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.

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  • Mathematics

    Math­e­mat­ics is the study of num­bers and other ideal ob­jects that can be de­scribed by ax­ioms.