Infinitely many monkeys?

[LA_Price]

Less than a second is included under the word eventually. Also infinetely many copies would be produced, but that wouldn't really matter as whether you get one or an infinitetly many is the same under the above statement.

These are really small points to argue. I think Mandelbrot's idea of fractality is much more important as it pertains to the idea of infinity.

[almostbusto]

a better way to frame the problem in my opinion:

i believe it has been proven that not only is Pi irrational, but that the decimal expansion of Pi contains all possible finites sequences of digits. Therefore, at some point in the binary expansion of Pi there exists a string of ones and zeros equivalent to a Microsoft word file that contains the written works of Shakespeare. There is also a jpeg image of your high school yearbook photo too. which i think is much more mind blowing.

[bunny]

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i believe it has been proven that not only is Pi irrational, but that the decimal expansion of Pi contains all possible finites sequences of digits.

[/ QUOTE ]
Anyone able to provide a citation for this? It seems wrong to me.

[Sephus]

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Less than a second is included under the word eventually.

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obviously i had no issue with the literal meaning of the sentence.

[almostbusto]

[ QUOTE ]
[ QUOTE ]
i believe it has been proven that not only is Pi irrational, but that the decimal expansion of Pi contains all possible finites sequences of digits.

[/ QUOTE ]
Anyone able to provide a citation for this? It seems wrong to me.

[/ QUOTE ]

a quick googling gave me this:
http://sprott.physics.wisc.edu/pickover/pimatrix.html

doesn't look like the best source. however, it does point at out that such numbers are called transcendental numbers. so even if you don't believe pi is transcendental, then just imagine some other transcendental number.

[Piers]

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do you mean countably infinite, i.e. There is a one to one correspondance with the natural numbers. when I hear uncountably infinite the best example I can think of is the real numbers.


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I think it would be just as tricky for me to feed aleph zero monkeys as aleph one, irrespective of your opinion of the continuum hypothesis. So I don’t think it matters, although admittedly a countable number monkeys might appear superficially more acceptable.

[bunny]

[ QUOTE ]
[ QUOTE ]
[ QUOTE ]
i believe it has been proven that not only is Pi irrational, but that the decimal expansion of Pi contains all possible finites sequences of digits.

[/ QUOTE ]
Anyone able to provide a citation for this? It seems wrong to me.

[/ QUOTE ]

a quick googling gave me this:
http://sprott.physics.wisc.edu/pickover/pimatrix.html

doesn't look like the best source. however, it does point at out that such numbers are called transcendental numbers. so even if you don't believe pi is transcendental, then just imagine some other transcendental number.

[/ QUOTE ]
It seems to me that he is basing his claim on the fact that the expansion of pi is infinite, non-repeating with all digits occuring with equal frequency. He seems to make an intuitive leap that this means all finite sequences will eventually occur which is just plain wrong (eq 0.123456789112233445566778899111... has the above properties but never contains the finite sequence 28).

[PairTheBoard]

[ QUOTE ]
[ QUOTE ]
[ QUOTE ]
[ QUOTE ]
i believe it has been proven that not only is Pi irrational, but that the decimal expansion of Pi contains all possible finites sequences of digits.

[/ QUOTE ]
Anyone able to provide a citation for this? It seems wrong to me.

[/ QUOTE ]

a quick googling gave me this:
http://sprott.physics.wisc.edu/pickover/pimatrix.html

doesn't look like the best source. however, it does point at out that such numbers are called transcendental numbers. so even if you don't believe pi is transcendental, then just imagine some other transcendental number.

[/ QUOTE ]
It seems to me that he is basing his claim on the fact that the expansion of pi is infinite, non-repeating with all digits occuring with equal frequency. He seems to make an intuitive leap that this means all finite sequences will eventually occur which is just plain wrong (eq 0.123456789112233445566778899111... has the above properties but never contains the finite sequence 28).

[/ QUOTE ]

If every "next digit" of pi was truly random then I think a probability argument could be made that any Fixed Sequence of length N has a 1/10^N chance of coming up next. Thus, since you have infinitely many 1/10^N chances of it coming up next it almost surely comes up over and over again.

It would be like flipping a coin infinitely many times. If you did, you would almost surely see 1 googolplex of heads flipped in a row somewhere in the infinite sequence of flips. Not only that, but you would see it infinitely many times, with probability 1.

However, I don't think it's clear that every next digit of pi behaves as if it is completely random. The conjecture might still be provable but I don't think it's obvious.

PairTheBoard

[ADDboy]

[ QUOTE ]
[ QUOTE ]
i believe it has been proven that not only is Pi irrational, but that the decimal expansion of Pi contains all possible finites sequences of digits.

[/ QUOTE ]
Anyone able to provide a citation for this? It seems wrong to me.

[/ QUOTE ]

I'm pretty sure that this is open for pi, but it is known to be true for almost all numbers in the sense that if you pick a number uniformly from (0,1) then it has this property.

[bunny]

Thanks. It's kinda ironic that I have no problem believing this is almost always true for numbers selected at random as you say but doubt it's true for pi in particular. [img]/images/graemlins/smile.gif[/img]