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We see all the time conjectures and proofs that specific (real) numbers are (more often than not) irrational. I'm wondering that apart from the mathematical curiosity motivating such proof attempts, are there technical reasons to consider the rationality or irrationality of certain real numbers. That is, are there real numbers whose rationality or irrationality if proven will have important consequences to mathematics (apart from the fact in itself that the numbers are rational or irrational)?

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  • $\begingroup$ One thing to consider for example is that all rational numbers can be "constructed". This is not true for most irrationals. That kind of thing you are looking for? $\endgroup$ – user88595 Apr 6 '14 at 13:57
  • $\begingroup$ I'm thinking of something like is there an instance of the following kind of implication: "x: rational $\Rightarrow$ 'Very-important-theorem-A' holds" $\endgroup$ – DancefloorTsunderella Apr 6 '14 at 14:01
  • $\begingroup$ Are you sure you mean irrational and not transcendent? Irrational just means they aren't a fraction, while transcendent means that a number isn't a zero of any polynomial with rational coefficients. AFAIK proving a number to be transcendent is usually harder than merely proving it irrational. $\endgroup$ – fgp Apr 6 '14 at 14:02
  • $\begingroup$ Well, we could consider both irrationality and transcendence. I'm just wondering if we're trying to prove numbers irrational (and transcendent) just for the fun of it, or are there practical applications for such results. $\endgroup$ – DancefloorTsunderella Apr 6 '14 at 14:04
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Yes. If you prove that the real part of the first zero of the Riemann zeta function off the critical line is irrational, you will get the Fields medal.

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    $\begingroup$ But what about all those retired engineers who have already solved the problem in a few pages? They won't get the Fields medal. $\endgroup$ – Casteels Apr 6 '14 at 14:07
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    $\begingroup$ Well, if I prove that, I won't get the Fields medal. $\endgroup$ – Daniel Fischer Apr 6 '14 at 14:12
  • $\begingroup$ @DanielFischer, Dancefloor has only been around for 1 year and 2 months. Plenty young to get the medal. $\endgroup$ – Mikhail Katz Apr 6 '14 at 14:13
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    $\begingroup$ @Daniel, I have an idea: let's prove it and publish it in a Physics journal, then we can get a Nobel prize. $\endgroup$ – Mikhail Katz Apr 6 '14 at 14:17
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    $\begingroup$ Let's publish it in a Mathematics journal, then I take the Abel prize, and you can have the Fields medal ;) $\endgroup$ – Daniel Fischer Apr 6 '14 at 14:20

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