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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

Sahiba Arora
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  • Try looking into the Peano Postulates -- the basic motivating idea is asking yourself why things like $2+2=4$ hold instead of $2+2=5$, and what exactly is meant by that statement. It all depends on context; when working in the 'integers modulo 1", both of the above statements are true, but in the standard integers one of them is clearly false. What addition, multiplication, exponentiation, etc. all mean depends entirely on what you have defined them to mean in the structure you wish to talk about -- in the rational numbers under rational multiplication, we can't find a square root of two. – Alec Rhea Aug 07 '17 at 10:40
  • instead of Rudin try to read others books as Understanding analysis of Abbott or any other great introductory books of real analysis. Analysis I of Tao or the book of analysis of Pugh are also very good choices. – Masacroso Aug 07 '17 at 11:29

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

Max Herrmann
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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.

Shane O Rourke
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