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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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marked as duplicate by J. M., joriki, t.b., Nate Eldredge, Gerry Myerson Aug 10 '12 at 12:04

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Yes, your teacher was being circular there. Anyway, have a look at this. –  J. M. Aug 9 '12 at 6:25
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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 9 '12 at 6:26
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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 9 '12 at 6:36
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A real number is a number whose square is a positive number, not necessarily an integer. –  lab bhattacharjee Aug 9 '12 at 6:47
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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 9 '12 at 7:05

1 Answer 1

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

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