Re: Standardized definitions

In article <32190234.1215170789924.JavaMail.jakarta@xxxxxxxxxxxxxxxxxxxxxx>,
"G.E. Ivey" <george.ivey@xxxxxxxxxxxxx> wrote:
On Jul 4, 1:39 pm, amzoti <amz...@xxxxxxxxx> wrote:
On Jul 3, 9:55 pm, pauldepst...@xxxxxxx wrote:
Are there any authorized "standard definitions" for terms like e, pi
etc? My impression is that there isn't and that a wide variety of
definitions are acceptable so long as they are clear and precise and
lead to the correct value.

So, if a student is asked to prove that the sum (over all positive
integers n) of 1/n^2 is pi^2/6, what is to prevent the student from
defining pi to be sqrt(6) * sqrt(sum 1/n^2) ? Is there any
implicit rule that says which definitions of pi are acceptable and
which aren't?

Paul Epstein


1.http://numbers.computation.free.fr/Constants/Pi/pi.html


2.http://numbers.computation.free.fr/Constants/constants.html

see the same constants pages for others.

There are many definitions for these on the web -
google is your
friend.

Google is a good friend, but a good friend also
listens to others. I
don't think you read my post carefully enough. Of
course, various
websites give various definitions!

The point I was getting at is: Take the following
three definitions
of pi. 1) The circumference of a circle with unit
diameter. 2) The
least positive real z such that cos z (defined via
power series) =
-1. 3) sqrt(6) * sqrt(sum 1/n^2).

My opinion is that the mathematical community would
object to
definition 3 but would find definitions 1 and 2 to be
fine. Why??

What is it about definition 3 that makes it not ok?

What's wrong with it, just as others have said, is that it does
not give the same value the others two doefinitions do. Definitions
1 and 2 can be shown to define the same VALUE. 3 does not. It is
the value that is crucial, not the particular definition.

"sum 1/n^2", accepting the notation, is pi^2/6. Thus, the value of
sqrt(6) * sqrt(sum 1/n^2) is pi.

Furthermore, I don't see any posts in which others claims that
this does not give the same value as the other two definitions.

Rob Johnson <rob@xxxxxxxxxxxxxx>
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