[Slightly OT]: More on ints and floats

[James Gregory]

On Tue, 2003-04-08 at 18:27, Michael Hudson wrote:
> Tim Daneliuk <tundra at tundraware.com> writes:
> 
> > I am unclear, though, on whether a mathematician understands 1.0 to
> > mean 1.0000000... (a single number) or the _neighborhood_ around 1
> > with one digit precision.
> 
> I'm a mathematician, albeit not a numerical analyst, and I don't think
> considering a float as the interval of real numbers to which it is the
> closest floating point approximation is a particularly helpful point
> of view.

To my mind, 1.0 is a member of R, 1 is a member of Z. But I do crazy
things like think of "1/2" as the number that yields 1 when multiplied
by 2. Not 0.5. Nor the quantity of apple you get when sharing an apple
with one other person.

But it's context dependent, and that's one of the properties that having
all "numbers" as objects affords you. It means that my software can tell
if a particular number is a member of GF(5) (The Galois Field of 5
elements), and then it can infer that "1/2" actually means 3 (3 * 2 = 6,
6 % 5 = 1, as per the above definition).

What does 1/2 mean to you in Z? Does it mean the same thing in R?

(Z is the set of all integers, R is the set of all reals in case it
wasn't clear)

James.