The rational numbers are either whole numbers or fractions of whole numbers, like \(0,\) \(1\), \(2\), \(\frac{1}{2}\), \(\frac{97}{3}\), \(-17\), \(\frac{-85}{1993},\) and so on. The set of rational numbers is written \(\mathbb Q.\)

Irrational numbers like \(\pi\) and \(e\) are not included in \(\mathbb Q;\) the rational numbers are only those numbers which can be written as \(\frac{a}{b}\) for integers \(a\) and \(b\) (where \(b \neq 0\)).

Formally, \(\mathbb{Q}\) is the underlying set of the field of fractions of \(\mathbb{Z}\) (the ring of integers). That is, each \(q \in \mathbb Q\) is an expression \(\frac{a}{b}\), where \(b\) is a nonzero integer and \(a\) is an integer, together with certain rules for addition and multiplication. The rational numbers are the last intermediate stage on the way to constructing the real numbers, but they are also very interesting and important in their own right.

One intuition about the rational numbers is that once we’ve created the real numbers, then a real number \(x\) is a rational number if and only if it may be written as \(\frac{a}{b}\), where \(a, b\) are integers and \(b\) is not \(0\). add division as a requisite

Examples

• The integer \(1\) is a rational number, because it may be written as \(\frac{1}{1}\) (or, indeed, as \(\frac{2}{2}\) or \(\frac{-1}{-1}\), or \(\frac{a}{a}\) for any nonzero integer \(a\)).

• The number \(\pi\) is not rational (proof).

• The number \(\sqrt{2}\) (being the unique positive real which, when multiplied by itself, yields \(2\)) is not rational (proof).

Properties

• There are infinitely many rationals. (Indeed, every integer is rational, because the integer \(n\) may be written as \(\frac{n}{1}\), and there are infinitely many integers.)

• There are countably many rationals (proof). Therefore, because there are uncountably many real numbers, almost all real numbers are not rational.

• The rationals are dense in the reals.

• The rationals form a field (proof). Indeed, they are a subfield of the real numbers.

Construction

Instead of taking the reals and selecting a certain collection which we label the “rationals”, it is possible to construct the rationals given access only to the natural numbers; and from the rationals we may construct the reals. In some sense, this approach is cleaner than starting with the reals and producing the rationals, because the natural numbers are very intuitive objects but the real numbers are less so. We can be closer to satisfying some deep existential unease if we can build the reals out of the much-simpler naturals.

As an analogy, being able to produce a block of wood given access to a wooden table is much less satisfying than the other way round, and we run into blocks of wood “in the wild” so we are pretty convinced that there actually are such things as blocks of wood. On the other hand, we almost never see wooden tables in nature, so we can’t be quite as sure that they’re real until we’ve built one ourselves.

Similarly, everyone recognises broadly what a counting number is, and they’re out there in the wild, but the rational numbers are somewhat less “natural” and their existence is less intuitive.

add simpler lenses