TITLE: What is the best base to use?
QUESTION [11 upvotes]: When I typed this question in google I found this link: http://octomatics.org/ Just from the graphic point of view: this system seems to be easier (when he explains that you can overlap the line). He also talks about how base 8 is useful for computers. Is base 8 really the best,what makes one base better than the other. Is there some mathematical argument that we can use to decide which is the best base to use? Thank you for your insight, I think the answer has to be entangled with other areas of knowledge as cognitive sciences or simply how large are the numbers that are used more often but: Arithmetic is something that almost everybody does so it is probably worth a while to try to see if we are making it as easy as possible. Thank you very much in advance.

REPLY [2 votes]: It depends what you find 'best'. Personally I don't care whether it's easy to convert to binary, since computers do that just fine for any base, so I never need to.
What is a little bit more annoying for humans is fractions that don't terminate. So I think to get the best base, you just multiply prime numbers from low to high until you feel the base gets too big. E.g.
1 * 2 = 2
1 * 2 * 3 = 6
1 * 2 * 3 * 5 = 30
1 * 2 * 3 * 5 * 7 = 210
....

Using base 30, fractions like 1/2, 1/3, 1/5, 1/6, 1/8, 1/9, 1/10 or 1/15 factorizes nicely because the denominator's prime factors are also prime factors of your base. In that respect, base 12, or any $2^N$ base, waste numbers because they have double prime factors. The same fractions that terminate in base 12 also terminate in base 6.
Base 210 is definitely too big since you need to remember like 21.000 multiplications. Base 30  isn't really fun at around 400 either, but that's doable. The best bet may be 6. A quarter is not ideal though (3/20).