# recurring digits

• Apr 27th 2009, 06:25 AM
anthp1234
recurring digits
In an irrational number what is the longest possible sequence of a single digit that can occur?
• Apr 27th 2009, 01:40 PM
CaptainBlack
Quote:

Originally Posted by anthp1234
In an irrational number what is the longest possible sequence of a single digit that can occur?

In almost all irrationals a string of $\displaystyle 1$'s will occur that is longer than $\displaystyle n$ for all $\displaystyle n$ in $\displaystyle \mathbb{N}$.

CB
• Apr 28th 2009, 01:06 PM
Media_Man
There is no such limit
Just by using the word "irrational" you automatically attach absolutely no general rules to the decimal expansion of the number. Here is a simple proof.

Define $\displaystyle x_n$ as follows:

The first $\displaystyle n$ digits of $\displaystyle x_n$ are $\displaystyle 1$. After this string of $\displaystyle 1$'s terminates, the next digits are exactly the digits of $\displaystyle \pi$.

$\displaystyle x_{20}=.111111111111111111113141592653589793238462 643383279502884197...$

This number is automatically irrational because it has an infinite non-repeating decimal expansion, and yet n can take on whatever value you choose, whether it be 20 or 20 trillion.
• Apr 28th 2009, 10:06 PM
anthp1234
OK just take
Ok, just consider http://www.mathhelpforum.com/math-he...7a9dac6d-1.gif, what is the longest string of a single recurring digit in http://www.mathhelpforum.com/math-he...7a9dac6d-1.gif?
• Apr 28th 2009, 11:19 PM
CaptainBlack
Quote:

Originally Posted by anthp1234
Ok, just consider http://www.mathhelpforum.com/math-he...7a9dac6d-1.gif, what is the longest string of a single recurring digit in http://www.mathhelpforum.com/math-he...7a9dac6d-1.gif?

Already answered in 2nd post in this thread, \pi is believed to be normal (though not proven so), so there is no longest string of a single digit in its' decimal expansion.

CB