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Re: ot: on a base being a power of another

Followups directed to: comp.programming

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Meredith Montgomery <mmontgomery@levado.to> writes:

> This is totally not C related, but I think you guys master the subject,
> so perhaps it isn't so off-topic.

Yes, it really is off-topic!  Since you are thinking about algorithms,
comp.programming is probably the closest fit.  I've set followup-To.
Maybe a good thread there will revive that group.

> It's of course very useful to know how to convert numerals from one base
> to another when we deal with C programming.  I can convert from any base
> to any base.  What I'm trying to understand now is why it is so easy to
> go from, say, base two to base eight or base sixteen or, say, base three
> to base nine.

When the two bases are positive integer powers of some common number,
the conversion is simply a matter of grouping digits.  You clearly get
this as you go on to give explicit examples, so I am not sure why you are
asking why it's easy.

> To go from base two to base sixteen, you take four digits in base two
> and map them to base sixteen --- from right to left.  (We can, say,
> left-pad the last group of digits if they don't have four digits.)  I
> notice that 2^4 = 16 and it is the exponent here that dictates how many
> digits I grab each time --- from right to left.
>
> So if I need to convert base nine to base three, I'll take each digit
> (from right to left) and replace it with its corresponding base three
> digits, making to sure to always write them as two digits because 
> 3^2 = 9.
<cut>

> Why does this work?  I don't know.

I don't know what sort of answer that question needs.  It seems obvious
to me that groups of k > 1 base b digits are just the digits in base b^k.

<cut>
> Definition.  To express a number N in a certain base b, we need to
> find the coefficients a0, a1, ..., ak such that
>
>   N = ak b^k + a(k-1) b^(k-1) + ... + a0 b^0.

You probably intended to say more, though this definition is interesting
in it's own right.  For example, using b=10, N=20 can be written with

  a0 = 20
  a0 = 0, a1 = 2
  a0 = -10, a1 = 3

and so on.  Also, your definition permits b to be something other than a
positive integer.  Base e, anyone?  Base -2?

And permitting (as you do by default) the digits to be rational or even
irrational numbers is also interesting!

> Can you solve this problem or point me somewhere?

I'm not sure what the problem is.  You started by saying you can convert
between any two bases, but maybe you meant "one can convert" and you
don't know a general algorithm to do it.  Is that the problem you are
asking about?

-- 
Ben.

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Thread

ot: on a base being a power of another Meredith Montgomery <mmontgomery@levado.to> - 2021-11-20 18:08 -0300
  Re: ot: on a base being a power of another Ben Bacarisse <ben.usenet@bsb.me.uk> - 2021-11-20 22:22 +0000
  Re: ot: on a base being a power of another Bart <bc@freeuk.com> - 2021-11-20 22:27 +0000
  Re: ot: on a base being a power of another James Kuyper <jameskuyper@alumni.caltech.edu> - 2021-11-20 18:08 -0500
    Re: ot: on a base being a power of another Meredith Montgomery <mmontgomery@levado.to> - 2021-11-23 15:43 -0300
      Re: ot: on a base being a power of another Bart <bc@freeuk.com> - 2021-11-23 19:57 +0000
      Re: ot: on a base being a power of another "james...@alumni.caltech.edu" <jameskuyper@alumni.caltech.edu> - 2021-11-23 16:53 -0800
        Re: ot: on a base being a power of another Meredith Montgomery <mmontgomery@levado.to> - 2021-11-24 11:14 -0300
  Re: ot: on a base being a power of another Manfred <noname@add.invalid> - 2021-11-21 05:41 +0100

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