Real number

A real num­ber is any num­ber that can be used to rep­re­sent a phys­i­cal quan­tity.

In­tu­itively, real num­bers are any num­ber that can be found be­tween two in­te­gers, such as \(0,\) \(1,\) \(-1,\) \(\frac{3}{2},\) \(\frac{-7}{2},\) \(\pi,\) \(e\), \(100 \cdot \sqrt{2},\) and so on. The set of real num­bers is writ­ten \(\mathbb R.\) You can think of \(\mathbb R\) as \(\mathbb Q\) ex­tended to in­clude the ir­ra­tional num­bers like \(\pi\) and \(e\) which can be found be­tween ra­tio­nal num­bers but which can­not be com­pletely writ­ten out in dec­i­mal no­ta­tion.

Defi­ni­tions of the real numbers

The most com­monly used defi­ni­tions of the real num­bers are con­struc­tions as ex­ten­sions of the ra­tio­nal num­bers, which in­volve ei­ther Cauchy se­quences or Dedekind cuts.

Cauchy sequences

Broadly speak­ing, a Cauchy se­quence is a se­quence where as the se­quence goes on, all the el­e­ments past that point get closer and closer to­gether. In the real num­bers, ev­ery Cauchy se­quence con­verges to a real num­ber. How­ever, in the set of ra­tio­nal num­bers, not all Cauchy se­quences con­verge to a ra­tio­nal num­ber. In the set of ra­tio­nals, a Cauchy se­quence which does not con­verge to a ra­tio­nal num­ber can­not re­ally be said to “con­verge” at all: the set of ra­tio­nals is “miss­ing some of the points” that would be re­quired to make ev­ery Cauchy se­quence con­verge.

For ex­am­ple, the se­quence of frac­tions of con­sec­u­tive Fibonacci num­bers \(1/1, 21, 32, 53, 85, \ldots\) gets closer and closer to \(\frac{1 + \sqrt{5}}{2}\), but can­not be said to con­verge to that num­ber be­cause it is not in the set of ra­tio­nal num­bers.

For each of these non-con­ver­gent Cauchy se­quences, we define a new ir­ra­tional num­ber to “fill in the gap”, and for the Cauchy se­quences that do con­verge, we define a real num­ber equal to that ra­tio­nal num­ber.

Dedekind cuts

A Dedekind cut of a to­tally or­dered set is a par­ti­tion of that set into two sets so that ev­ery el­e­ment in the first set is less than ev­ery el­e­ment in the sec­ond set, and the sec­ond set has no small­est el­e­ment. The lat­ter re­stric­tion re­quires that the set also be a perfect set (have no iso­lated points), in the sense used in topol­ogy.

In the real num­bers, such a par­ti­tion will always have the first set hav­ing a great­est el­e­ment, which is known as the least-up­per-bound prop­erty. How­ever, in the ra­tio­nal num­bers, we might come across a par­ti­tion where the first set does not have such an el­e­ment.

For ex­am­ple, define a Dedekind cut \((A, B)\) of the ra­tio­nal num­bers such that \(B = \{x \in \mathbb{Q} \ | \ x > 0 \wedge x^2 > 2\}\) and \(A\) is the com­ple­ment of \(B\). In plainer lan­guage, \(B\) con­sists of all the num­bers greater than \(\sqrt{2}\), but be­cause \(\sqrt{2}\) doesn’t ex­ist in the space of ra­tio­nal num­bers, we can’t use that to for­mu­late our defi­ni­tion. Ob­vi­ously ev­ery el­e­ment of \(A\) is less than ev­ery el­e­ment of \(B\), but \(A\) has no great­est el­e­ment ei­ther, be­cause we can cre­ate a se­quence of num­bers in \(A\) that gets big­ger and big­ger (as it ap­proaches \(\sqrt{2}\)) but never stops at a max­i­mum value.

For each of these “strict cuts” where nei­ther set has a “bound­ary el­e­ment”, we define a new ir­ra­tional num­ber to “fill in the gap”, just like with the Cauchy se­quences. For the Dedekind cuts where one of the sets does have a least or great­est el­e­ment, we define a real num­ber equal to that ra­tio­nal num­ber.

This defi­ni­tion has the ad­van­tage that each real num­ber is rep­re­sented by a unique Dedekind cut, un­like the Cauchy se­quences where mul­ti­ple se­quences can con­verge to the same num­ber.

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  • Mathematics

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