[Slightly OT]: More on ints and floats

[Tim Daneliuk]

Lulu of the Lotus-Eaters wrote:
> Tim Daneliuk <tundra at tundraware.com> wrote previously:
> |As I understand it, integers and floats are distinct mathematical
> |entities. A colleague of mine claims, that insfar as we use them in
> |computing, ints are merely a proper subset of floats.
> 
> Floats are peculiar creatures.  They really are not much like Real or
> Rational numbers, even though it is often convenient to pretend they
> are.  The trick about floating point numbers is that although they are
> extremely useful for representing real-life (fractional) quantities,
> operations on them do not obey the arithmetic rules we learned in middle
> school:  associativity, transitivity, commutativity; moreover, many very
> ordinary-seeming numbers can be represented only approximately with
> floating point numbers.  For example:
> 
>      >>> 1./3
>      0.33333333333333331
>      >>> .3
>      0.29999999999999999
>      >>> 7 == 7./25 * 25
>      0
>      >>> 7 == 7./24 * 24
>      1
> 
> Tim knows this, of course, but it is worth emphasizing.  Moreover, a
> quote the Timbot found for this list a couple years ago is worth
> reiterating:
> 
>   Many serious mathematicians have attempted to analyze a
>   sequence of floating point operations rigorously, but found
>   the task so formidable that they have tried to be content
>   with plausibility arguments instead.
>   -- Donald Knuth (_The Art of Computer Programming_, Third Edition,
>      Addison-Wesley, 1997; ISBN:  0201896842, vol.  2, p.  229):
> 
> Yours, Lulu...

As a side note, this question was ultimately motivated by an entirely
different sort of "pure math" question: Does (integer) 1 = 1.0?
Engineers, physicists, and computer scientists all would
say "no" because of the precision problem with the latter number.
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.


Thanks to all who took the time to respond...


-- 
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Tim Daneliuk     tundra at tundraware.com
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