# Randomness of pi

1. Apr 3, 2014

### 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.

2. Apr 3, 2014

### 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.

Last edited: Apr 3, 2014
3. Apr 4, 2014

### SteveL27

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

4. Mar 31, 2015

### Zuri Rohm

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

5. Mar 31, 2015

### micromass

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".

6. Mar 31, 2015

### Zuri Rohm

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

7. Mar 31, 2015

### micromass

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.

8. Mar 31, 2015

### Zuri Rohm

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

9. Mar 31, 2015

### micromass

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

10. Mar 31, 2015

### Zuri Rohm

Thank you very much.

11. Mar 31, 2015

### Zuri Rohm

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.

12. Mar 31, 2015

### phinds

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.

13. Apr 11, 2015

### lavinia

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.

14. Apr 11, 2015

### WWGD

Ah, I see, this is _for all n_ .

15. Apr 11, 2015

### micromass

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

16. Apr 11, 2015

### lavinia

Yeah. I meant for every n so it would be

.1001000100000001 ....

17. Apr 11, 2015

### WWGD

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

18. Apr 11, 2015

### WWGD

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

19. Apr 11, 2015

### WWGD

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

20. Apr 11, 2015

### micromass

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