# Definition of a Real number [duplicate]

Possible Duplicate:
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 MyersonAug 10 '12 at 12:04

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