# True Definition of the Real Numbers

I've found lots of resources that say this is a real number if it's not rational, but what is a real number, really? I mean what is the definition of a real number? If nothing else, anyone know of a resource where I could find out myself?

Thanks!

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A real number may be defined to be an equivalence class of Cauchy sequence of rational numbers. Alternatively, it can be thought of as a Dedekind cut. – Eric Naslund Dec 17 '12 at 23:45

There is no "true" definition of the real numbers because there are several ways to think of the real numbers either as mathematical notions (i.e. we don't really care what are the objects which represent the numbers, we just care about the structure) and there are concrete ways to construct the real numbers, e.g. as sets of rational numbers or equivalence classes of sequences.

The structure of the real numbers is unique. It is an order field which is order-complete. It is also the unique complete Archimedean field. This means that if we construct any other field which is ordered and order complete, then we built something which is isomorphic to the real numbers.

Generally speaking, if we accept the rational numbers as "atomic" (namely, objects whose existence we take for granted, and do not investigate further) then the real numbers can be constructed either as particular sets of rationals, called Dedekind cuts, or as equivalence classes of Cauchy sequences.

It is a nontrivial task (at least without seeing it a couple of times before) to prove that either definition gives us this structure we seek. That complete ordered field. It is even less trivial to actually prove the uniqueness of that structure. I won't go into either subjects.

In either definition we can find the rationals are embedded into the real numbers, and in most cases we think about the rationals as being part of the real numbers as much as we think about integers being rational numbers.

One final remark is that if one prefers not to accept the rational numbers as atomic then it is possible to construct them from the integers, and we can construct those from the natural numbers, and in fact we can construct those just from the empty set.

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When you say true, dude, what does that really mean? – Will Jagy Dec 18 '12 at 20:03
Will, in which part? – Asaf Karagila Dec 18 '12 at 20:04
What is the meaning of Christmas? As an Israeli I always thought it meant giving the non-Jewish Russians a day off midweek, and the non-observant Jews have a senseless reason to drink without feeling guilty. Hellenism is fun. – Asaf Karagila Dec 18 '12 at 20:20
And Ramanujan's Birthday the next day. If there is a next day. – Will Jagy Dec 18 '12 at 20:28
Will, to paraphrase Pearl Jam, we're still alive. – Asaf Karagila Dec 30 '12 at 20:31

We start with natural numbers $$\mathbb N=\{0,1,2,...\}$$ then expand to ad them negative numbers to get the set of whole numbers $$\mathbb Z=\{...,-2,-1,0,1,2,...\}$$. Next step is definition of division of numbers and if we divide two whole numbers result is not always whole number then we define the rationals or fractions $$\mathbb Q=\{\frac{a}{b}:a,b\in\mathbb Z,b\neq 0\}$$ Cantor proved that all three sets are equivalent or countable. Historically problem came when ancient greeks want to find diagonal of square of size 1 that is $\sqrt2$ then Pitagora or Euclid proved that $\sqrt2$ can not be written as ratio of two whole numbers. So $\sqrt2$ is not a rational numbers. All numbers that can not be written as ratio of whole numbers we call irrational numbers. The set of irrational numbers we denote by $$\mathbb I$$ Cantor proved that the set $\mathbb I$ is uncountable finally the set of real numbers is $$\mathbb R=\mathbb Q\cup\mathbb I$$

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I'm not sure I like this idea of giving the irrational numbers their own notation. As far as I know, the irrational numbers cannot be made into an interesting algebraic structure in their own right, and hence probably don't deserve a notation other thand $\mathbb{R} \setminus \mathbb{Q}.$ Just my 2 cents. – goblin 2 days ago

Here is Whitehead & Russell's definition of real number in Principia Mathematica, whose simplicity has not been rivalled by any other constructions of reals.

*1. A ratios is $\mu/\nu$ where $\mu$ and $\nu$ are cardinal numbers (natural numbers).

*2 Let $H$ be the relation "less than" over the field of all ratios.

*2.1 A segment of $H$ is a set of ratios whose members are less than some members (not necessarily all) of another set of ratios.

*3 A real number is not really a number; it is a segment of $H$.

*3.1 Some segments of $H$ are all the ratios less than a single term in $H$, they are called rationals. E.g. "All the ratios less than 1/2."

*3.2 Some other segments of $H$ cannot be defined as the lessor ratios of a single term, but can only be defined as the lessor ratios of another set. They are called irrationals. E.g. "All the ratios whose square is less than 2."

For more details see this post or Chapter XXXIII of The Principles of Mathematics by Bertrand Russell.

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How is this different from the Dedekind Cut construction? – Nishant Dec 19 '14 at 20:39
@Nishant - Almost exactly the same. Does modern construction by Dedekind cuts still use an axiom? – George Chen Dec 20 '14 at 0:15
I think it uses most ZF axioms. – Nishant Dec 20 '14 at 0:54
Between two cuts that has nothing in between, Dedekind postulated that there is one and only one number. Is this axiom still being used by modern theory? If not, then construction by Dedekind cuts as we know it today must be using W&R's idea which in turn gave much credit to Cantor. Of course, no one disputes that Dedekind is the giant who shoulders them all. – George Chen Dec 20 '14 at 12:37
But there are no two distinct cuts with nothing in between...in fact given any two distinct cuts, there are infinitely many rationals in one and not the other. – Nishant Dec 20 '14 at 15:01