# 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.

Children:

Parents:

• Mathematics

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

• Ap­proved, but the sum­mary could do with a bit of im­prove­ment, make it some­thing that a non-math­e­mat­i­cian will get some­thing out of. Give ex­am­ples of things that are and are not real num­bers.

• I’m pretty tired right now, but this defi­ni­tion seems kind of cir­cu­lar to me. It in­volves an in­finite sum, and in­finite sums are defined in terms of limits. But a limit of ra­tio­nal num­bers is defined in terms of the set of real num­bers. Maybe it would be bet­ter to pre­sent the defi­ni­tion of real num­bers that one would find in a real anal­y­sis text.

• Hey Kevin. I think I ac­ci­den­tally clicked ig­nore on your query. I’ll look into that defi­ni­tion. I think po­ten­tially this defi­ni­tion isn’t cir­cu­lar, but you might be right. I don’t know quite how to put it, but it seems like this crit­i­cism could be di­rected at any al­gorithm that at­tempts to point to­wards all real num­bers. The al­gorithm, it seems, would have to keep ges­tur­ing closer and closer to the real num­ber it is try­ing to in­di­cate. It’s not as if the al­gorithm can take the set of real num­bers for granted, and then say that we just need to find the num­ber in the set of real num­bers that satis­fies the con­di­tion that we keep get­ting closer and closer to it, and then fi­nally we add that to the set of real num­bers. And yet, any al­gorithm that doesn’t have a set in mind that it can pull the an­swer from, I think it will in­volve some­thing along the lines of point­ing to­wards a num­ber that doesn’t yet come from a well-defined spot. Other­wise, the al­gorithm can’t cre­ate the well-defined spot. That might all be ut­ter non­sense though.

• I un­der­stand what you’re say­ing and I think it’s a good point. The prob­lem is that you’re de­vel­op­ing an al­gorithm (a non-ter­mi­nat­ing one) that finds real num­bers rather than pro­vid­ing a defi­ni­tion of them. It turns out that pro­vid­ing a defi­ni­tion of real num­bers is not a sim­ple as it may at first seem. This pre­sen­ta­tion is some­what similar con­struc­tive anal­y­sis, in which a real num­ber is defined as reg­u­larly con­verg­ing se­quence of ra­tio­nal num­bers; im­por­tantly, con­struc­tive anal­y­sis does not define real num­bers as in­finite sums of these se­quences, be­cause as I’ve said, that would be a cir­cu­lar defi­ni­tion.

If you want to learn more about rigor­ous foun­da­tions for real num­bers and re­lated top­ics, I think that the book Calcu­lus by Michael Spi­vak is a very ap­proach­able and well re­spected in­tro­duc­tion to the topic.

• Is an in­finite sum of ra­tio­nals iso­mor­phic with a reg­u­larly con­verg­ing se­quence of ra­tio­nals (some­thing along the lines of 12, 58, 1116, etc.), where each ra­tio­nal in the se­quence is the sum of all the ad­dends up un­til then? I agree it is prob­a­bly worth putting up a differ­ent defi­ni­tion any­way. I’m not sure I’ll be able to do that for a lit­tle bit, since I haven’t stud­ied real anal­y­sis yet, but if you want to do that sooner, go for it. This is a fun con­ver­sa­tion!

• This defi­ni­tion of the real num­bers has a big­ger prob­lem with it than just cir­cu­lar logic — it also runs into the 0.9999… = 1 para­dox. The sets $$\mathbb{N} \setminus \{1, 2, 3, 4, 5\}$$ and the set $${5}$$ both en­code the num­ber $$1/8$$.

Nor­mally the real num­bers are defined us­ing ei­ther Dedekind cuts or Cauchy se­quences of ra­tio­nal num­bers. Could we please use one of those defi­ni­tions in­stead, as they’re the stan­dard ones used by most math­e­mat­i­ci­ans?

• Sorry about my in­ac­tivity on Ar­bital, and thanks for go­ing ahead and fix­ing it!

• This is not a very good sum­mary, since it re­lies on the reader un­der­stand­ing what a “com­plete num­ber line” is.

• I thought so too when I wrote it up; I put it there as a place­holder for a Wikipe­dia-style ini­tial defi­ni­tion once we find one that’s more suit­able, be­cause I’m hav­ing a hard time think­ing of one.