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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!

Asaf Karagila
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Jonathon
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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
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    The "lots of resources" are wrong: the rational numbers are also real. – user Mar 04 '19 at 23:01

1 Answers1

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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.

To read more:

  1. Completion of rational numbers via Cauchy sequences
  2. question about construction of real numbers
  3. Constructing $\mathbb R$
  4. Why does the Dedekind Cut work well enough to define the Reals?
  5. Construction of $\Bbb R$ from $\Bbb Q$
  6. In set theory, how are real numbers represented as sets?
Community
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Asaf Karagila
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    When you say true, dude, what does that really mean? – Will Jagy Dec 18 '12 at 20:03
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    Will, in which part? – Asaf Karagila Dec 18 '12 at 20:04
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    Asaf, I'm just giving you trouble. I had hoped the word dude would be enough to signal that. And, of course, maybe you are just returning the favor. – Will Jagy Dec 18 '12 at 20:10
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    Oh, "dude" certainly signaled that. I'm just wondering, because I only used the word "true" once and it was in quotation marks. – Asaf Karagila Dec 18 '12 at 20:12
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    I noticed that. It was in the title of the original question, the True Definition, as in The True Meaning of Christmas. So perhaps I ought to have written When we say true, dude...Meanwhile, you did use quotation marks. Your street cred is safe. – Will Jagy Dec 18 '12 at 20:19
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    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
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    Or possibly hedonism – Will Jagy Dec 18 '12 at 20:24
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    Well, combine the two of them with the impending doom which awaits us in four days... and we should be having End of the World Debauchery parties. – Asaf Karagila Dec 18 '12 at 20:25
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    And Ramanujan's Birthday the next day. If there is a next day. – Will Jagy Dec 18 '12 at 20:28
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    Maybe that is why the world ends. – Asaf Karagila Dec 18 '12 at 20:32
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    Will, to paraphrase Pearl Jam, we're still alive. – Asaf Karagila Dec 30 '12 at 20:31
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    I think you're right. Mind you, I've felt better. – Will Jagy Dec 30 '12 at 21:16
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    It is very interesting to know that the whole edifice of analysis is built with empty sets. +1 I also like the idea that once you a define a set that contains "nothing" you have created "something" (namely the set that contains nothing). – Paramanand Singh Jul 28 '17 at 04:40
  • I know that you are only stating the universally accepted view of the matter (and that you are stating it correctly, of course!), but do you (or does anybody) really believe that there are lots of numbers $\pi$? Isn't there just one number $\pi$? – Calum Gilhooley Aug 05 '17 at 13:17
  • I'm guessing an answer that just states that we can take the rational numbers for granted works well for experts because it's so easy to show that for each integer p and positive integer q where gcf(p, q) = 1 and $p \div q$ is not an integer, we can invent the number $p \div q$ and for any integer p and nonzero integer q, $p \div q$ means the number you get by reducing it so that the gcd of them is 1 and the second number is positive and then taking their invented quotient. Now we can invent +, $\times$, and $\leq$ on them and show that it's an ordered field. – Timothy Dec 13 '18 at 20:15