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Completion of rational numbers via Cauchy sequences
What is a Real Number?

Today my teacher first defines irrational numbers saying its the set R-Q , then she says union of rational and irrational numbers is the set of real numbers.

She uses real numbers to define irrationals and then vice versa, which i find quite ridiculous

So what exactly are real numbers?

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user151203
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  • Yes, your teacher was being circular there. Anyway, have a look at this. – J. M. ain't a mathematician Aug 09 '12 at 06:25
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    There is no reason to think that (in the "then says" part) the teacher is defining the reals to be the union of the set of the rationals and irrationals. It is just a minor observation. The set of reals numbers is likely not properly defined at all, at least if this is a high school course. – André Nicolas Aug 09 '12 at 06:26
  • So, no more $\pi$? :( – Salech Alhasov Aug 09 '12 at 06:35
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    so we keep on using real numbers everywhere and like what , we dont even get to know its proper definition in school life? :O – user151203 Aug 09 '12 at 06:36
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    A real number is a number whose square is a positive number, not necessarily an integer. – lab bhattacharjee Aug 09 '12 at 06:47
  • If you want a solid 'definition' of the real numbers, you might want to check out 'Dedekind Cuts.' – RougeSegwayUser Aug 09 '12 at 06:54
  • Wolfram says 'A set partition of the rational numbers into two nonempty subsets S1 and S2 such that all members of S1 are less than those of S2 and such that S1 has no greatest member.' what does it mean by no greatest member?? hows that even possible unless all are the same. – user151203 Aug 09 '12 at 07:05
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    People used real numbers for thousands of years before anyone came up with what we would accept today as a proper definition. It's more subtle (and less important, for day-to-day stuff) than you might think. – Gerry Myerson Aug 09 '12 at 07:05
  • "no greatest member" - think about the set ${{.9,.99,.999,.9999,\dots}}$. It has no geatest member. – Gerry Myerson Aug 09 '12 at 07:06
  • ah! got it..then all the irrationals will inevitably be covered .. wow an ingenious idea really – user151203 Aug 09 '12 at 07:08
  • @GerryMyerson "It's more subtle (and less important, for day-to-day stuff) than you might think" The definition is actually surprisingly short and clear: The real numbers are a complete ordered field. This defines the real numbers up to a natural isomorphism. – Filippo Sep 16 '21 at 11:59
  • @Filippo, you're joking, aren't you? Do you think the user who posted the question nine years ago would have had any clue what a "complete ordered field" might be? or what a "natural isomorphism" might be? At the level at which the user was at the time, it would take weeks to explain those concepts in a way the user might understand. What's "short and clear" to you and me is very, very subtle to someone operating at ground level. – Gerry Myerson Sep 16 '21 at 13:01
  • @GerryMyerson Well, my maths professor "defined" the real numbers by constructing them from the natural numbers (which in turn are defined up to some isomorphism) and to me this made the real numbers very difficult to grasp. I think it's very important to tell students about the characterization as an ordered field (with some special property). Hence my comment. – Filippo Sep 16 '21 at 15:52
  • Looking back, I'd say that the construction from the natural numbers is merely a proof of existence (based on the axioms of set theory). – Filippo Sep 16 '21 at 15:57

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The "standard" way to construct the real numbers in analysis would be the completion of $\mathbb{Q}$ with respect to cauchy sequences. I.e. add just as many elements to the rational numbers, such that every cauchy sequence has a limit in your space. However, there are many ways to construct the real numbers, you might wanna check out http://en.wikipedia.org/wiki/Construction_of_the_real_numbers . Considering the rational numbers, the most common way to construct them (as far as I'm concerned) is algebraically, which is outlined here: http://en.wikipedia.org/wiki/Rational_number#Formal_construction

roman
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