Infinite Monkeys Pi and Phi

In summary, the conversation discusses the possibility that the first billion decimal digits of pi would occur within the decimal expansion of phi, and whether this would occur an infinite number of times. It also brings up the concept of "normal numbers" and how it relates to the infinite monkey theorem. Ultimately, it is not known whether pi or phi are normal numbers, but it is speculated that they are, making them subject to the infinite monkey theorem.
  • #1
Diffy
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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?
 
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  • #2
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.
 
  • #3
Diffy said:
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.
 
  • #4
Diffy said:
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.
 
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  • #5
As CRGreathouse and D H pointed out, "irrational" is not enough. You need "normal number" and it is not know whether [itex]\pi[/itex] or [itex]\phi[/itex] is a normal number.
 
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  • #6
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?
 
  • #7
Sounds like it Daniel,

Thanks to everyone else for entertaining the idea!
 
  • #8
Daniel Y. said:
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).
 
  • #9
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.
 
  • #10
CRGreathouse said:
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.
 

1. What is the concept of "Infinite Monkeys Pi and Phi"?

The concept of "Infinite Monkeys Pi and Phi" is based on the idea that an infinite number of monkeys typing randomly on an infinite number of typewriters will eventually produce the complete works of Shakespeare. This concept is used to illustrate the concept of randomness and probability.

2. How does "Infinite Monkeys Pi and Phi" relate to pi and phi?

In this concept, the monkeys are typing random sequences of letters, which can also be represented as random numbers. The digits of pi and phi are also infinite and considered to be random. Therefore, the monkeys typing random sequences can also be seen as producing the digits of pi and phi.

3. Is there any scientific basis for "Infinite Monkeys Pi and Phi"?

While this concept is often used as a thought experiment, it does have some scientific basis. It relates to the mathematical concept of randomness and the probability of certain events occurring. It also demonstrates the concept of infinity and how even seemingly impossible events can occur in an infinite universe.

4. Can "Infinite Monkeys Pi and Phi" be applied to other mathematical concepts?

Yes, the concept can be applied to other mathematical concepts such as the Fibonacci sequence or the Golden Ratio. It can also be used to explore the concept of chaotic systems and how small changes in initial conditions can lead to significantly different outcomes.

5. What can we learn from "Infinite Monkeys Pi and Phi"?

This concept teaches us about the power of randomness and how it can lead to unexpected outcomes. It also highlights the limitations of our understanding of infinity and how even seemingly impossible events can occur in an infinite universe. Additionally, it reminds us to question our assumptions and think creatively when exploring scientific concepts.

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