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 Subject: RE: exponents Author: Alan Cooper Date: Oct 18 2004
On Oct  4 2004, lanius wrote:
> On Oct  2 2004, Alan Cooper wrote:

>> Once whole number exponents have been fully understood,
>> then the definitions of negative and fractional
>> powers can be seen as natural and convenient choices of
>> notation (which they are) rather than facts of nature
>> (which they are not).
>> The failure to clearly identify such distinctions is
>> actually one of the most confusing things about our subject,
>> and perhaps explains the sense of frustration and anger
>> with which many people leave it.

> So Alan, this is very intriguing. Would you
> please explain it further. What do you mean by not a "fact of
> nature".

Sorry for the delayed reply. (I have been distracted by other matters for a
couple of weeks). But in fact your following paragraph captures pretty well what
I mean.

> I know that at one point I lumped rules, conventions,
> and notations all into one basket. You couldn't have 4/6, for
> example (as if that weren't a perfectly good number). How do we help
> students differentiate the mathematically true, what I think you are
> calling the facts of nature, from conventions or symbolic
> expedients, however convenient they might be.  . . .

Exactly! The "fact" that powers with negative exponents are the reciprocals of
the ones with positive exponents is really just a convenient definition rather
than something that *must* be true. It would even have been possible to define
them differently (eg as negatives); though then of course then the laws of
exponents would not extend to all cases, so I don't mean to imply that there
isn't a "fact" which underlies our conventional definition.

Alan