Getting to Third Base: e
Several years ago, I received a phone call at 11:30 pm...from a retired math teacher. He asked: "If you make the number base an irrational number, will the nonrepeating decimals become terminating or repeating decimals? Will the irrationals become rational?"
The question is fascinating...though I first wondered if that is what all retired math teachers are destined to do or think about on a Saturday night!
Number bases, number words, and number symbols form the foundation of our numeration scheme...yet, in trying to find a response to his question, I found some interesting trivia lurking nearby:
A primitive tribe used distinct sets of number words when counting different types of objects--one for men, one for women, one for measure, one for long objects, etc.
A Northwest Native American tribe used base four, based on using the spaces between fingers of an open hand
Measuring efficiency of a number base by the size of a number (i.e. number of digits) needed to record a numerical value versus the number of symbols needed for a number base, the most efficient number base is base e (see Brian Hayes' article "Third Base" (2001) for a proof).
Yes...base e does exist! On the Internet, you can find a world of interesting bases: base -10, base pi, base phi, base Fibonacci, base factorials, base i, etc.
Note...I have not provided an answer to the original question...intentionally. You can make the same search, as it is a fun and revealing adventure.