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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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    Yes, your teacher was being circular there. Anyway, have a look at [this](http://en.wikipedia.org/wiki/Construction_of_the_real_numbers).2012-08-09
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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.2012-08-09
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    So, no more $\pi$? :(2012-08-09
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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? :O2012-08-09
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    A real number is a number whose square is a positive number, not necessarily an integer.2012-08-09
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    If you want a solid 'definition' of the real numbers, you might want to check out 'Dedekind Cuts.'2012-08-09
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    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.2012-08-09
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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.2012-08-09
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    "no greatest member" - think about the set $\{{.9,.99,.999,.9999,\dots\}}$. It has no geatest member.2012-08-09
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    ah! got it..then all the irrationals will inevitably be covered .. wow an ingenious idea really2012-08-09

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