Re: OT: Need help with numeric bases
| From: | Apollo Hogan <apollo@...> |
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| Date: | Tuesday, February 25, 2003, 5:55 |
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On Mon, 24 Feb 2003, Nik Taylor wrote:
> Peter Clark wrote:
> > Also, how does one go about converting the base of non-integer numbers?
>
> Division. For example, to convert, say, .125 into duodecimal, divide 1
> by 8, using a duodecimal multiplication table, so, just as the first
> step in long division in decimal would be 8 divided by 10 (1), so you
> divide 8 by (duodecimal) 10 to get 1 with a remainder of 4, then 8 by
> (duodecimal) 40 (= decimal 48) you get 6 with a remainder of 0, so .125
> in decimal = .16 in duodecimal
One can also do it with Multiplication: to convert x to base b, just
iteratively apply the following steps: (Assuming 0<=x<1 already)
1) multiply by b
2) the result will have an integer part between 0 and b-1 (inclusive);
that is the next digit of your base-b rep
3) replace x with its fractional part
For example, to convert 0.125 to base 7, we get:
rep x x*7
0. 0.125 0.875
0.0.. 0.875 6.125
0.06.. 0.125 0.875
etc. Here we get a repeat already, so we see that 0.125 (base 10) is
written as 0.0606060606... in base 7.
A more interesting example, write PI in base 12: (We first strip off the
integer part 3)
rep x x*12
3. 0.1415926.. 1.69911184..
3.1 0.69911184.. 8.38934211..
3.18 0.38934211.. 4.67210540..
3.184 0.67210540.. 8.06526483..
3.1848
etc. ad nauseum; hence PI = 3.1848... in base 12.
Cheers,
--Apollo
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