John M. Flanigan <johnf@hawaii.edu> The equation is the final arbiter.
Math Resource Instructor --Werner Heisenberg
Kapi'olani Community College The scoreboard is the final arbiter.
Honolulu, Hawaii --Bill Walton
On Mon, 23 Jun 1997, Steve Kraisler wrote:
> I was rereading your message of a while back and enjoyed thinking about
> number concepts.
> We say that the number 2 is a)integer b)rational c)real. What that means I
> think
> is the number 2 is a member of the set of intergers, the set of rationals
> etc.
>
> Is 2 a complex number? Well, construct the set of complex numbers-that is
> give closure rules and operation rules and
> then see if 2 is a member of the set.
>
> i believe the question "Is 2(the real number) and 2 (the complex number)
> the same ?" a piece of platonic nonsense.
>
>
> >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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> Steve Kraisler Internet: stevek@forum.swarthmore.edu
> Mathematics Dept. stevek@icdc.com
> Edison High School Fidonet: steve_kraisler@f941.n273.z1.fidonet.org
> Philadelphia, PA 19140 Voice: (215)-324-9440 (X3390)
> Fax: (215)-242-9279
> WWW: http://forum.swarthmore.edu/~stevek
>
>
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