|
|
| View previous topic :: View next topic |
| Author |
Message |
Guest
|
 Posted: Wed Jul 09, 2008 7:42 am Post subject: Re: Standardized definitions |
 |
|
On Jul 9, 12:26 am, Angus Rodgers <twir...@bigfoot.com> wrote:
...
| Quote: |
What's wrong with defining, say, pi/2 = int_0^1 dx/sqrt{1 - x^2}
(with perhaps a preliminary rough argument about arc length, say
by using similar triangles to argue that ds/dx = 1/y, where y > sqrt{1 - x^2}, or else just computing ds/dx = sqrt{1 + (dy/dx)^2},
or else pi/4 = int_0^1 sqrt{1 - x^2)dx (which is what I assumed
Timothy was suggesting). These two definitions are equivalent:
e.g. use integration by parts to prove int_0^1 sqrt{1 - x^2)dx > int_0^1 (x^2)dx/sqrt{1 - x^2}, and then write 1/sqrt{1 - x^2} > (1 - x^2)/sqrt{1 - x^2}, giving:
int_0^1 (x^2)dx/sqrt{1 - x^2}
= int_0^1 dx/sqrt{1 - x^2) - int_0^1 sqrt{1 - x^2)dx
whence:
int_0^1 dx/sqrt{1 - x^2) = 2*int_0^1 sqrt{1 - x^2)dx
(And then you could use the binomial theorem for a rational
exponent, and term-by-term integration, to derive some not-
very-rapidly-convergent series for pi.)
I'm not saying that either of these is the "best" definition of
pi; in fact, I think that for most purposes the most convenient
definition is the one usually given in rigorous analysis texts,
i.e. pi is the smallest positive real zero of the sine function.
But either definition is workable. In fact, section 12.5 of
John Stillwell, /Mathematics and Its History/ (2nd ed. 2002)
mentions that the first ten pages of the book by C. L. Siegel,
/Topics in Complex Function Theory, Vol. 1: Elliptic Functions
and Uniformization Theory/ (1969) (which unfortunately seems
to be hard to get hold of) illuminate Euler's discovery of
the addition theorem for the lemniscatic sine function by
proving the similar (but simpler) addition theorem for the
usual (circular) sine function defined by inverting the integral
int_0^x dt/sqrt{1 - t^2}.
|
As I understand it, e is almost always defined as the sum over all
nonnegative integers n of 1/n! If there are any other common
definitions, they are trivially equivalent. However, in contrast, pi
is defined differently in different texts. This suggests that no
definition is without its disadvantages. The concept of "integral" is
less elementary than the preliminary concepts needed to understand the
"least positive zero of sin" definition. Furthermore it is less
natural than the circumference definition. So, whether you want a
natural definition, or an elementary definition, it's not a great
choice. That's "what's wrong with it".
Paul Epstein |
|
| Back to top |
|
 |
| |
Ads |
Advertising
Sponsor
|
|
Angus Rodgers
Guest
|
| Posted: Wed Jul 09, 2008 11:04 am Post subject: Re: Standardized definitions |
|
|
On Wed, 9 Jul 2008 00:42:12 -0700 (PDT), pauldepstein@att.net wrote:
| Quote: |
On Jul 9, 12:26 am, Angus Rodgers <twir...@bigfoot.com> wrote:
...
What's wrong with defining, say, pi/2 = int_0^1 dx/sqrt{1 - x^2}
(with perhaps a preliminary rough argument about arc length, say
by using similar triangles to argue that ds/dx = 1/y, where y =
sqrt{1 - x^2}, or else just computing ds/dx = sqrt{1 + (dy/dx)^2},
or else pi/4 = int_0^1 sqrt{1 - x^2)dx (which is what I assumed
Timothy was suggesting). These two definitions are equivalent:
[...]
I'm not saying that either of these is the "best" definition of
pi; in fact, I think that for most purposes the most convenient
definition is the one usually given in rigorous analysis texts,
i.e. pi is the smallest positive real zero of the sine function.
But either definition is workable. [...]
As I understand it, e is almost always defined as the sum over all
nonnegative integers n of 1/n! If there are any other common
definitions, they are trivially equivalent.
|
Is it trivially equivalent to e being the unique real number such
that int_1^e dt/t = 1? Or even to e = lim_{n -> oo} (1 + 1/n)^n?
I'm not saying either of these is very hard to prove, but neither
seems "trivial".
| Quote: |
However, in contrast, pi
is defined differently in different texts. This suggests that no
definition is without its disadvantages. The concept of "integral" is
less elementary than the preliminary concepts needed to understand the
"least positive zero of sin" definition. Furthermore it is less
natural than the circumference definition. So, whether you want a
natural definition, or an elementary definition, it's not a great
choice. That's "what's wrong with it".
|
Unfortunately you snipped the context. I was specifically replying
to Michael's assertion that pi as "area of a unit circle" could not
be defined rigorously without (1) defining "rectifiable curve", and
(2) proving the Jordan Curve Theorem (or perhaps only something about
winding numbers?), and (3) proving that for 0 < x < right angle,
sin x < x < tan x. Of course, much of this does have to be done if
one insists on every word in the phrase being rigorously defined,
but FWIW, I didn't think that that was the intended interpretation,
and I was suggesting another interpretation. I wasn't claiming
rhetorically that there was "nothing wrong" with the definition.
It's workable, but inconvenient.
You /seem/ to be suggesting that pi is unusual in having different
definitions in different texts. But many definitions and theorems
are different in different texts, even when they are given the same
names. (For instance, I posted at length, about three years ago, on
the topic of many different versions of "Fatou's lemma" I'd seen.)
This still bothers me a lot, but I'm holding off from writing too
much on mere psychology.
--
Angus Rodgers
(twirlip@ eats spam; reply to angusrod@)
Contains mild peril |
|
| Back to top |
|
|
| |
Ads |
Advertising
Sponsor
|
|
Guest
|
| Posted: Thu Jul 10, 2008 12:51 am Post subject: Re: Standardized definitions |
|
|
On Jul 9, 6:44 pm, Angus Rodgers <twir...@bigfoot.com> wrote:
| Quote: |
On Wed, 9 Jul 2008 00:42:12 -0700 (PDT), pauldepst...@att.net wrote:
On Jul 9, 12:26 am, Angus Rodgers <twir...@bigfoot.com> wrote:
...
What's wrong with defining, say, pi/2 = int_0^1 dx/sqrt{1 - x^2}
(with perhaps a preliminary rough argument about arc length, say
by using similar triangles to argue that ds/dx = 1/y, where y > >> sqrt{1 - x^2}, or else just computing ds/dx = sqrt{1 + (dy/dx)^2},
or else pi/4 = int_0^1 sqrt{1 - x^2)dx (which is what I assumed
Timothy was suggesting). These two definitions are equivalent:
[...]
I'm not saying that either of these is the "best" definition of
pi; in fact, I think that for most purposes the most convenient
definition is the one usually given in rigorous analysis texts,
i.e. pi is the smallest positive real zero of the sine function.
But either definition is workable. [...]
As I understand it, e is almost always defined as the sum over all
nonnegative integers n of 1/n! If there are any other common
definitions, they are trivially equivalent.
Is it trivially equivalent to e being the unique real number such
that int_1^e dt/t = 1? Or even to e = lim_{n -> oo} (1 + 1/n)^n?
I'm not saying either of these is very hard to prove, but neither
seems "trivial".
However, in contrast, pi
is defined differently in different texts. This suggests that no
definition is without its disadvantages. The concept of "integral" is
less elementary than the preliminary concepts needed to understand the
"least positive zero of sin" definition. Furthermore it is less
natural than the circumference definition. So, whether you want a
natural definition, or an elementary definition, it's not a great
choice. That's "what's wrong with it".
Unfortunately you snipped the context. I was specifically replying
to Michael's assertion that pi as "area of a unit circle" could not
be defined rigorously without (1) defining "rectifiable curve", and
(2) proving the Jordan Curve Theorem (or perhaps only something about
winding numbers?), and (3) proving that for 0 < x < right angle,
sin x < x < tan x. Of course, much of this does have to be done if
one insists on every word in the phrase being rigorously defined,
but FWIW, I didn't think that that was the intended interpretation,
and I was suggesting another interpretation. I wasn't claiming
rhetorically that there was "nothing wrong" with the definition.
It's workable, but inconvenient.
You /seem/ to be suggesting that pi is unusual in having different
definitions in different texts. But many definitions and theorems
are different in different texts, even when they are given the same
names. (For instance, I posted at length, about three years ago, on
the topic of many different versions of "Fatou's lemma" I'd seen.)
This still bothers me a lot, but I'm holding off from writing too
much on mere psychology.
--
|
Sorry for snipping the context. I'm including your whole post (bar
the signature) to make sure I don't do this again. I think that pi is
unusual in that it is a number encountered early in one's mathematical
education that has several standard definitions. In other words, this
several-definitions property is somewhat unusual for common
mathematical constants even though I agree with you that it's common
in mathematics generally. I don't agree with you that e also has this
property of several standard definitions. You do give two definitions
of e but they seem very non-standard (as definitions) to me. Usually,
the sum of 1/n! is the definition and lim (1+1/n)^n = e is a lemma/
claim/theorem.
Paul Epstein |
|
| Back to top |
|
|
| |
Ads |
Advertising
Sponsor
|
|
Guest
|
| Posted: Thu Jul 10, 2008 12:53 am Post subject: Re: Standardized definitions |
|
|
On Jul 9, 6:59 pm, Angus Rodgers <twir...@bigfoot.com> wrote:
| Quote: |
On Wed, 09 Jul 2008 11:44:21 +0100, I wrote:
On Wed, 9 Jul 2008 00:42:12 -0700 (PDT), pauldepst...@att.net wrote:
As I understand it, e is almost always defined as the sum over all
nonnegative integers n of 1/n! If there are any other common
definitions, they are trivially equivalent.
Is it trivially equivalent to e being the unique real number such
that int_1^e dt/t = 1? Or even to e = lim_{n -> oo} (1 + 1/n)^n?
I'm not saying either of these is very hard to prove, but neither
seems "trivial".
Sorry, I forgot to give citations.
For the former definition: G. H. Hardy, /A Course of Pure Mathematics/
(10th ed. 1967), pages 399 and 405.
For the latter definition: J. M. Hyslop, /Real Variable/ (1960), pages
65, 66 and 69.
--
Angus Rodgers
(twirlip@ eats spam; reply to angusrod@)
Contains mild peril
|
Fortunately, I'm in a very generous mood today, and I can find it in
my heart to forgive you for initially omitting the citations.
Paul Epstein |
|
| Back to top |
|
|
| |
Ads |
Advertising
Sponsor
|
|
Guest
|
| Posted: Thu Jul 10, 2008 12:59 am Post subject: Re: Standardized definitions |
|
|
On Jul 9, 7:17 pm, Angus Rodgers <twir...@bigfoot.com> wrote:
| Quote: |
(Sorry about the flurry of posts all in reply to the same thing.)
On Wed, 9 Jul 2008 00:42:12 -0700 (PDT), pauldepst...@att.net wrote:
On Jul 9, 12:26 am, Angus Rodgers <twir...@bigfoot.com> wrote:
...
What's wrong with defining, say, pi/2 = int_0^1 dx/sqrt{1 - x^2}
(with perhaps a preliminary rough argument about arc length, say
by using similar triangles to argue that ds/dx = 1/y, where y > >> sqrt{1 - x^2}, or else just computing ds/dx = sqrt{1 + (dy/dx)^2},
[...]
[...] The concept of "integral" is
less elementary than the preliminary concepts needed to understand the
"least positive zero of sin" definition.
True, especially as the integral in question is either a Lebesgue
integral (I had to remind myself of several theorems I'd forgotten)
or an improper Riemann integral (or, I suppose, a gauge integral).
Furthermore it is less natural than the circumference definition.
But it /is/ the circumference definition!
--
|
If you mean "it is the circumference definition" in a pure-
mathematical and theoretical sense, then surely you would say that
your integral definition is the same as (for example) the definition
of being sqrt(6) * sum (1/n^2). After all, they can be shown to be
equivalent.
And if those two definitions are the same, then the whole concept of
"alternative definitions" has no meaning. Therefore, I assumed that
we were talking about definitions from a pedagogical standpoint rather
than a theoretical one.
Surely, you would agree that there's a huge pedagogical difference
between explaining to an average ten-year-old what the circumference
of a circle intuitively means, and giving pi as the circumference/
diameter ratio, as opposed to your integral definition.
Paul Epstein |
|
| Back to top |
|
|
| |
Ads |
Advertising
Sponsor
|
|
Guest
|
| Posted: Thu Jul 10, 2008 4:07 am Post subject: Re: Standardized definitions |
|
|
On Jul 10, 11:41 am, Gerry Myerson <ge...@maths.mq.edi.ai.i2u4email>
wrote:
| Quote: |
In article
09a1092d-28ba-4f31-bad2-09b1db079...@26g2000hsk.googlegroups.com>,
pauldepst...@att.net wrote:
I don't agree with you that e also has this property of several
standard definitions. You do give two definitions of e but they seem
very non-standard (as definitions) to me. Usually, the sum of 1/n!
is the definition and lim (1+1/n)^n = e is a lemma/ claim/theorem.
When I teach intro calculus, the definition I give for e is
that it's the number a for which the derivative of a^x is a^x.
That is, I prove that the derivative of a^x is C_a a^x, where C_a
is some constant depending on a; I give evidence that C_2 < 1
and C_3 > 1; I assert that this evidence suggests that there's
some number a, 2 < a < 3, such that C_a = 1; and then I define e
to be that number a. We only get to power series some time later,
so what you propose as the standard definition is not available;
it comes in as a theorem at the appropriate time.
--
Gerry Myerson (ge...@maths.mq.edi.ai) (i -> u for email)
|
Interesting way of teaching it. When you say "evidence", do you mean
plugging in a small value of epsilon into a computer and using finite
difference for the derivative -- f'(x) is approx (f(x + epsilon) -
f(x - epsilon)) / (2 * epsilon) ? |
|
| Back to top |
|
|
| |
Ads |
Advertising
Sponsor
|
|
Gerry Myerson
Guest
|
| Posted: Thu Jul 10, 2008 8:41 am Post subject: Re: Standardized definitions |
|
|
In article
<09a1092d-28ba-4f31-bad2-09b1db079752@26g2000hsk.googlegroups.com>,
pauldepstein@att.net wrote:
| Quote: |
I don't agree with you that e also has this property of several
standard definitions. You do give two definitions of e but they seem
very non-standard (as definitions) to me. Usually, the sum of 1/n!
is the definition and lim (1+1/n)^n = e is a lemma/ claim/theorem.
|
When I teach intro calculus, the definition I give for e is
that it's the number a for which the derivative of a^x is a^x.
That is, I prove that the derivative of a^x is C_a a^x, where C_a
is some constant depending on a; I give evidence that C_2 < 1
and C_3 > 1; I assert that this evidence suggests that there's
some number a, 2 < a < 3, such that C_a = 1; and then I define e
to be that number a. We only get to power series some time later,
so what you propose as the standard definition is not available;
it comes in as a theorem at the appropriate time.
--
Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email) |
|
| Back to top |
|
|
| |
Ads |
Advertising
Sponsor
|
|
Gerry Myerson
Guest
|
| Posted: Thu Jul 10, 2008 11:03 am Post subject: Re: Standardized definitions |
|
|
In article
<72f73567-a7d3-466e-8d0b-0d27104d23f4@a70g2000hsh.googlegroups.com>,
pauldepstein@att.net wrote:
| Quote: |
On Jul 10, 11:41 am, Gerry Myerson <ge...@maths.mq.edi.ai.i2u4email
wrote:
When I teach intro calculus, the definition I give for e is
that it's the number a for which the derivative of a^x is a^x.
That is, I prove that the derivative of a^x is C a a^x, where C a
is some constant depending on a; I give evidence that C 2 < 1
and C 3 > 1; I assert that this evidence suggests that there's
some number a, 2 < a < 3, such that C a = 1; and then I define e
to be that number a. We only get to power series some time later,
so what you propose as the standard definition is not available;
it comes in as a theorem at the appropriate time.
Interesting way of teaching it. When you say "evidence", do you mean
plugging in a small value of epsilon into a computer and using finite
difference for the derivative -- f'(x) is approx (f(x + epsilon) -
f(x - epsilon)) / (2 * epsilon) ?
|
Yes, pretty much.
--
Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email) |
|
| Back to top |
|
|
| |
Ads |
Advertising
Sponsor
|
|
Rob Johnson
Guest
|
| Posted: Fri Jul 11, 2008 3:41 am Post subject: Re: Standardized definitions |
|
|
In article <g55sli$43re@odds.stat.purdue.edu>,
hrubin@odds.stat.purdue.edu (Herman Rubin) wrote:
| Quote: |
In article <20080705.051057@whim.org>, Rob Johnson <rob@trash.whim.org> wrote:
In article <g4m5ic$19o$1@aioe.org>,
Joshua Cranmer <Pidgeot18@gmail.com> wrote:
Rob Johnson wrote:
"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.
I suggest we drop the word "definitions" almost completely
and use instead "characterizations". It does not matter if
pi is defined as the ratio of the circumference of a circle
in a Euclidean plane to its diameter, or 2/c, where c is the
limit of the directrix as it approaches the x-axis, or the
integral of dx/(1+x^2) over the real line, or any other of
the multitude of expressions which equal it.
|
Again, a mis-snip. I was replying to a misquote of a message in
which I was arguing a point of fact (sqrt(6) * sqrt(sum 1/n^2) is pi)
and it was misconstrued that I was replying to the proof-by-
redefinition aspect of the thread.
I agree with you. The word "definition" carries a connotation of
definitiveness that is not always intended. Many books define a
term and then go on to prove the equivalence of two definitions of
that term. A definition should be the definitive characterization
of a term and all other equivalent characterizations should be called
just that; characterizations. However, one book's definition, may be
another book's equivalent characterization.
Rob Johnson <rob@trash.whim.org>
take out the trash before replying
to view any ASCII art, display article in a monospaced font |
|
| Back to top |
|
|
| |
Ads |
Advertising
Sponsor
|
|
|
|
You cannot post new topics in this forum
You cannot reply to topics in this forum
You cannot edit your posts in this forum
You cannot delete your posts in this forum
You cannot vote in polls in this forum
|
|
FAQ
Search
Memberlist
Usergroups
Register
Profile
Log in to check your private messages
Log in
