The Real Number System

Calculus rests on one quiet assumption: that the number line has no holes. The idea of a limit only makes sense if the value a sequence is "closing in on" actually exists. The rationals \mathbb{Q} are not enough — they are riddled with gaps. The real numbers \mathbb{R} are exactly \mathbb{Q} with every gap filled.

First, what \mathbb{R} shares with \mathbb{Q}: both are ordered fields. You can add, subtract, multiply and divide (a field), and there is a consistent order \le compatible with that arithmetic (an ordered field). What sets \mathbb{R} apart is one extra property — completeness — and the whole point of this page is to see, concretely, why \mathbb{Q} fails it.

A gap you can name: \sqrt{2}

The diagonal of a unit square has length \sqrt{2}, a number whose square is 2. It sits somewhere on the line — between 1.41 and 1.42, between 1.414 and 1.415, and so on forever. Yet no fraction lands on it. We prove this by contradiction.

Step 1 — assume the opposite. Suppose \sqrt{2} were rational. Then we could write it as a fraction in lowest terms,

\sqrt{2} = \frac{p}{q}, \qquad p, q \in \mathbb{Z},\ q \ne 0,

where p and q share no common factor (any fraction can be reduced to this form, so this costs us nothing).

Step 2 — clear the square root. Square both sides and multiply through by q^2:

2 = \frac{p^2}{q^2} \quad\Longrightarrow\quad p^2 = 2q^2.

Step 3 — read off that p is even. The right-hand side 2q^2 is 2 times an integer, so p^2 is even. But the square of an odd number is odd ((2k+1)^2 = 4k^2+4k+1), so for p^2 to be even, p itself must be even. Write p = 2k.

Step 4 — substitute and simplify. Put p = 2k back into p^2 = 2q^2:

(2k)^2 = 2q^2 \quad\Longrightarrow\quad 4k^2 = 2q^2 \quad\Longrightarrow\quad q^2 = 2k^2.

Step 5 — read off that q is even too. By exactly the same reasoning as Step 3, q^2 = 2k^2 is even, so q is even.

Step 6 — the contradiction. We have shown both p and q are even — they share the factor 2. But Step 1 fixed the fraction in lowest terms, with no common factor. This is impossible. The only faulty link in the chain was the assumption that \sqrt{2} is rational, so that assumption is false:

\sqrt{2} \notin \mathbb{Q}.

Watch the gap survive any zoom

The rationals are dense: between any two of them lies another, so they pack the line with no visible holes. Step through the figure to scatter rationals just below \sqrt{2} (where (p/q)^2 < 2) and just above it (where (p/q)^2 > 2). They crowd in from both sides — yet the point \sqrt{2} itself, marked in the centre, is claimed by none of them. Refresh to redraw at a fresh random zoom level: however far you magnify, the gap is always there.

Calling \mathbb{R} a field is shorthand for a short list of axioms governing + and \times: both are commutative and associative, multiplication distributes over addition, there are identities 0 and 1, every element has an additive inverse, and every nonzero element has a multiplicative inverse. \mathbb{Q} satisfies all of these too — as do the complex numbers.

Ordered adds a relation \le that is total and plays nicely with arithmetic: if a \le b then a + c \le b + c, and if also 0 \le c then ac \le bc. This is why \mathbb{C} is not an ordered field — you cannot order it compatibly with multiplication. Both \mathbb{Q} and \mathbb{R} are ordered fields; the watershed between them is the next property.

Density and completeness are different things, and \mathbb{Q} shows they come apart. Dense means: between any two rationals lies a third — concretely, \tfrac{a+b}{2}. So the rationals leave no interval-sized hole; you can approximate any real to any precision with a fraction.

But the set

S = \{\, x \in \mathbb{Q} : x > 0,\ x^2 < 2 \,\}

is nonempty and bounded above (by 2, say), yet has no least upper bound in \mathbb{Q}: any rational upper bound can be nudged a little smaller and still bound S, because the would-be boundary \sqrt{2} is missing. In \mathbb{R} that least upper bound exists and equals \sqrt{2}. Filling such gaps — guaranteeing every bounded set its boundary point — is exactly the completeness axiom, first made precise by Dedekind and Cauchy.