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?
> 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?
To clarify, it seems to me that 2, 3, 4, etc. do have fairly standard definitions (for example 2 is defined as 1 + 1). However, pi does not appear to have a standard definition. Some would define it as the circumference of a circle of unit diameter, but others would define it as the smallest real solution of cos z = -1 where cos z is defined in power-series form.
> 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?
> On Jul 4, 12:55 pm, pauldepst...@att.net 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?
> To clarify, it seems to me that 2, 3, 4, etc. do have fairly standard > definitions (for example 2 is defined as 1 + 1). However, pi does not > appear to have a standard definition. Some would define it as the > circumference of a circle of unit diameter, but others would define it > as the smallest real solution of cos z = -1 where cos z is defined in > power-series form.
There isn't a standard definition. Only a standard value. Definitions that are not standard velued, are substandard.
In article <f6203d20-0f65-4072-a132-ff23e3069...@h1g2000prh.googlegroups.com>,
pauldepst...@att.net 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?
Context. What a student is & isn't permitted to do depends on the context within which the question is asked. The context will tell you what's to be assumed & what's to be proved.
-- Gerry Myerson (ge...@maths.mq.edi.ai) (i -> u for email)
> > 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?
> 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?
> > On Jul 4, 12:55 pm, pauldepst...@att.net 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?
> > To clarify, it seems to me that 2, 3, 4, etc. do have fairly standard > > definitions (for example 2 is defined as 1 + 1). However, pi does not > > appear to have a standard definition. Some would define it as the > > circumference of a circle of unit diameter, but others would define it > > as the smallest real solution of cos z = -1 where cos z is defined in > > power-series form.
> There isn't a standard definition. Only a standard value. > Definitions that are not standard velued, are substandard.
William seems to be saying that we cannot have 4 = 2 + 2 because we already have 4 = 3 + 1.
There is nothing in mathematics, or logic, even in or everyday usage, that prevents a definiendum from having multiple definiens.
> 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.
Equivalent mathematical definitions are equal. None is better than the other (objectively).
> 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?
Usually, in cases where the lecturer wishes the student to do the exercise using a specific method, (s)he would do well to guide the student in the desired direction, as in: "Use Parseval's theorem to show that pi^2/6 = sum 1/n^2."
On Fri, 4 Jul 2008, Virgil wrote: > William Elliot <ma...@hevanet.remove.com> wrote: > > On Thu, 3 Jul 2008 pauldepst...@att.net wrote:
> > > On Jul 4, 12:55 pm, pauldepst...@att.net 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.
> > There isn't a standard definition. Only a standard value. > > Definitions that are not standard valued, are substandard.
> William seems to be saying that we cannot have 4 = 2 + 2 because we > already have 4 = 3 + 1.
I did not. 4 = 2 + 2 and 4 = 3 + 1 are both standard value definitions because they both yield the standard value for 4.
> 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) ?
At a certain level, students are inclined to do things like this. But one aspect of "mathematical maturity" is recognizing that when such a question is asked, this definition is not the one intended for pi.
> On Jul 4, 1:39 pm, amzoti <amz...@gmail.com> wrote: > > On Jul 3, 9:55 pm, pauldepst...@att.net 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?
> > 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?
> Paul Epstein
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.
