How and where to learn the properties of numbers and really understand them?

Question

I am a university student and was reading through the first chapter of Rudin's Principles of Mathematical Analysis to improve my mathematical skills, but I was immediately stumped when the author started talking about rational numbers and the proof that p^2 = 2 is not satisfied by any rational number p, and then he goes on about why it is so using an argument about odd and even numbers.

This sort of made me realise how poor my understanding is of numbers, as well as many other basic fundamental axioms in math, when I could not even get past the first paragraph without getting confused... XDDD

I want to relearn the numbers and other basic math properties from scrath -- what book would be best to do that? I want a rigorous book like rudin which will teach me how to think properly like a mathematician. Thanks for the help

Answer 1

Terence Tao's "Analysis I". Title says "Analysis", but it really starts off with a rigorous treatment of sets and numbers.

Answer 2

It is partly a matter of taste, but I would recommend Spivak's book Calculus. As well as giving a rigorous treatment of calculus, this book has appendices that deal specifically with the construction of the real numbers. The definition is given and the various properties that we expect the real numbers to satisfy are deduced from the definition. Moreover he gives enough discussion to see what the issues are and to explain what he is doing in proofs. I find it an enjoyable read.

Other items of interest that you won't find in other Calculus books include a proof that $\pi$ is an irrational number. He also includes recommendations for further reading.

As for the particular example you raise (there is no rational number $p$ such that $p^2=2$), this amounts to the statement that $\sqrt{2}$ is irrational, which has been explained in other MSE questions. See Square root of $2$ is irrational.