[Maths-Education] concentric and congruent circles
Hugh.Burkhardt at nottingham.ac.uk
Hugh.Burkhardt at nottingham.ac.uk
Fri Jan 29 12:23:20 GMT 2010
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"
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