Between any two distinct real numbers, however close, there is a rational number — and there is an irrational one too. The rationals are therefore spread everywhere on the line, and so are the numbers that are not rational. Neither fact is about size: there are vastly more irrationals than rationals, and both kinds sit in every interval you can name.

A fine net laid over a beach touches everywhere, in the sense that no matter how small a patch you mark out, some knot of the net falls inside it. That does not mean the net covers the sand. Density is about being everywhere, not about filling anything up.

A set $D$ is dense in $\mathbb{R}$ when every open interval, however short, contains a point of $D$. Both $\mathbb{Q}$ and its complement are dense. The proof for $\mathbb{Q}$ is the Archimedean property twice: once to make a step $\tfrac1n$ shorter than the gap, and once to find where the first multiple of that step lands past the left end.

• Between any two reals there is a rational, and by applying that to $a-\sqrt2$ and $b-\sqrt2$ and adding back, an irrational as well.

However close two reals are, a fraction fits between them. The Archimedean property supplies the denominator: some whole number exceeds $1/(b-a)$, and steps that small cannot stride over the gap.

Make the step small enough, then count. The Archimedean property supplies a natural n with 1/n below the gap, which is the same as saying nb exceeds na by more than one. A stretch of length more than one has an integer strictly inside it: take the least integer m above na, and the step back from m is at most one, so m has not yet reached nb. Dividing by n puts the fraction where it was wanted.

Between $\sqrt{2}$ and $\sqrt{2} + 0.001$ lies a rational.

Both ends are irrational, and a fraction still fits between them.

$\mathbb{Q}$ is dense in itself but not complete.

One property says the marks are everywhere, the other that nothing is missing between them.

Dense means every interval, however short, holds a point of the set. It says nothing about how many points there are.

• Dense means every interval holds a point of the set, however short the interval.
• Both the rationals and the irrationals are dense, and every interval holds infinitely many of each.
• Density is not covering, and it has nothing to do with cardinality.
• The proof is the Archimedean property used twice.

Great work! You now know how to:

• Say what dense means, and test it one interval at a time
• Produce a rational between two given reals
• Get the irrational case from the rational one by shifting
• Keep density apart from covering and from cardinality