...
> >
> The problem is that, unlike the integer-base case, this representation
> using a non-integer base is not unique. A trivial example, using base
> 2.5:
>
> (2.5)^3 = 2 * (2.5)^2 + 1 * (2.5)^1 + 0.625, so the number
> 15.625 has, at least, two representations in "base" 2.5, which
> contradicts the whole idea of a _base_.
>
Another way of looking at the above is that 2.5 is a root
of x^3 - 2 x^2 - x - (1/4)x = 0. (Since 0.625 = (1/4)*2.5)
Note that this depends on the fact that 2.5 is algebraic over
the rationals--there can be no such equation with a transcendental
number like pi as a root.
But one can get uniqueness in any event, by using a greedy
algorithm to pick the coefficients of 2.5. For example, you have
that 4*(2.5)^3 = 8*(2.5)^2 + 5*(2.5), or that 4000.00 equals 850.00
in base 2.5. That's nice, but neither is the natural representation
of this number (62.5 in base 10). For in base 10, I calculate:
62.5 = (2.5)^4 + (2.5)^3 + (2.5)^2 + 1 + (2.5)^-1 + (2.5)^-2 + ...
Thus the natural representation of 4000 base 2.5 is really
11101.110... base 2.5 (I know it's not as nice, but it DOES yield
uniqueness.)
---David
[EMAIL PROTECTED]
> >The d(i) are what we call in base ten "digits," but since that term is
> >sometimes considered only applicable in base ten,
> >
> No! bit = binary digits. Digits come from the latin word for fingers
>
> Alberto Monteiro
>
>
>
>
