Is it possible to convert base 7 to base 19 directly without first converting to base 10 ?

If so, what is the algorithm ?

  • $\begingroup$ Yes­­­­­­­­­­­­ $\endgroup$ – evil999man Apr 12 '14 at 11:13
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    $\begingroup$ Yes. 10 isn't special; it's just the number after 9. It only seems special because we've chosen it as the usual base we work in. $\endgroup$ – user2357112 supports Monica Apr 12 '14 at 11:13
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    $\begingroup$ The algorithm is the same one you'd use to convert from base $7$ to base $10$, except for the obvious. $\endgroup$ – Jack M Apr 12 '14 at 11:15
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    $\begingroup$ Same exact algorithm you'd use for converting from base 10 to base 19, except with all the 10s switched out for 7s. $\endgroup$ – user2357112 supports Monica Apr 12 '14 at 11:15
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    $\begingroup$ @Cemre Why didn't you Google first what will you ask before posting your question here? Anyway, answer of your question may refer to this link. I hope this helps. $\endgroup$ – Tunk-Fey Apr 12 '14 at 11:22

You have two options:

  1. Work in base nineteen. Calculate the powers of $7$ in base nineteen, multiply them by the base-7 digits of the original number, add together.

  2. Work in base seven. Start with your original number, successively divide by nineteen (that is, $25_7$), and remember the remainders; these become your base-nineteen digits.

The first method has the advantage that you don't need to do long division; on the other hand you need a nineteen-by-nineteen multiplication table. And long division by a two-digit number is not that cumbersome.


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