Re: OT: mathematicians (Was: Re: Results of Poll by Email No. 27)

Nokta Kanto scripsit:

> It hardly seems complete. What is -1? The largest set contained by the
> empty set? I didn't think mathematicians would like a system that couldn't
> be extrapolated to a more general number space.
The typical approach is to build up numbers other than natural numbers
in pieces.  We can define the ordered pair (a,b) as {a, {a, b}}.  Then +5
can be rendered (0, 5) and -5 as (5, 0).  Going from integers to rationals
is easy:  5/2 is (+5, +2).  Real numbers are sets of rationals; pi is
{3/1, 31/10, 314/100, 3141/1000, 31415/10000, ...}.  Finally, complex
numbers are ordered pairs of reals: 4+5i is ({4/1}, {5/1}}.

It's all artificial, but it can afford to be:  as long as the various
rules of arithmetic are defined so that the meanings of operators are
preserved, all is well.  We have to distinguish between numeric equality
and set identity, of course:  (1, 0) = 1, but they are not the same set.

--
John Cowan           http://www.ccil.org/~cowan              cowan@ccil.org
To say that Bilbo's breath was taken away is no description at all.  There
are no words left to express his staggerment, since Men changed the language
that they learned of elves in the days when all the world was wonderful.
        --_The Hobbit_

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