The real number

Definition

A real number is a point on the continuous line. Integers, fractions, roots and $\pi$π all hold a place there, with no position left empty.

It is this absence of gaps that defines the set $\mathbb{R}$R, far more than the list of numbers it contains.

What each widening brings

Each floor repairs a lack in the one below. The last is of another nature: it is no longer an operation that fails, but a sequence with nowhere to land.

The gap in the rationals

No fraction has 2 as its square. Suppose $p/q$p/q is in lowest terms with $(p/q)^2 = 2$(p/q)2=2: then $p^2 = 2q^2$p2=2q2, so $p$p is even, so $p = 2m$p=2m, so $4m^2 = 2q^2$4m2=2q2, so $q$q is even too. Both are even, against the assumption of lowest terms.

Yet $\sqrt{2}$2​ exists: it is the diagonal of the unit square. The rationals therefore leave a gap at a perfectly constructible place.

Completeness

The property that sets $\mathbb{R}$R apart fits in one sentence: every non-empty set bounded above has a least upper bound.

Take the set of rationals whose square is less than 2. It is non-empty, it is bounded above — but within $\mathbb{Q}$Q it has no smallest upper bound: a tighter one can always be found. In $\mathbb{R}$R that smallest upper bound exists, and it is $\sqrt{2}$2​.

This property characterises $\mathbb{R}$R: up to isomorphism, it is the only complete totally ordered field.

A larger infinity

The rationals can be numbered: they can be arranged in a list indexed by the integers. The reals cannot — no list exhausts them.

The infinity of the reals is therefore strictly vaster than that of the rationals, even though the latter are everywhere dense among the former: between two distinct reals, however close, a rational always slips in.

Summary

The real is the rational completed: taking a limit always lands somewhere.