All Rational Approximations of Pi Are Useless

Re:All Rational Approximations of Pi Are Useless by pfortuny on Tuesday October 02, 2012 @06:01AM (#4601296)

Sorry to nag but the 'goodness' of 22/7 is due to the size of the denominator, not to the number of digits you have to memorize (actually 2, not 3, but anyway). You get less than a .001 relative error with a denominator less than 10, that is why it is a good approximation: nothing to do with memory.

The thing is: as pi is transcendental, there are very very good rational approximations in that sense (this is an old theorem due to Liouville): http://mathworld.wolfram.com/LiouvillesApproximationTheorem...., there is no 'memorizing' going on there.

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Re:All Rational Approximations of Pi Are Useless by lutusp on Tuesday October 02, 2012 @06:21AM (#4601325)

> The thing is: as pi is transcendental, there are very very good rational approximations in that sense (this is an old theorem due to Liouville) ...

Expressed another way, for every estimate of Pi's value, however large, there are two integers that, expressed as a ratio, will produce the same result.

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Re:All Rational Approximations of Pi Are Useless by pfortuny on Tuesday October 02, 2012 @06:42AM (#4601357)

No, no, not at all: the meaning of the theorem is that there are amazingly accurate approximations for small sized denominators, that is.

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Re:All Rational Approximations of Pi Are Useless by lutusp on Tuesday October 02, 2012 @01:03PM (#4602989)

I wasn't trying to summarize Liouville's Theorem, I was expressing a different idea. I didn't make that clear.

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