Re: operations between different numbers
- From: Jim Spriggs <jim.sprigs@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxx>
- Date: Sat, 24 Sep 2005 04:27:31 +0000 (UTC)
sizors_1@xxxxxxxxxxx wrote:
>
> I'm wondering, can you add, subtract, multiply, etc. between different
> kinds of numbers? For instance, 3 * 1/8 = 3/8, but are we multiplying
> 3 the *natural number* (or integer) by 1/8 *the rational number*, or,
> technically speaking, are we really multiplying the rational equivalent
> of 3 (i.e., 3/1) by 1/8? I guess what I'm asking is, have
> mathematicians defined operations between different number sets?
Yes, sort of. Starting from the integers
..., -4, -3, -2, -1, 0, 1, 2, 3, 4, ...
one can define rational numbers as classes of pairs of integers. The
rational p/q is defined as the class that the pair (p, q) goes into
along with all the pairs (m, n) such that pn = qm. So, for example
2/3 is defined to be {(2, 3), (-2, -3), (4, 6), (-4, -6), ...}.
The operations of addition and multiplication on the rational numbers
are then defined in terms of those sets [*]. This doesn't define
5 * 2/3 or -5 + 2/3
because 5 and -5 are not rational numbers _according to the definition_
but it does define
5/1 * 2/3 or -5/1 + 2/3.
Since the integers can be "embedded" in the rationals, for example
5 corresponds to the set {(5, 1), (-5, -1), (10, 2), (-10, -2), ...},
one just forgets that when doing rational arithmetic the integer 5 has
to be replaced by the rational 5/1 and one pretends that the integer 5
is the same as the rational 5/1.
Similar subterfuges go on when one defines integers as being sets of
ordered pairs of natural numbers, real numbers as being Dedekind cuts of
rationals (or whatever), and complex numbers as being ordered pairs of
real numbers. As one ascends one really leaves the previous system
behind but in practise an embedding allows one to pretend that one
hasn't.
To continue the story a little... ordered pairs can be defined in terms
of sets, natural numbers can be defined in terms of sets, Dedekind cuts
can be defined as sets; so all one is left with is sets and operations
on them. All of the sets are built up from the empty set by repeated
operations of power set and union. So even seemingly concrete
mathematical entities like numbers are built up out of thin air!
[*] The operations are defined as follows. Let [p, q] be the class of
pairs that p/q is defined to be, then
[p, q] + [m, n] is defined to be [pn + qm, qn]
and
[p, q] * [m, n] is defined to be [pm, qn].
The properties of the rationals can then be deduced from the properties
of the integers.
--
I don't know who you are Sir, or where you come from,
but you've done me a power of good.
.
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