Re: integer exponentiation with negative exponent

[Coda Highland <chighland@...>]

On Thu, May 22, 2014 at 9:08 AM, Roberto Ierusalimschy
<roberto@inf.puc-rio.br> wrote:
>> Let's add some rigor. In order to be consistent, Arithmetic is build
>> on top of a algebraic field.
>
> Why? Wolfram MathWorld, for instance, says "Arithmetic is the branch
> of mathematics dealing with INTEGERS or, more generally, numerical
> computation" (my emphasis).
>
> Number theory (integers, too) is ususally called "higher Arithmetic".
>
> -- Roberto
>

I'm not certain I agree with Wolfram in this sense. In my mathematical
upbringing, arithmetic was more or less "everything lower than
algebra." In practice this meant computations on the rationals, not on
the integers, as my schooling introduced decimals and fractions very
early on -- probably first or second grade. Solving for variables or
using functions took you out of "arithmetic" and into "algebra", and
dealing with irrational numbers wasn't introduced until "geometry".

Despite MathWorld's statement I think that division is generally
considered to be an arithmetic operation.

/s/ Adam