[Maths-Education] concentric and congruent circles
John Bibby
johnbibbyjohnbibby at gmail.com
Fri Jan 29 12:37:39 GMT 2010
Hugh's comment reminds me that pedagogically it can be useful to give
definitions which invites exploration. Thus I like to define a prime as a
nunmber that has exactly two factors. This invites the exploration for ANY
number - I wonder how many factors it has? And why is this equivalent to the
usual definition - and what is unique about 1?
It also invites progress to the Laplacian (?) formula for the number of
factors for any integer: namely (a+1)*(b+1)*... where the number's prime
decomposition is p1^a * p2^b etc. So (to continue the exploration) if you
look at the squares 1,4,9,16 .... , how many factors does each one have -
and why? - and why are none of them prime?
(Actually I find primes much more interesting than concentric circles - its
a pity that Euclid is remembered as a geometer, not a number theorist.)
JOHN BIBBY
On 29 January 2010 12:23, <Hugh.Burkhardt at nottingham.ac.uk> wrote:
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> This is a good opportunity to make the point, to students as well as to
> ourselves, that:
> definitions in mathematics are human constructs **, involving
> choices
> there is usually a cost benefit analysis involved in standard
> definitions, for example:
>
> Why do we say that a square is a rectangle (when, in informal language, it
> is obviously not an oblong)? Because excluding it would put a hole in a
> continuum, as geometry software shows (There may be other better reasons;
> that it fits the definition is not enough, see next example) Saying
> concentric circles of the same radius aren't concentric circles has the same
> problem.
>
> Why is 1 not a prime? (It has no factors except one and itself.) We
> choose explicitly to exclude it because letting it in would ruin the unique
> prime factorisation theorem (You can multiply by as many 1s as you choose)
>
> Pure mathematicians choose (for reasons that seem to me artificial) to
> demand that "functions" are unique mappings, thus excluding square roots,
> arcsin, etc and much of complex numbers. Other mathematicians don't want
> to lose these, distinguishing "single valued functions" etc.
>
> It is rare that students are taught about this vital aspect of choice in
> definitions. Once you see it, it clears up a lot of things. (Units are
> similarly matters of convenience -- but that's another topic)
>
> Hugh Burkhardt
>
>
> ** As Kronecker (I think) said:
> "Die ganzen Zahlen hat Gott gemacht; alles andere is Menschenwerk"
>
--
Best wishes for 2010
JOHN BIBBY
(NB: this is my only current email address now.)
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