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Is there a way to perform these operations in a particular base b but 10? I can convert it back to base 10 and perform the operation, however, I think it's kinda odd by doing this way.


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up vote 4 down vote accepted

If what you are interested in are paper-and-pencil methods, then the same algorithms that one uses in base 10 work in base $b$, provided you remember to perform all operations in base $b$. (Of course, it may be that the proliferation of calculators has atrophied some people's ability to do it even in base $10$...)

For example, if you want to add $573641$ and $561373$ in base $9$ (I just made them up), then we add can add them right to left, with carries; note that in base $9$, $7+4 = 12$, $6+3=10$, etc. So we get: $$\begin{array}{r} 573641\\\ \underline{+\ 561373}\\\ 1245124 \end{array}$$ Similar with subtraction, multiplication, and long division.

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Thanks a lot, I do want to use paper and pencil. – Chan Feb 4 '11 at 4:12
@Chan: Great, then. Multiplication is a bit difficult, because if you know the multiplication tables in base 10, you'll have a hard time remembering things like the fact that, if you are working in base $7$, then $6\times 5$ is $42$, but if you keep that in mind, the good old base 10 algorithms work. – Arturo Magidin Feb 4 '11 at 4:16
I find when I do this I convert back to base 10 for each single digit calculation (except for binary). I did a math puzzle years ago that was so based in 6's that it made sense to do all the arithmetic in base 6. – Ross Millikan Feb 4 '11 at 5:00
@Ross: I suspect with enough practice one would be able to do it almost without thinking for any particular base, but yes, that's pretty much what I do. Except, as you say, base 2, though I would extend that to base $2^k$, which I do by converting to binary and back (the latter via groupings, which is pretty easy). – Arturo Magidin Feb 4 '11 at 5:02
I don't do hex enough to know that 7*8=38 or even that A+B=15. But I know many people can. It is said that the stage calculators worked in base 100 and knew the multiplication table for that. – Ross Millikan Feb 4 '11 at 5:08

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