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Originally posted by XanthosNZWhy? Are you sure? Irrationality certainly doesn't require that.
I should clarify:
The digits of pi contain every possible sequence of digits including e to any number of significant digits you wish.
EDIT. Looks like it's an open question - http://en.wikipedia.org/wiki/Pi#Open_questions - seen as how it's not known even whether the numbers 0-9 each occur infinitely in pi's decimal expansion. Unless you have more uptodate mathematical knowledge, Xanthos?
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Originally posted by TommyCYes, the number
Why? Are you sure? Irrationality certainly doesn't require that.
0.1100010000000000000000010000...
(where the nth 1 is in the n! place)
has been proven to be transcendental (and therefore irrational) and there are plenty of integer sequences it does not contain.
I could be wrong, but I thought that questions regarding the decimal expansion of pi were quite difficult. I'm not sure what Xanthos says in his last post has actually been proven mathematically. (Edit: although it's very plausible and empirical evidence certainly backs it up.)
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Originally posted by TommyCPi is irrational so it therefore contains an infinite number of digits. Pi also does not repeat in any way. Therefore it must contain every finite string somewhere within it.
Why? Are you sure? Irrationality certainly doesn't require that.
EDIT: To emperically prove it you'd have to show that the digits of pi are in fact random. This is pretty much impossible to prove but it can be shown that pi passes every test of randomness you wish to throw at it.
Originally posted by XanthosNZSee edit to previous post - you are incorrect, it is not (yet) a fact that pi contains every sequence. In fact, according to wikipedia, it's not known whether the numbers 0 to 9 do occur infintiely in it (an obvious prerequisite for your claim.)
Pi is irrational so it therefore contains an infinite number of digits. Pi also does not repeat in any way. Therefore it must contain every finite string somewhere within it.
EDIT: To emperically prove it you'd have to show that the digits of pi are in fact random. This is pretty much impossible to prove but it can be shown that pi passes every test of randomness you wish to throw at it.
I don't follow your logic, btw. Surely it would be easy to construct a non-repeating, irrational number without certain digits in it?
Originally posted by TommyCYes. Just use 0 and 1s in any non-repeating fashion.
Surely it would be easy to construct a non-repeating, irrational number without certain digits in it?
eg. 0.101001000100001000001000....
But the actual example I gave above was transcendental (unnecessary I know!).
Originally posted by TommyCYou could only construct such a number by instituting rules into the number (9 can never follow 6). If it can be shown that pi is random then obviously such rules do not exist. The fact that spigot algorithm exists for pi is an extremely strong indicator that this is true.
See edit to previous post - you are incorrect, it is not (yet) a fact that pi contains every sequence. In fact, according to wikipedia, it's not known whether the numbers 0 to 9 do occur infintiely in it (an obvious prerequisite for your claim.)
I don't follow your logic, btw. Surely it would be easy to construct a non-repeating, irrational number without certain digits in it?
Originally posted by XanthosNZA mathematical statement is either proven or not. Establishing truth is not an exercise in offering up impressionistic evidence to the subjectivities of a jury; it is a watertight activity. Your previous claims could be reasonably modified by the world "probably", however.
You could only construct such a number by instituting rules into the number (9 can never follow 6). If it can be shown that pi is random then obviously such rules do not exist. The fact that spigot algorithm exists for pi is an extremely strong indicator that this is true.
Originally posted by TommyCI agree. I should have prefaced my statement with a probably. But I can never keep track of what has and hasn't been mathematically proven.
A mathematical statement is either proven or not. Establishing truth is not an exercise in offering up impressionistic evidence to the subjectivities of a jury; it is a watertight activity. Your previous claims could be reasonably modified by the world "probably", however.
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Originally posted by XanthosNZMy undergraduate degree was in mathematics and frankly, I couldn't even understand what was meant to be proven or not most of the time, let alone anything more ambitious.
I agree. I should have prefaced my statement with a probably. But I can never keep track of what has and hasn't been mathematically proven.
Btw, I was thinking about these 'normal numbers' (new to me as of this morning) - numbers whose decimal part contains all number sequences - just now on the tube. It's kind of anti-intuitive that they exist until you remember the cardinality of the natural numbers, & that the size of the set of all finite number sequences must be aleph zero too. But I was wondering more specifically, what is precise definition of a 'random decimal' and is it sufficient for normality? Or just necessary? And across all bases . . . ? EDIT. And how to prove 'random', unless deliberately constructed as such from definition (ie unlike most important numbers)?
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Originally posted by GammastyleYou cannot measure the diameter and circumference of a circle to arbitrary precision, so that method is out (and in any case, pi is a mathematical constant, not a physical one...so any "real world" way of calculating pi isn't going to work).
Being a non-math guy. How does one calculate pi. Is there a formula?. Do they measure a radius and diameter more and more exact? I don't understand how they keep getting more and more digits. Please explain.
There are on the other hand plenty of formulae for pi in the form of infinite series (sums) and products.
One formula you could use to work out pi to arbitrarily high precision is
pi = 4 x (1 - 1/3 + 1/5 - 1/7 + ... )
although the convergence here is very, very slow (ie. you'd have to look at loads of terms in the sum just to get a few decimal places).
There are other methods which give very fast convergence....