a + bi : a, b \in \mathbb{R}\}\), where \(i^2 = -1\), closes the last major gap in \(\mathbb{R}\): not every polynomial with real coefficients has a real root. Gained, and this is a genuinely striking payoff for adjoining a single new element: the fundamental theorem of algebra guarantees every nonconstant polynomial has a root in \(\mathbb{C}\) — no further extension is ever needed to solve a polynomial equation. Lost: order. \(\mathbb{R}\) has a total order compatible with its arithmetic (\(a < b \Rightarrow a + c < b + c\)); no such order exists on \(\mathbb{C}\) that plays nicely with multiplication (if it did, \(i^2 = -1\) would force a contradiction between \(i > 0\) and \(i < 0\) cases). You can compare real numbers by size; you can only compare complex numbers by distance from the origin, which is a different, weaker kind of structure.

Complex numbers were trusted least, longest: “imaginary” was originally a term of mild derision, and full acceptance — largely through Gauss’s and Hamilton’s work giving \(\mathbb{C}\) a solid geometric and algebraic footing — didn’t arrive until the late 18th and 19th centuries, roughly 250 years after they first appeared (Cardano, 1545) as an uncomfortable but useful bookkeeping device inside the formula for solving cubic equations.

7.6 The pattern, named

Five extensions, one shape: a specific unsolved equation motivates a new construction, built from equivalence classes over the previous system, that gains solvability and loses some structural property the old system had for free:

Extension	Solves	Gains	Loses
\(\mathbb{N} \to \mathbb{Z}\)	\(x + a = b\)	additive inverses	—
\(\mathbb{Z} \to \mathbb{Q}\)	\(ax = b\)	multiplicative inverses	—
\(\mathbb{Q} \to \mathbb{R}\)	least upper bounds	completeness	countability
\(\mathbb{R} \to \mathbb{C}\)	\(x^2 = -1\)	every polynomial has a root	a compatible order

This is the generalization theme from the map in its purest form — and it’s about to reappear one level of abstraction up. Each of \(\mathbb{N}\), \(\mathbb{Z}\), \(\mathbb{Q}\), \(\mathbb{R}\), \(\mathbb{C}\) is, at bottom, a set equipped with operations satisfying certain properties. Naming what “a set with operations satisfying certain properties” means in general — independent of which numbers, or non-numbers, fill in the set — is exactly the subject of What Is a Mathematical Structure?, next.