[Maths-Education] concentric and congruent circles
Shelley Walsh
shelleywalsh at mac.com
Fri Jan 29 13:10:37 GMT 2010
Interesting seeing the various perspectives on 1 being a prime. I
usually take the historical perspective that when people first
starting being interested in primes and composites 1 was not
considered to be a number. I think there is an extent that we all can
identify with of one not being a number in the sense that the other
whole numbers are. It's singular, not plural, even our grammar makes
the distinction. One is the unit out of which all the numbers are made
of. That's why it isn't composite either. It's a different kind of
critter. It's not in the game.
On 29 Jan 2010, at 12:37, John Bibby wrote:
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> 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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