Can the Infinite Monkey Theorem Predict Pi's Digits in Phi's Expansion?

[Diffy]

I guess one could use any irrational numbers here, but phi and pi are favorites.

I am sure that most people are aware of the infinite monkey theorem. If not use http://en.wikipedia.org/wiki/Infinite_monkey_theorem as a reference.

By using this theorem, could one say that the the first billion decimal digits of pi (in order) almost certainly would show up somewhere in the decimal digits of phi? Of course where this phenomenon would occur would start at some unimaginably enormous number.

I assume this would be true. Since phi is irrational, the digits in its decimal expansion have no pattern, so essentially they are random.

To take this one step further, could one say that the said pattern of the first billion digits of pi, would occur within the decimal expansion of phi an infinite number of times?

 

[mathman]

To take this one step further, could one say that the said pattern of the first billion digits of pi, would occur within the decimal expansion of phi an infinite number of times?

Maybe: Since we don't know, it becomes a matter of probability.

 

[CRGreathouse]

I assume this would be true. Since phi is irrational, the digits in its decimal expansion have no pattern, so essentially they are random.

This does not follow. The number 0.01101010001010001010001000001... where the nth decimal place is 1 if n is prime and 0 otherwise never has a 5 in it, but it's still irrational.

Under the likely but hard-to-prove conjecture that pi is 10-normal, it contains the initial k digits of phi in order infinitely many times (for any k). Likewise for phi and pi.

 

[D H]

By using this theorem, could one say that the the first billion decimal digits of pi (in order) almost certainly would show up somewhere in the decimal digits of phi?

No. In applying the infinite monkey theorem you are assuming phi is normal. Consider for example, the irrational number whose decimal expansion is 0.101001000100001... The probability of finding the sequence 314159 in the decimal expansion of this number is zero. The number I have chosen is not normal. We don't know if phi, or pi, or e is normal.

EDIT:
Dang. CRGreathouse beat me to the punch.

 

[HallsofIvy]

As CRGreathouse and D H pointed out, "irrational" is not enough. You need "normal number" and it is not know whether \pi or \phi is a normal number.

 

[Daniel Y.]

So Pi and Phi are not proven to be 'normal numbers', but if one wanted to speculate, it seems likely that they are normal, and thus subject to the Infinite Monkey Theorem?

 

[Diffy]

Sounds like it Daniel,

Thanks to everyone else for entertaining the idea!

 

[CRGreathouse]

So Pi and Phi are not proven to be 'normal numbers', but if one wanted to speculate, it seems likely that they are normal, and thus subject to the Infinite Monkey Theorem?

Essentially, yes.

Actually, it's one step better. The infinite monkey 'theorem' says that a random string of digits will eventually produce any given digit string with probability 1. If the number was normal, it will certainly contain any given digit string. All certain events have probability 1, but not all events of probability 1 are certain. For example, if you choose a 'random' real number, the probability that it's irrational is 1, but there are rational real numbers (just vanishingly few compared to the irrationals).

 

[CRGreathouse]

D H, I'm extremely amused to see that we posted at almost the same time, with almost the same conclusion, using the same argument with essentially similar irrational decimal expansions.

 

[HallsofIvy]

D H, I'm extremely amused to see that we posted at almost the same time, with almost the same conclusion, using the same argument with essentially similar irrational decimal expansions.

I've edited my response to include reference to D H.