Princeton Companion to Mathematics

Metadata

• Author: sites.math.rutgers.edu
• Title: Princeton Companion to Mathematics

Page Notes

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Highlights

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• : it is easy to find twoirrational numbers a and b such that a + b is rational,or such that ab is rational (in both cases one could takea = √2 and b = −√2), but is it possible for ab to berational? Here is an elegant proof that the answer is yes.Let x = √2√2 . If x is rational then we have our example.But x√2 = √2 2 = 2 is rational, so if x is irrational thenagain we have an example.Now this argument certainly establishes that it ispossible for a and b to be irrational and for a b tobe rational. However, the proof has a very interestingfeature: it is nonconstructive, in the sense that it doesnot actually name two irrationals a and b that work.Instead, it tells us that either we can take a = b = √2or we can take a = √2√2 and b = √2. Not only does itnot tell us which of these alternatives will work, it givesus absolutely no clue about how to find out. — Updated on 2024-04-26 12:20:48
  • This is a great observation that I don’t see myself being philosophically troubled by. Proofs of these for prove what is asked: does something exist that works? It is not asking for an example (directly).
• we prove things not only to be surethey are true but also to gain an idea of why they aretrue — Updated on 2024-04-26 12:21:47

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• : it is easy to find twoirrational numbers a and b such that a + b is rational,or such that ab is rational (in both cases one could takea = √2 and b = −√2), but is it possible for ab to berational? Here is an elegant proof that the answer is yes.Let x = √2√2 . If x is rational then we have our example.But x√2 = √2 2 = 2 is rational, so if x is irrational thenagain we have an example.Now this argument certainly establishes that it ispossible for a and b to be irrational and for a b tobe rational. However, the proof has a very interestingfeature: it is nonconstructive, in the sense that it doesnot actually name two irrationals a and b that work.Instead, it tells us that either we can take a = b = √2or we can take a = √2√2 and b = √2. Not only does itnot tell us which of these alternatives will work, it givesus absolutely no clue about how to find out. — Updated on 2024-04-26 12:20:48

  • This is a great observation that I don’t see myself being philosophically troubled by. Proofs of these for prove what is asked: does something exist that works? It is not asking for an example (directly).

• we prove things not only to be surethey are true but also to gain an idea of why they aretrue — Updated on 2024-04-26 12:21:47

• Roughly speaking, the axiom ofchoice says that we are allowed to make an arbitrarynumber of unspecified choices when we wish to forma set. — Updated on 2024-04-27 18:24:31

• the general view in mainstream math-ematics is that, although there is nothing wrong withusing the axiom of choice, it is a good idea to signalthat one has used it, to draw attention to the fact thatone’s proof is not constructive. — Updated on 2024-04-27 18:31:34