density of the rationals

The rationals are densely sprinkled along the real line: between any two distinct real numbers, however close, you can always find a fraction. There are no gaps so narrow that no rational fits inside.

Precisely: if a is less than b, there exists a rational number q with a less than q and q less than b. The proof uses the Archimedean property to pick a denominator n so large that 1 over n is smaller than b minus a, after which some multiple of 1 over n must land strictly between a and b.

Density is why the rationals can approximate any real to any desired accuracy, and why they form a countable yet dense subset of the reals. The same argument shows the irrationals are dense too: between any two reals there is also an irrational, for instance by translating a rational by the square root of 2 over a large integer.

Be careful not to confuse dense with complete. The rationals are dense in the reals but are not complete; density says they get arbitrarily close to every real, while completeness would require them to actually contain every limit, which they do not.