Hi everyone, I am studying calculus right now from a book called ''The Principles of Mathematical Analysis'' by Walter Rudin
The subject is Real and Complex number systems. And I didn't understand some paragraphs.I attached a picture, you can see there the paragraph.I understood the first 3 paragraphs where he proves that p2=2 there is no rational p that satisfies because if p=m/n=2 , a rational number m can be divided by 4 and n can be divided 2 therefore it's not a rational number according to the definition of rational number.
Now the part that I didn't really understand is after that , the paragraph that starts ''We examine the situation closely..'' and the rest of the text. I didn't understand how he came to those 2 equations that defined p and q. And how he proved that there is a q>p in group A and in group B there is q<p.
I would appreciate if someone can help me understand this part.Thanks.
Re: Need help
They are just using the constraints listed (i.e. p^2 < 2). Note that (p^2 - 2)/(p+2) = (p-2). It is just a clever way of picking a q such that you can pick a p that satisfies the properties for q (making q < p or p > q).
They picked a q in terms of a p where by they can apply the constraints and prove the result. The particular decomposition they used was picked solely to prove what they were intending to prove.