Stumbling Upon Pi: Could a Billion 3's be Hidden Inside?

[rmberwin]

Let's say we randomly select integers to construct a potentially infinite number, for example 3588945... There is a non-zero chance that eventually we will obtain any possible finite series of numbers, say a billion 3's in a row. It is known that pi is indistinguishable from a random series of digits. Does this mean that a billion 3's in a row are contained in pi? Think Carl Sagan's novel Contact.

 

[micromass]

The answer is: we don't know. The property you're referring to is called that of a normal number.

http://en.wikipedia.org/wiki/Normal_number

So roughly speaking, a normal number is a random collection of digits. Every digit appears as many times as the other. Every pair of digits appears as many times as other pairs, and so on. In particular, a sequence of a billion three's will appear infinitely many times. Furthermore, if you take all the works of Shakespeare and express it in binary, then somewhere in the normal number, this work of Shakespeare will appear.

Allmost all numbers are normal, but very few numbers have been shown normal. One example is the Champernowne's constant:

$$0.12345678910111213141516...$$

http://en.wikipedia.org/wiki/Champernowne_constant

At this point of time, it remains unproven whether $\pi$, $e$ or $\sqrt{2}$ is normal. It probably is, but it might not be so. We'll have to wait until a mathematician finds a proof or disproof.

 

[SteveL27]

Let's say we randomly select integers to construct a potentially infinite number, for example 3588945... There is a non-zero chance that eventually we will obtain any possible finite series of numbers, say a billion 3's in a row. It is known that pi is indistinguishable from a random series of digits. Does this mean that a billion 3's in a row are contained in pi? Think Carl Sagan's novel Contact.

Tragically, we failed to recognize the aliens because they sent us the digits of Tau.

 

[Zuri Rohm]

Let's say we randomly select integers to construct a potentially infinite number, for example 3588945... There is a non-zero chance that eventually we will obtain any possible finite series of numbers, say a billion 3's in a row. It is known that pi is indistinguishable from a random series of digits. Does this mean that a billion 3's in a row are contained in pi? Think Carl Sagan's novel Contact.

I think that it is possible to find any amount of any number as many times as you like seeing that Pi is an irrational number.

 

[micromass]

I think that it is possible to find any amount of any number as many times as you like seeing that Pi is an irrational number.

Irrationality has nothing to do with it. For example, the number

$$0.11010010001000010000010000001...$$

is irrational but does not contain "any amount of any number as many time as you want".

 

[Zuri Rohm]

Irrationality has nothing to do with it. For example, the number

$$0.11010010001000010000010000001...$$

is irrational but does not contain "any amount of any number as many time as you want".

Thank you, but if if irrationality does not matter then what type of number is it in this situation?

 

[micromass]

Thank you, but if if irrationality does not matter then what type of number is it in this situation?

What you're thinking of is probably a "normal number". Any normal number is irrational, but not every irrational number is normal. It is currently unkown whether $\pi$ is normal.

 

[Zuri Rohm]

Irrationality has nothing to do with it. For example, the number

$$0.11010010001000010000010000001...$$

is irrational but does not contain "any amount of any number as many time as you want".

Does binary code exist after the decimal point? (Not saying that what you typed is binary code)

 

[micromass]

Does binary code exist after the decimal point?

Could you expand on that question, since I have no idea what you mean.

 

[Zuri Rohm]

What you're thinking of is probably a "normal number". Any normal number is irrational, but not every irrational number is normal. It is currently unkown whether $\pi$ is normal.

Thank you very much.

 

[Zuri Rohm]

Could you expand on that question, since I have no idea what you mean.

I mean, since Binary Code is a string of 1's and 0's is it just a code or is it a long number that can go into decimals.

 

[phinds]

I mean, since Binary Code is a string of 1's and 0's is it just a code or is it a long number that can go into decimals.

Base 2 number system works just like the base 10 number system. You can have fractions, decimals, whole numbers, etc. That is, it is NOT a "code" it is a radix number system.

 

[lavinia]

I think that it is possible to find any amount of any number as many times as you like seeing that Pi is an irrational number.

A number is rational if its decimal expansion is ultimately repeating.
So it is easy to make irrational numbers.

For instance the number whose 2^nth place is 1 and all other places are zero is irrational.

 

[WWGD]

A number is rational if its decimal expansion is ultimately repeating.
So it is easy to make irrational numbers.

For instance the number whose 2^nth place is 1 and all other places are zero is irrational.

Ah, I see, this is _for all n_ .

 

[micromass]

Isnt this the number $\frac {1}{10^{2^n}}$ ?

He meant a number for each $n$, not a particular one.

 

[lavinia]

Isnt this the number $\frac {1}{10^{2^n}}$ ?

Yeah. I meant for every n so it would be

.1001000100000001 ...

 

[WWGD]

He meant a number for each $n$, not a particular one.

Yes, I just realized that, should have been obvious to me.

 

[WWGD]

I can delete my dumb post if you all want, to get rid of the noise.

 

[WWGD]

But an interesting point, lavinia, maybe, is that your irrational number is a(n) (obviously infinite) sum of Rationals.

 

[micromass]

But an interesting point, lavinia, maybe, is that your irrational number is a sum of Rationals.

Every real number can be written as the (infinite) sum of rationals

 

[WWGD]

Yes, I know, just to comment that Rationals are not close under infinite sums, and Irrationals are not even under finite sums: x-x =0. But the issue too is the speed of the convergence to decide if the limit is Rational or Irrationals.

 

[Matterwave]

It is known that pi is indistinguishable from a random series of digits.

I don't think this is known. I'm not sure if this statement is equivalent to "pi is a normal number", but either way I don't think this is proven.

 

[lavinia]

Here is a thought/question.

There are simple algorithms that generate infinite series of rational numbers that converge to π. By going out far enough on one of these series, one obtains a rational number whose first n decimal places are the same as those for π. So it would seem then that the decimal expansion of π is not random in some sense because it is computable.

It would also seem that for most numbers there is no such algorithm - proof? - so most numbers are truly random but π is not.

If statistically, the decimal expansion of π were indistinguishable from a random sequence, then I am reminded of mechanical systems (chaotic systems) which statistically appear random but in fact are completely determined.

- I imagine that the concept of an algorithm needs defining; perhaps an inductive rule that requires only a finite amount of initial data. But I am out of my element here.

- I just found this Wikipedia link on what are called "computable numbers". The definition involves Turing machines. Perhaps someone can explain what it means but it seems to mean something like there exists a computer program that generates a series that converges to the number.

http://en.wikipedia.org/wiki/Computable_number

In any case, by this definition,π is "computable", as also is e but in total there are only countably many computable numbers. Interestingly, the computable numbers from a subfield of the real numbers.

 

[lavinia]

But an interesting point, lavinia, maybe, is that your irrational number is a(n) (obviously infinite) sum of Rationals.

Yes if by obviously you mean an inductive algorithm. I had the same thought. Look at my post about computable numbers just above.

 

[MostlyHarmless]

I don't think this is known. I'm not sure if this statement is equivalent to "pi is a normal number", but either way I don't think this is proven.

I believe what is meant by this is that the probability of finding a number at a position in pi is the same as if you just randomly assembled the numbers and looked at that same position.