Decimal numbers come in two varieties:

- Finite decimals which have nothing but trailing zeros after some point (e.g. [imath]\frac 1 2 = 0.5[/imath])
- Infinite (or "unbounded") decimals which continue on indefinitely given some definition of how they are constructed (e.g. [imath]\frac 1 3 = 0.33333.....[/imath] or for those mathematicians out there more correctly [imath]0.\overline{3}[/imath])

Now, definitionally, Rational numbers have repeating sequences of digits, so the fraction [imath]\frac 1 3[/imath] repeats the three on and on endlessly. Conversely, irrational numbers however have random sequences that do not repeat at all.

Here's where my pondering came in: for any particular decimal number, you can create a distribution map of the digits in the decimal representation. For the finite representations, this is just counting appearances so for the number [imath]\frac 5 {32} = 0.15625[/imath] you have the distribution of digits:

0: 0

1: 1

2: 1

3: 0

4: 0

5: 2

6: 1

7: 0

8: 0

9: 0

With unbounded numbers, you have an infinite number of digits, so things get a bit more interesting. In order to "compute" the distribution, you need to start working with ratios of each digit to the others. With unbounded rational numbers, this is relatively easy since you can take just the "repeating sequence" and count the occurrences of each digit and then divide by the length of the sequence, so the number [imath]\frac {41} {333} = 0.123123123... = 0.\overline{123}[/imath] has the distribution:

0: [imath]\frac 1 3[/imath]

1: [imath]\frac 1 3[/imath]

2: [imath]\frac 1 3[/imath]

3: 0

4: 0

5: 0

6: 0

7: 0

8: 0

9: 0

On the other hand this lovely fraction: [imath]\frac 1 {81} = 0.0123456790123… = 0.\overline{0123456789}[/imath] has the perfectly Poisson Distribution curve of:

0: [imath]\frac 1 {10}[/imath]

1: [imath]\frac 1 {10}[/imath]

2: [imath]\frac 1 {10}[/imath]

3: [imath]\frac 1 {10}[/imath]

4: [imath]\frac 1 {10}[/imath]

5: [imath]\frac 1 {10}[/imath]

6: [imath]\frac 1 {10}[/imath]

7: [imath]\frac 1 {10}[/imath]

8: [imath]\frac 1 {10}[/imath]

9: [imath]\frac 1 {10}[/imath]

**What are the distributions of digits in Irrational Numbers?**

Can we make any interesting generalizations? Since the digits in an Irrational number are random, they *could* all be Poisson Distributed, but are they? How's about [imath]\pi[/imath] or

*e*?

Any references to cool Abstract Algebra or Number Theory proofs that are relevant (or have solved this one completely!)?

Anyone want to contribute some code to test any theories?

The creator of the universe works in mysterious ways. But he uses a base ten counting system and likes round numbers,

Buffy