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 objectsone 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.
