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 Posted: Fri Jul 04, 2008 4:55 am Post subject: Standardized definitions |
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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 |
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| Posted: Fri Jul 04, 2008 5:08 am Post subject: Re: Standardized definitions |
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On Jul 4, 12:55 pm, pauldepst...@att.net wrote:
| Quote: |
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.
Paul Epstein |
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amzoti
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| Posted: Fri Jul 04, 2008 5:39 am Post subject: Re: Standardized definitions |
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On Jul 3, 9:55 pm, pauldepst...@att.net wrote:
| Quote: |
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. |
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Guest
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| Posted: Fri Jul 04, 2008 6:39 am Post subject: Re: Standardized definitions |
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On Jul 4, 1:39 pm, amzoti <amz...@gmail.com> wrote:
| Quote: |
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?
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?
Paul Epstein |
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| Posted: Fri Jul 04, 2008 7:16 am Post subject: Re: Standardized definitions |
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On Jul 4, 7:55 am, pauldepst...@att.net wrote:
| Quote: |
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).
| Quote: |
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." |
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William Elliot
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| Posted: Fri Jul 04, 2008 10:41 am Post subject: Re: Standardized definitions |
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On Thu, 3 Jul 2008 pauldepstein@att.net wrote:
| Quote: |
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. |
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Virgil
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| Posted: Fri Jul 04, 2008 11:02 am Post subject: Re: Standardized definitions |
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In article <Pine.BSI.4.58.0807032239250.441@vista.hevanet.com>,
William Elliot <marsh@hevanet.remove.com> wrote:
| Quote: |
On Thu, 3 Jul 2008 pauldepstein@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.
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. |
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William Elliot
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| Posted: Fri Jul 04, 2008 11:02 am Post subject: Re: Standardized definitions |
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On Fri, 4 Jul 2008, Virgil wrote:
| Quote: |
William Elliot <marsh@hevanet.remove.com> wrote:
On Thu, 3 Jul 2008 pauldepstein@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. |
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Gerry Myerson
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| Posted: Fri Jul 04, 2008 11:02 am Post subject: Re: Standardized definitions |
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In article
<f6203d20-0f65-4072-a132-ff23e3069ef2@h1g2000prh.googlegroups.com>,
pauldepstein@att.net wrote:
| Quote: |
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 (gerry@maths.mq.edi.ai) (i -> u for email) |
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Rob Johnson
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| Posted: Sat Jul 05, 2008 12:36 am Post subject: Re: Standardized definitions |
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In article <32190234.1215170789924.JavaMail.jakarta@nitrogen.mathforum.org>,
"G.E. Ivey" <george.ivey@gallaudet.edu> wrote:
| Quote: |
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?
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@trash.whim.org>
take out the trash before replying
to view any ASCII art, display article in a monospaced font |
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Joshua Cranmer
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| Posted: Sat Jul 05, 2008 2:43 am Post subject: Re: Standardized definitions |
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Rob Johnson wrote:
| Quote: |
"sum 1/n^2", accepting the notation, is pi^2/6. Thus, the value of
sqrt(6) * sqrt(sum 1/n^2) is pi.
|
The entire crux of the matter is that the student is being asked to
prove that "sum 1/n^2" is "pi^2/6"; although it is the same value, the
definition of pi as that expression does not actually succeed in solving
the problem at hand. |
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Angus Rodgers
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| Posted: Sat Jul 05, 2008 4:56 am Post subject: Re: Standardized definitions |
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On Thu, 3 Jul 2008 21:55:28 -0700 (PDT), pauldepstein@att.net wrote:
| Quote: |
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?
|
Funnily enough, from A. F. Beardon, /Limits: A New Approach to Real
Analysis/ (Springer 1997), p. 89f.:
"How should we define pi? We could use either of the known formulae
pi/4 = 1 - 1/3 + 1/5 - 1/7 + ...,
pi^2/6 = 1/1^2 + 1/2^2 + 1/3^2 + 1/4^2 + ...
[...] or we could use geometry and define pi as the angle sum of a
triangle, or perhaps as the length of the circumference of a circle
divided by the length of its diameter. In fact, the best way is to
avoid lengths and angles altogether and to define pi /in terms of
the zeros of the function sin/ [...]"
I know this doesn't address your question! But I thought it was
amusing to see a textbook actually mentioning the possibility of
defining pi using Euler's solution to the Basel problem.
In a quite similar vein, Apostol, /Mathematical Analysis/ (2nd ed.,
Addison-Wesley 1974), p. 19, does actually define the exponential
function of a complex variable by using another famous identity of
Euler's, thus:
"If z = x + iy, we define e^z = e^{x + iy} to be the complex number
e^z = (e^x)(cos y + i*sin y)"
(where, I must say, cos and sin do not actually seem to have been
defined earlier in the book).
--
Angus Rodgers
(twirlip@ eats spam; reply to angusrod@)
Contains mild peril |
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Michael Press
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| Posted: Sat Jul 05, 2008 8:18 am Post subject: Re: Standardized definitions |
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In article <Pine.BSI.4.58.0807032239250.441@vista.hevanet.com>,
William Elliot <marsh@hevanet.remove.com> wrote:
| Quote: |
On Thu, 3 Jul 2008 pauldepstein@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.
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.
|
A coherent development of numbers circumscribes
possible definitions. One development defines
complex numbers, then defines exp(z) as a power
series. Now we _prove_ that there is a unique
real number pi such that
{w in C: exp(w) = 1} = 2.pi.i.Z.
--
Michael Press |
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Guest
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| Posted: Sun Jul 06, 2008 8:13 am Post subject: Re: Standardized definitions |
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On Jul 3, 9:55 pm, pauldepst...@att.net wrote:
| Quote: |
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?
|
The Metamath website defines e to be exp(1):
http://us.metamath.org/mpegif/df-e.html
where exp is defined via its Taylor series:
http://us.metamath.org/mpegif/df-ef.html
Meanwhile, pi is defined to the the smallest positive number
whose sine is zero:
http://us.metamath.org/mpegif/df-pi.html
where sine is defined from exp above, via a form of Euler's
identity, viz. sin(x) = (exp(ix)-exp(-ix))/2i:
http://us.metamath.org/mpegif/df-sin.html
Angus Rodgers hints at such a definition later in this thread. |
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Guest
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| Posted: Mon Jul 07, 2008 1:37 am Post subject: Re: Standardized definitions |
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On Jul 6, 8:06 pm, Angus Rodgers <twir...@bigfoot.com> wrote:
| Quote: |
On Sun, 6 Jul 2008 01:13:22 -0700 (PDT), lwal...@lausd.net wrote:
On Jul 3, 9:55 pm, pauldepst...@att.net wrote:
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 whichdefinitionsof pi are acceptable and
which aren't?
The Metamath website defines e to be exp(1):
http://us.metamath.org/mpegif/df-e.html
where exp is defined via its Taylor series:
http://us.metamath.org/mpegif/df-ef.html
Meanwhile, pi is defined to the the smallest positive number
whose sine is zero:
http://us.metamath.org/mpegif/df-pi.html
where sine is defined from exp above, via a form of Euler's
identity, viz. sin(x) = (exp(ix)-exp(-ix))/2i:
http://us.metamath.org/mpegif/df-sin.html
Angus Rodgers hints at such a definition later in this thread.
That definition of pi is the nearest to a "standard" one that I
know, but what I was hinting at was some nice way that Bourbaki
seems to have of defining the function R --> U, t |--> e^{it} as
a homomorphism from the additive group of real numbers to the
multiplicative group of complex numbers of unit modulus. As you
might expect with Bourbaki, there's rather too much preliminary
material for me (or anyone not prepared to put some time into it)
just to pull the relevant section out of context. However, it's
in /Elements of Mathematics: General Topology, Part 2/ (I have a
photocopy of part of the 1966 English translation), Chapter VIII,
section 2, subsection 1 [gawd! - what was that Wade said about a
CPA poring over the tax code?], "The multiplicative group U". A
reference back to Chapter V, section 3, Theorem 2 proves that U
is isomorphic (as a topological group) to T = R/Z, but it doesn't
single out any particular isomorphism. However, they immediately
go on to point out that there are only two such isomorphisms
(basically because, from Chapter VII, section 2, Proposition 6 -
of which, dammit, I don't possess a photocopy! - T only has two
automorphisms). They pick g: T --> U to be such that g(1/4) = i,
and then they define the homomorphism bold{e} = g o phi: R --> U,
where phi: R --> T is the canonical homomorphism. Then (finally!)
pi is defined by:
lim_{x -> 0} (bold{e}(x/(2pi)) - 1)/x = i
which looks like a neat way of saying that the homomorphism R --
U, x |--> bold{e}(x/(2pi)) preserves "arc length" locally. Well,
I like it - even though I don't understand it! But I think I'll
have to join the League for Fighting Chartered Accountancy. :-)
--
|
In my experience, I would say that at undergrad level and higher,
definitions of the flavour sin z = 0 or exp(2zi) = 1 etc. are the most
common. [I'm doing a rough summary -- those definitions are not
complete.]
The most natural definition is probably "the circumference of a circle
with unit diameter." When I learnt the definition of pi just over
thirty years ago, that definition was by far the most common in
British elementary schools. My guess is that it still is. However,
there's quite a lot of work involved to translate the elementary-
school defintion into rigorous elementary mathematics because the
concept of "length" is quite difficult to develop rigorously. Much
more work is involved in making "length" rigorous than in proving
convergence and zeroes-existence of series like sin z and exp(z) - 1.
In the undergrad world, where rigour is more highly-valued than at
lower levels, the sin z = 0 type definitions therefore tend to be
preferred over the circumference or area definitions.
Paul Epstein |
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