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
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