as for the theorem to construct R from Q I am reading Rudin's principle of mathematics analysis and I am confused about the  theorem 1.19 in chapter 1. How can he construct R from Q. I don't know what are he doing,  how can he construct the irrational number? I haven't seen it. Can anyone explain to me?
 A: Rudin's Theorem 1.19 states the existence of a field containing the rational numbers, and then tells you he is calling it the field of real numbers. But he does not go into any details there. Instead, he states that if you are interested, look at the 'bit tedious' proof found in the Appendix.
It turn out that NO HARM IS DONE - one can take a synthetic approach in describing the real numbers. Not only do we have the existence of these numbers, we also have uniqueness - only one such ordered number field exists (up to an isomorphism) satisfying the lub property. 
So you can take it on faith - by studying and understanding the synthetic axioms, you are not 'missing anything', in the sense that there are no properties of the real numbers which you do not know about.
To develop this ever so slightly, notice that in Proposition 1.18 you are told that from the axioms, $0 < 1$. Going further, you could show that our field contains something that acts and feels just like the set of integers. Continuing, since you have multiplicative inverses in a field, you 'must have' the rational numbers. And, if you have considerable amount of time on your hands, you can prove that it can only be 'completed' in one way - $\mathbb{R}$.
If you need a way to get a visual of a real number, think about infinite decimal expansions and look at Wikipedia 0.999999999999.... The article also explains how Dedekind cuts forces you to accept that $.9999999999... = 1$.

