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I only know that the rational and irrational are together called real numbers. Rational numbers can be written in the form $\frac {p}{ q}$ ,where $p \in \mathbb Z$ and $q \in \mathbb Z$.

Are there any additional information which I left? If 'yes' then please tell me the alternative definitions.How to answer it in an interview? Thank you in advance.

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    p and q are co prime – User8976 Jul 24 '16 at 17:02
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    Related: http://math.stackexchange.com/questions/261074/true-definition-of-the-real-numbers – Arpit Kansal Jul 24 '16 at 17:03
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    real numbers are by definition limits of converging (Cauchy) sequences of rationals. think to the decimal expansion – reuns Jul 24 '16 at 17:05
  • The real numbers can be constructed by means of Dedekind cuts. If you'd like to learn more about this, you can read about it in any good real analysis textbook. – Danny Jul 24 '16 at 17:14
  • @Danny seriously, your advice is to look at the Dedekind cuts ?.. – reuns Jul 24 '16 at 17:23
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    @user1952009 Why is that bad advice? – Danny Jul 24 '16 at 17:29
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    @user1952009: I would say that Dedekind cuts are a more natural definition of $\mathbb R$ than Cauchy sequences, mainly because an irrational number has one and only one Dedekind cut. But it's really just a question of personal preference. – TonyK Jul 24 '16 at 17:42
  • @TonyK yes you are right, let's learn the Dedekind cuts (a purely theorical object) before the infinite decimal expansion and the series expressions for $\sqrt{2},\ln 2, e,\pi,\ldots$ – reuns Jul 24 '16 at 17:50
  • @user1952009: Students already know what a decimal expansion is before they have to learn a rigorous definition of $\mathbb R$, so it's just a question of conceptual simplicity. And in my book, a subset of $\mathbb Q$ is a simpler concept than an equivalence class of Cauchy sequences of rationals. (And whatever your view on the matter, there is no need for sarcasm here.) – TonyK Jul 24 '16 at 18:50
  • @TonyK the sarcasm is because you can't understand such a theoretical object as the Dedekind cuts before knowing the Cauchy sequences construction of $\mathbb{R}$ – reuns Jul 24 '16 at 19:13
  • @user1952009: Yes I can. And did, in fact. – TonyK Jul 24 '16 at 19:15

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What you need to notice first for this is that some sequences of $ \mathbb{Q} $ do not converge to a value in $ \mathbb{Q} $. In fact, for every real number, you can find a sequence of $ \mathbb{Q} $ that converges to that number (for example the different fractional series converging to $ \pi $). The idea is to create the set of "limits" of sequences of rational numbers.

However, you can't just write $ \displaystyle\mathbb{R} = \{\lim_{n\rightarrow+\infty}u_n, u\in\mathbb{Q}^\mathbb{N}\} $, since it doesn't mean anything until you've defined $ \mathbb{R} $, because the limit itself isn't defined.

What you need is to create an equivalence relation over the "converging" subset of $ \mathbb{Q}^\mathbb{N} $ and identify the equivalence classes as elements of $ \mathbb{R} $. But first you'll need Cauchy sequences, which can be used to show that sequences have a converging behaviour but without specifying their limits.

We say that a sequence $ u $ is a Cauchy sequence iff $ \forall\varepsilon>0, \exists N, \forall n \ge N, \forall m \ge N, |u_n - u_m |<\varepsilon $ It is easy to show that if $ u $ converges in $ \mathbb{Q} $, then it is a Cauchy sequence.

Let $ A $ denote the subset of $ \mathbb{Q}^\mathbb{N} $ of Cauchy sequences. Let $ \sim $ be a relation on $ A $, with $ \displaystyle u \sim v \iff \lim_{n\rightarrow+\infty}u_n - v_n = 0 $ (we can define this properly because $ 0 \in \mathbb{Q} $ and $ u - v \in \mathbb{Q}^\mathbb{N} $).

It is then easy to show that $ \sim $ is and equivalence relation :

  • Reflexivity : $ \displaystyle\forall u \in A, \lim_{n\rightarrow+\infty}u_n - u_n = 0 $
  • Symmetry : $ \displaystyle \forall (u, v) \in A^2, \lim_{n\rightarrow+\infty}u_n - v_n = 0 \implies \lim_{n\rightarrow+\infty}v_n - u_n = 0 $
  • Transitivity : if $ u \sim v $ and $ v \sim w $, then we have $ \displaystyle \lim_{n\rightarrow+\infty}u_n - w_n = \lim_{n\rightarrow+\infty}u_n - v_n + v_n - w_n = 0 $.

Then $ \mathbb{R} = \{[u], u \in A\}$ is the set of equivalence classes of $ \sim $, and every Cauchy sequence of rational numbers defines a real number.

This construction is due to Cantor if I remember correctly. There are some other constructions using Dedekind cuts for example, but this is easy enough to understand (but no so much to use).

Josselin Poiret
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