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vilidice wrote:The best way for a layman to approach it is probably with triangles (really most of the circle stuff "comes from" trig anyway):
but using trig Lambert's proof is probably the easiest to work through:
http://en.wikipedia.org/wiki/Proof_that ... irrational
a more extensive discussion is found here:
http://paramanands.wordpress.com/2011/0 ... rts-proof/
tomandlu wrote:I realise that there's no way to ask this question without it seeming really stupid, but I'll give it a go anyway... is there some intrinsic quality of a circle that dictates that Pi must be an irrational number or is it just 'bad luck'?
SunAvatar wrote:tomandlu wrote:I realise that there's no way to ask this question without it seeming really stupid, but I'll give it a go anyway... is there some intrinsic quality of a circle that dictates that Pi must be an irrational number or is it just 'bad luck'?
It's not a stupid question at all. There is proof that π is irrational (in fact several proofs are on Wikipedia), but this is incidental to what you are really asking. Ultimately what it comes to is this: almost all real numbers are irrational, in the sense that if you randomly plop down an infinite string of decimal digits, one after another, you will almost certainly not get a rational number. Since there's no particular reason that π must be rational, it would be a coincidence bordering on the miraculous if it were. It turns out that no such miracle occurs, and so π is irrational.
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gmalivuk wrote:No more than it's a coincidence that there are rational numbers. And similarly, a random set of three integers is highly unlikely to be a Pythagorean triple. (In the sense that, as n goes to infinity, the fraction of integer points in the box [-n,n]^{3} that are Pythagorean triples goes to zero.)
Proginoskes wrote:Actually, it goes deeper than that. There are solutions to a^2 + b^2 = c^2 in positive integers, but no solutions to
a^3 + b^3 = c^3 in positive integers. It's not the existence of rational numbers; it's the fact that three of them "conspire" to make an equation like
a^2 + b^2 = c^2 possible.
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