Rational number

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.