On 12/22/23 11:22 PM, The Horny Goat wrote:
> On Thu, 21 Dec 2023 08:59:12 -0800, Paul S Person
> <psperson@old.netcom.invalid> wrote:
>
>> On Wed, 20 Dec 2023 23:35:11 -0800 (PST), Andrew McDowell
>> <mcdowell_ag@sky.com> wrote:
>>
>> <snippo>
>>
>>> I believe that school geometry is educationally very useful, because it allows people to learn and practice with mathematical proof. I was entranced by proof, perhaps because almost every other form of argument I had seen before was, at best, occupying a position that might need to be abandoned in the light of further evidence or the other side of the story - and few enough paid even that much attention to the facts of the case.
>>
>> The teacher of the Geometry course I took in High School (long, long
>> ago ... but not far, far away) /explicitly stated/ that the purpose of
>> the course was to teach Deductive Reasoning.
>>
>> We had to list and justify each and every step in our proofs. A
>> skipped step or a wrong justification produced a failure.
>>
>> I suppose you had to do as well.
>>
>> When I eventually read Euclid, as part of the set known as The Great
>> Books of the Western World, I found that we had covered only a part of
>> the material. But that's not as bad as it seems -- a lot of what we
>> didn't cover was covered in algebra and what we got of number theory.
>>
>> Still, his proof that you can multiply two irrational numbers and get
>> a rational result (that is, that a rectangle with two irrational sides
>> can have a rational area) was interesting.
>
> Why? (pi + 1) is an irrational number which is obviously exactly one
> more than pi and the proof that e^*(i*pi)+1 = 0 is a 2nd year vector
> calculus problem.
>
> (My junior high math teacher put it on the blackboard one day without
> proof in response to a question from a student about "what's the best
> part of mathematics?" and said "come back and discuss it with me in
> ten years when you've done university level math". Since after
> university graduation I signed up for teacher's training I did just
> that "Hey Mr Peters, you remember the day in grade 9 when you
> said...")

Heh - over many years, I've worn out two t-shirts with that equation on
it. Hm...now that I think about it, I should start looking for #3.

Tony

On Thursday 21 December 2023 at 16:59:20 UTC, Paul S Person wrote:
> On Wed, 20 Dec 2023 23:35:11 -0800 (PST), Andrew McDowell
> <mcdow...@sky.com> wrote:
>
> <snippo>
> >I believe that school geometry is educationally very useful, because it allows people to learn and practice with mathematical proof. I was entranced by proof, perhaps because almost every other form of argument I had seen before was, at best, occupying a position that might need to be abandoned in the light of further evidence or the other side of the story - and few enough paid even that much attention to the facts of the case.
> The teacher of the Geometry course I took in High School (long, long
> ago ... but not far, far away) /explicitly stated/ that the purpose of
> the course was to teach Deductive Reasoning.
>
> We had to list and justify each and every step in our proofs. A
> skipped step or a wrong justification produced a failure.
>
> I suppose you had to do as well.
>
> When I eventually read Euclid, as part of the set known as The Great
> Books of the Western World, I found that we had covered only a part of
> the material. But that's not as bad as it seems -- a lot of what we
> didn't cover was covered in algebra and what we got of number theory.
>
> Still, his proof that you can multiply two irrational numbers and get
> a rational result (that is, that a rectangle with two irrational sides
> can have a rational area) was interesting.

Is that Euclid or your high school teacher?

I'm not sure if the clever bit is simply casting
it in geometry or... Let me see, a square of
unit side has a diagonal of length square root
of 2. A square whose side is sqrt(2) has...
diagonal length 2 and area 2 (?), which if it's
right may be a coincidence. Then I suppose
we want the proof that sqrt(2) is irrational.
<https://en.wikipedia.org/wiki/Irrational_number>
which considers the matter more generally,
records that (long before Euclid) that may have been
fighting talk amongst followers of Pythagoras.

On Fri, 22 Dec 2023 20:22:13 -0800, The Horny Goat <lcraver@home.ca>
wrote:

>On Thu, 21 Dec 2023 08:59:12 -0800, Paul S Person
><psperson@old.netcom.invalid> wrote:
>
>>On Wed, 20 Dec 2023 23:35:11 -0800 (PST), Andrew McDowell
>><mcdowell_ag@sky.com> wrote:
>>
>><snippo>
>>
>>>I believe that school geometry is educationally very useful, because it allows people to learn and practice with mathematical proof. I was entranced by proof, perhaps because almost every other form of argument I had seen before was, at best, occupying a position that might need to be abandoned in the light of further evidence or the other side of the story - and few enough paid even that much attention to the facts of the case.
>>
>>The teacher of the Geometry course I took in High School (long, long
>>ago ... but not far, far away) /explicitly stated/ that the purpose of
>>the course was to teach Deductive Reasoning.
>>
>>We had to list and justify each and every step in our proofs. A
>>skipped step or a wrong justification produced a failure.
>>
>>I suppose you had to do as well.
>>
>>When I eventually read Euclid, as part of the set known as The Great
>>Books of the Western World, I found that we had covered only a part of
>>the material. But that's not as bad as it seems -- a lot of what we
>>didn't cover was covered in algebra and what we got of number theory.
>>
>>Still, his proof that you can multiply two irrational numbers and get
>>a rational result (that is, that a rectangle with two irrational sides
>>can have a rational area) was interesting.
>
>Why? (pi + 1) is an irrational number which is obviously exactly one
>more than pi and the proof that e^*(i*pi)+1 = 0 is a 2nd year vector
>calculus problem.

That's not multiplying two irrational number together.

And let's see the /Euclidean geometrical/ proof of your monstrosity.

Noting that there is no line in Euclidean geometry with a length of
"0". Among other difficulties.

As to your question: because it never occurred to me that that could
be the case.

And I said it was "interesting", not that it revealed the secrets of
the Universe to me.
--
"Here lies the Tuscan poet Aretino,
Who evil spoke of everyone but God,
Giving as his excuse, 'I never knew him.'"

On Sat, 23 Dec 2023 04:04:36 -0800 (PST), Robert Carnegie
<rja.carnegie@excite.com> wrote:

>On Thursday 21 December 2023 at 16:59:20 UTC, Paul S Person wrote:
>> On Wed, 20 Dec 2023 23:35:11 -0800 (PST), Andrew McDowell
>> <mcdow...@sky.com> wrote:
>>
>> <snippo>
>> >I believe that school geometry is educationally very useful, because it allows people to learn and practice with mathematical proof. I was entranced by proof, perhaps because almost every other form of argument I had seen before was, at best, occupying a position that might need to be abandoned in the light of further evidence or the other side of the story - and few enough paid even that much attention to the facts of the case.
>> The teacher of the Geometry course I took in High School (long, long
>> ago ... but not far, far away) /explicitly stated/ that the purpose of
>> the course was to teach Deductive Reasoning.
>>
>> We had to list and justify each and every step in our proofs. A
>> skipped step or a wrong justification produced a failure.
>>
>> I suppose you had to do as well.
>>
>> When I eventually read Euclid, as part of the set known as The Great
>> Books of the Western World, I found that we had covered only a part of
>> the material. But that's not as bad as it seems -- a lot of what we
>> didn't cover was covered in algebra and what we got of number theory.
>>
>> Still, his proof that you can multiply two irrational numbers and get
>> a rational result (that is, that a rectangle with two irrational sides
>> can have a rational area) was interesting.
>
>Is that Euclid or your high school teacher?

Euclid.

As I said, I ran across this when I read his work later. It is the
next-to-last theorem in the chapter on Numbers, which are actually
line lengths measured by a line arbitrarily chosen to have length "1".
>I'm not sure if the clever bit is simply casting
>it in geometry or... Let me see, a square of
>unit side has a diagonal of length square root
>of 2. A square whose side is sqrt(2) has...
>diagonal length 2 and area 2 (?), which if it's
>right may be a coincidence. Then I suppose
>we want the proof that sqrt(2) is irrational.
><https://en.wikipedia.org/wiki/Irrational_number>
>which considers the matter more generally,
>records that (long before Euclid) that may have been
>fighting talk amongst followers of Pythagoras.

Well, you are close, but no cigar.

It is true that Euclid defines sqrt(2) as "rational in square" because
a square of side sqrt(2) would have a rational area. This does
complicate matters and may be one reason this part of Euclid was not
used in High School Geometry, since many of those taking it were also
taking Algebra and may have found juggling two different views of
"rational/irrational" in their minds a bit much.

But the proof is not about something so simple as a square root. It is
about numbers of the form
a + sqrt(b) [1]
and
a - sqrt(b)
The assertion (proved geometrically) is that the area of the rectangle
will be
a*a + sqrt(b)*sqrt(b)
which, of course, is (in algebra) simply
(c + d)*(c - d) = c*c + d*d

Note that, in geometry, a and b will always be > 0. They are, after
all, line lengths. In algebra, of course, no such restriction occurs.

Also note that I never said that /any/ two irrational numbers could do
this, only that some pairs of them could.

Also note that, whatever the algebraic form may be called, Euclid
proved it [2] long before algebra existed.

[1] Note that I am using the algebraic notation to avoid having to
look up and copy what Euclid used.

[2] Well, recorded the proof, anyway. My understanding is that Euclid
is very much a school textbook: it combines material from many other
geometricians into one convenient package and maybe adds a few
additional things as well, who can say?
--
"Here lies the Tuscan poet Aretino,
Who evil spoke of everyone but God,
Giving as his excuse, 'I never knew him.'"

On Sat, 23 Dec 2023 06:55:49 -0500, Tony Nance <tnusenet17@gmail.com>
wrote:

UH - > is Tony, >> is me

>> (My junior high math teacher put it on the blackboard one day without
>> proof in response to a question from a student about "what's the best
>> part of mathematics?" and said "come back and discuss it with me in
>> ten years when you've done university level math". Since after
>> university graduation I signed up for teacher's training I did just
>> that "Hey Mr Peters, you remember the day in grade 9 when you
>> said...")
>
>Heh - over many years, I've worn out two t-shirts with that equation on
>it. Hm...now that I think about it, I should start looking for #3.

Main thing I remember from the day in 2nd year calculus when the
professor provided that e^(i * pi) + 1 = 0 was the reaction from the
prof when I said "I first saw that formula in grade 9!"

He was a bit relieved when I told him that our 9th grade teacher had
said "those of you who go on in mathematics will see that formula
again - remember it and be impressed when you see the proof!"