Standardized definitions
paulde...@att.net
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
paulde...@att.net
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
amzoti
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.
William Elliot
There isn't a standard definition. Only a standard value.
Definitions that are not standard velued, are substandard.
Gerry Myerson
<f6203d20-0f65-4072...@h1g2000prh.googlegroups.com>,
paulde...@att.net wrote:
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)
paulde...@att.net
> 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
>
>
> 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
Virgil
William Elliot <ma...@hevanet.remove.com> wrote:
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.
toni.l...@gmail.com
> 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."
William Elliot
> William Elliot <ma...@hevanet.remove.com> wrote:
> > On Thu, 3 Jul 2008 paulde...@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.
> > >
>
> William seems to be saying that we cannot have 4 = 2 + 2 because we
> already have 4 = 3 + 1.
>
definitions because they both yield the standard value for 4.
G. A. Edgar
>
> 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.
--
G. A. Edgar http://www.math.ohio-state.edu/~edgar/
G.E. Ivey
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.
David C. Ullrich
>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) ?
The fact that pi _wasn't_ defined this way.
> Is there any
>implicit rule that says which definitions of pi are acceptable and
>which aren't?
When a student has a homework problem he's not allowed
to invent his own definitions for previously-defined terms
used in the statement of the problem - if a term has been
defined then _the_ definition is the definition.
In a given context a term has _a_ definition - the fact that
the definition _could_ have been given differently doesn't
change that.
>Paul Epstein
David C. Ullrich
"Understanding Godel isn't about following his formal proof.
That would make a mockery of everything Godel was up to."
(John Jones, "My talk about Godel to the post-grads."
in sci.logic.)
Frederick Williams
> 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) ?
Nothing, nor is there anything to prevent the marker giving the student
nought.
--
He is not here; but far away
The noise of life begins again
And ghastly thro' the drizzling rain
On the bald street breaks the blank day.
Rob Johnson
>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 <r...@trash.whim.org>
take out the trash before replying
to view any ASCII art, display article in a monospaced font
Joshua Cranmer
> "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.
Angus Rodgers
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
Michael Press
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
Rob Johnson
You snipped the message to which I was replying. In that message, the
claim was
In article <32190234.1215170789...@nitrogen.mathforum.org>,
"G.E. Ivey" <georg...@gallaudet.edu> wrote:
>> 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.
The claim was that this definition does not give the same value.
I was replying to that claim.
Angus Rodgers
Questions of definition have always worried me. I've thrown a few
remarks about the definition of the number e, and powers u^x and
logarithms log_u(y), into the old thread "Re-writing a variable to
the power of a fraction" (17 June). (I hope they're not excessive -
these things are a matter of taste, and I can't tell if my taste is
screwy.)
As for pi: I suspect there's a neat definition buried in Bourbaki's
/General Topology/, but you have to read through some stuff on
topological groups to get to it. I vaguely recall that it was
Weil's idea, but that may be wrong. (Sorry to be so vague, but I'm
only slowly working to fill the many yawning gaps in my knowledge.)
victor_me...@yahoo.co.uk
> Now we _prove_ that there is a unique
> real number pi such that
>
> {w in C: exp(w) = 1} = 2.pi.i.Z.
It's a bit difficult to prove that, since there are
two real numbers with that property :-)
Victor Meldrew
"I don't believe it!"
Michael Press
<bc19c90c-0fb7-4925...@y38g2000hsy.googlegroups.com>,
victor_me...@yahoo.co.uk wrote:
> On 5 Jul, 04:18, Michael Press <rub...@pacbell.net> wrote:
> > Now we _prove_ that there is a unique
> > real number pi such that
> >
> > {w in C: exp(w) = 1} = 2.pi.i.Z.
>
> It's a bit difficult to prove that, since there are
> two real numbers with that property :-)
Drat! Make that
unique real number pi > 0 such that
{w in C: exp(w) = 1} = 2.pi.i.Z.
--
Michael Press
lwa...@lausd.net
> 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.
Angus Rodgers
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. :-)
paulde...@att.net
> 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
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
Timothy Murphy
> 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.
I would have thought "area of a unit circle" was both simpler to understand
and easier to define rigorously.
Michael Press
Timothy Murphy <gayl...@eircom.net> wrote:
Try it. You always end up having to define a rectifiable curve,
(and that a closed curve has an inside and an outside.)
Our intuition that for 0 < x < right angle
sin x < x < tan x is the basis for elementary geometric mensuration;
but proving it is not elementary.
--
Michael Press
Angus Rodgers
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}.
paulde...@att.net
...
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
Angus Rodgers
>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.
Angus Rodgers
>On Wed, 9 Jul 2008 00:42:12 -0700 (PDT), paulde...@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.
Timothy Murphy
>> I would have thought "area of a unit circle" was both simpler to
>> understand and easier to define rigorously.
>
> Try it. You always end up having to define a rectifiable curve,
> (and that a closed curve has an inside and an outside.)
It doesn't seem to me to have anything to do with either of these.
Not only can the area of an open subset of R^2 be defined
in an elementary way (see eg Burkill's classic text on Lebesgue integration)
but the notion of area has strong intuitive content -
I mean it is taken for granted in school mathematics
that area and volume are well-defined concepts.
Angus Rodgers
On Wed, 9 Jul 2008 00:42:12 -0700 (PDT), paulde...@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!
paulde...@att.net
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
paulde...@att.net
> On Wed, 09 Jul 2008 11:44:21 +0100, I 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
paulde...@att.net
> (Sorry about the flurry of posts all in reply to the same thing.)
>
> >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
Gerry Myerson
<09a1092d-28ba-4f31...@26g2000hsk.googlegroups.com>,
paulde...@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)
paulde...@att.net
wrote:
> In article
> <09a1092d-28ba-4f31-bad2-09b1db079...@26g2000hsk.googlegroups.com>,
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) ?
Gerry Myerson
<72f73567-a7d3-466e...@a70g2000hsh.googlegroups.com>,
paulde...@att.net wrote:
> 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.
Angus Rodgers
In the thread I referred to earlier ("Re-writing a variable to the
power of a fraction", 17 June), I suggested arguing as follows (in
slightly changed notation):
(a^{x + h} - a^x)/h
= a^x * (a^h - 1)/h
= a^x * (int_1^a t^{h - 1} dt)
and the integral is (either by hand-waving - appealing to earlier,
less rigorous knowledge of calculus - or else by using compactness,
uniform continuity, the most elementary properties of the Riemann
integral, and an exp-free proof of d(x^h)/dx = hx^{h - 1} - as also
given in that thread - or is this overkill?) obviously a continuous,
increasing function of h (assuming that a > 1 - it's decreasing if
0 < a < 1), therefore, as h tends to 0, it tends to int_1^a dt/t.
Denoting this function of a by ln(a), we easily prove ln(ab) = ln(a) +
ln(b), and of course ln(1) = 0, and ln is continuous. (We can either
use the integral expression, as in Hardy's book, IIRC, or else argue
from ln(a) = lim_{h -> 0} (a^h - 1)/h, e.g. ln(a/b) = lim-{h -> 0}
b^{-h} * ((a^h - 1)/h - (b^h - 1)/h) = ln(a) - ln(b), etc.) Hence
(using only elementary, exp-free properties of real powers of real
numbers, as discussed in that thread) it follows that there exists a
unique real number e (> 1) such that ln(a) = log_e(a) for all a > 0
(where log_b is defined for all b > 0 as the inverse of x |--> b^x).
This seems to me to be about the most "natural" definition of e, and
it neatly ties together: an intuitive definition of real powers of
real numbers; logarithms as inverses of exponential functions; the
alternative definition of ln as an integral; and calculating the
derivatives of real powers of real numbers with respect to either
argument.
I hope I'm not sounding Messianic! If so, call me Mr. Natural. :-)
It's not something I've ever bothered to work out in detail (and
indeed, I only thought of most of it in response to that earlier
thread), but I honestly haven't run into anything suggesting that
there would be difficulties in a detailed development (/pace/ Wade!).
Not being a teacher, however, I have no idea how it would work out
pedagogically - except that, as a student, I would have liked to
have been taught it that way myself.
ge...@math.mq.edu.au
logarithms and exponentials before integrals, so it won't
work for me.
--
GM
Herman Rubin
>In article <g4m5ic$19o$1...@aioe.org>,
>Joshua Cranmer <Pidg...@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.
--
This address is for information only. I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Department of Statistics, Purdue University
hru...@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558
Rob Johnson
hru...@odds.stat.purdue.edu (Herman Rubin) wrote:
>In article <2008070...@whim.org>, Rob Johnson <r...@trash.whim.org> wrote:
>>In article <g4m5ic$19o$1...@aioe.org>,
>>Joshua Cranmer <Pidg...@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.