# Numbering in base other than 10

On what basis the numbering less than 20, the fraction 17/6 (base 10) does not generate a regular tithe?

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What is a regular tithe? –  Qiaochu Yuan Nov 5 '10 at 12:03
If I'm interpreting your question correctly, there are no repeating digits in the digit expansion of your fraction if the base is a multiple of the denominator (assuming of course the fraction is in lowest terms). –  Ｊ. Ｍ. Nov 5 '10 at 12:04
@Qiaochu: It's a repeating decimal. –  Paulo Argolo Nov 5 '10 at 12:46
In base 7 2.8333... is 2.555... is that a "regular tithe?" –  a little don Nov 5 '10 at 13:23
@J.M. 10 is not a multiple of 3, but 1/3 has repeating decimals (base 10). –  Willie Wong Nov 5 '10 at 14:56

## 1 Answer

The terminating decimals (though that term should be reserved for base 10, but I have no better) in any base are precisely the fractions whose denominator has every prime factor represented in the factorization of the base. So in base 10, it is fractions with denominators of the form 2^x*5^y, where x and y are allowed to be zero.

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Then why is 2.8333... 2.9BBB... in base 12? and it is 2.4555... base 6? –  a little don Nov 5 '10 at 13:28
17/6 (base 10) in base 12 is 2.A (which can be expressed as 2.9BBB.., but usually we use the terminating version if available) and 2.5 in base 6 (again you could use 2.4555...). Would you say in base 10 that 1/2 is .49999...? Though correct, it is not usual. If you use the infinite forms, then all fractions form a "regular tithe". And so do whole numbers-1=0.999..... –  Ross Millikan Nov 5 '10 at 13:33