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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!

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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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2 Answers 2

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?
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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
    
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
    
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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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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We start with natural numbers $$\mathbb N=\{0,1,2,...\}$$ then expand to ad them negative numbers TO get 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 antic greks want to find diagonal of quadratic of size 1 that is $\sqrt2$ then Pitagora or Euclid proved that $\sqrt2$ can not be writen as ratio of two whole numbers. So $\sqrt2$ is not a rational numbers. All numbers that can not be writen as ratio of whole numbers we call irrational numbers. The set of irational 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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