There are a lot of logical traps hiding in the process of trying to decide what
kind of number a given expression "is". Is 3+0i the same as 3? Is this a real
number? Is it an integer? Etc. etc.
The only logically satisfying way to resolve this sort of question that I have
ever come across is to realize that numbers do not exist independently but only
as part of particular number systems. Classically, the natural
numbers can be developed from simple set theory (zero is the empty set, one is
the set containing only the empty set, etc.) and the arithmetic operations can
be developed from this basis. Once you have the natural numbers you can
develop the _integers_ as equivalence classes of pairs of
natural nos. For example the pairs (4,6) and (0,2) and (100,102) all correspond
to the integer "negative two", for which we introduce the new symbol "-2". But
for the pairs (6,4) and (2,0) and such we re-use the old symbol "2". The
meaning attached to this symbol is _not the same_ as it was when we were
discussing the natural numbers, but we don't let this bother us! We can neatly
define arithmetic operations between these equivalence classes that are
isomorphic (correspond exactly) to the operations we previously defined for
natural numbers when we operate only with the integers that correspond to the
old natural numbers, and we just say that the natural numbers are "embedded"
in the integers. We never think of the classes of pairs again.
In the same way we can extend the set of integers to equivalence classes of
pairs of integers such that the second number in the pair is non-zero, and we
have created the rational numbers. Instead of writing, say, (1,2) we normally
write 1/2 and consider it "equal" to 3/6, etc. For the equivalence class
containing the pair (2,1) we again use the same old symbol 2, but we again mean
something new by it! Hence the old question: is 2 a fraction?
The process of extending the rationals to the reals is a lot more
sophisticated, having been worked out carefully not much more than 100 years
ago. Dedekind cuts or Cauchy sequences (equivalence classes once more), we
again arrive at a new entity that we like to label 2, but it is a much more
sophisticated construct...
We are now ready to construct the complex numbers. This time we can simply set
up pairs (a,b) of real numbers (no equivalence classes) with a special rule for
multiplication. We generally write the pair (a,b) as a+bi, although (beware!)
we also mean something different by the "+" here!! Because operations between
complex numbers of the form (a,0) or a+0i are isomorphic to operations with
real numbers of the form a, we like to call those special complex numbers
"real" and write them in short form, though we are still _considering_ them as
complex numbers capable of entering into operations with other complex numbers.
Thus the symbol "2" takes on one more new meaning.
So we see that the bare question of what kind of number 2 (or any expressed
number of your choice) really "is" is kind of a meaningless question. With all
these multiple layers of meaning floating around its a wonder we don't all go
nuts trying to keep things straight! Fortunately the human mind has the power
to "fuzz" reality, hiding inessential detail, and we get by from day to day.
Only when the need arises for absolute clarity do we need to descend to precise
detail.
Of course there is the important pedagogical question of what part of this
logical development we should share with our students, and when. Certainly any
grad student in math ought to be led to develop this perspective. But it would
be silly to try to be formal in grade school (it was tried 30 years ago
anyway). Somewhere in between there ought to be some comfortable middle ground
that would be helpful to present to students and get them to work with, but I
have not seen it yet. Mostly we just get the same old antique, confusing
definitions and exam questions that ask students to decide whether the integers
are a subset of the irrationals, or whatever. This is definitely an area where
the standard math curriculum needs to "grow up". Comments, anyone?
RWW Taylor
National Technical Institute for the Deaf
Rochester Institute of Technology
Rochester NY 14623
>>>> The plural of mongoose begins with p. <<<<
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