Numbering in base other than 10

Question

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

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. ain't a mathematician, Nov 05 '10 at 12:04
@J.M. 10 is not a multiple of 3, but 1/3 has repeating decimals (base 10). — Willie Wong, Nov 05 '10 at 14:56
@Willie: You misread; "no repeating digits if the base is a multiple of the denominator". — J. M. ain't a mathematician, Nov 05 '10 at 15:05
@Willie: Okay, maybe the wording was a tad too strong... Repeating 0s and 9s should be tacitly excluded, then (or in general, 0 and n-1 for base-n numbers). :) — J. M. ain't a mathematician, Nov 05 '10 at 15:22

score 3 · Accepted Answer · answered Nov 05 '10 at 13:11

3

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.

answered Nov 05 '10 at 13:11

Ross Millikan

374,822

Then why is 2.8333... 2.9BBB... in base 12? and it is 2.4555... base 6? – futurebird Nov 05 '10 at 13:28
4

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 05 '10 at 13:33

Numbering in base other than 10

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