1. ## recurring digits

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

2. 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

3. ## 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.

4. ## OK just take

Ok, just consider , what is the longest string of a single recurring digit in ?

5. Originally Posted by anthp1234
Ok, just consider , what is the longest string of a single recurring digit in ?
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