Properties of Fibonacci numbers are often proved by induction. Although
this results in technically correct proofs, I find that proofs by
induction usually give me very little insight. Over the years I've found
various alternative definitions of the Fibonacci numbers that I can try
out when I'm trying to understand a result. I recently realized that the
sequence of Fibonacci numbers can be viewed as the projection of a two
dimensional geometric sequence, and I'm writing about it here.

The final sections aren't done, but the draft has been sitting on my
hard disk for over a year, so I'm posting an incomplete version to
provide myself with motivation to finish.

It's at
<http://www.gbbservices.com/math/fib.html>

Hope you enjoy it,

Walter

> I'm new to the group.

Welcome!

  > I'm trying to get my brain wrapped around the concept of a
  > Googleplex. I *think* * understand a Google. It's one followed by
  > 100 zeros, right?

Yes.

  > The definition of a Googleplex is one followed by a google of zeros?
  > I don't get it. The definitions sound the same.

They're kind of the same: they all involve one followed by some
number of zeros. But, the NUMBER of zeros is QUITE different.
Here's an approach to this topic that might shed some light
on what's going on:

Ten can be defined as one followed by one zero (ie 10).
The corresponding "plex" number, call it a "tenplex",
is one followed by ten zeros, also known as 10 billion
(ie 10,000,000,000).

Going up in size, a hundred can be defined as one
followed by two zeros (100). A "hundredplex" is then one
followed by a hundred zeros, also known as a googol
(ie  10,000,000,000,000,000,000,000,000,000,000,
000,000,000,000,000,000,000,000,000,000,000,000,
000,000,000,000,000,000,000,000,000,000,000).

There would then be a thousand and a thousandplex,
and so on.

Thinking about these smaller analogous examples might
help you in getting a feel for the relationship between
a googol and a googolplex.

  > I've looked around on the web ...

Searching for "googol" rather than "google" will
get you MUCH better search engine results.

Just in case it doesn't come up in your web searching,
let me also mention that my essay
"10, 10^10, 10^10^10... Infinity"
<http://www.gbbservices.com/math/large.html>
touches on related topics.

Hope that helps,

Walter

--- skyblueredbirdgreenhouse wrote:
> Hello,
> I'm new to the group.
> I'm trying to get my brain wrapped around the concept of a
> Googleplex. I *think* * understand a Google. It's one followed by
> 100 zeros, right?
> The definition of a Googleplex is one followed by a google of zeros?
> I don't get it. The definitions sound the same.
> If anyone can help shed some light on this mysterious concept I'll
> be very grateful. I've looked around on the web for a site that will
> explain it but most searches turn up crap about the google search
> engine.
> Thanks in advance for any help!

Hello,
I'm new to the group.
I'm trying to get my brain wrapped around the concept of a
Googleplex. I *think* * understand a Google. It's one followed by
100 zeros, right?

	 Yes...

The definition of a Googleplex is one followed by a google of zeros?
I don't get it. The definitions sound the same.
If anyone can help shed some light on this mysterious concept I'll
be very grateful.

	 Okay, a google is 10^100, which is a 1 followed by 100 zeros.  A
googleplex is 10^(10^100), which is ten raised to a power 1 followed by
a hundred zeros in that power.,  Which is significantly larger than
10^100.  Here is some historical information on the subject that I found
on Ask Dr. Math.  It appears that Dr. Peterson was asked about the
origin of the terms and there is some interesting things he says about
it below.

                            Associated Topics || Dr. Math Home || Search
Dr. Math


              Googol, Kasner, and Milton Sirotta


              Date: 07/14/99 at 16:55:09
              From: Melanie
              Subject: Powers of 10

              Who coined the phrase "googleplex," and when? I have used
several
              search engines and they have the definition, but not the
origin.

              Thank you for your consideration,
              Melanie




              Date: 07/15/99 at 11:59:13
              From: Doctor Peterson
              Subject: Re: Powers of 10

              Hi, Melanie.

              We get questions about this frequently, so I did a little
extra
              research to find the details beyond what's in our archives.

              Here's a page on the Web that tells about the origin of the
words
              googol and googolplex:

              How Many? A Dictionary of Units of Measurement
              http://www.unc.edu/~rowlett/units/dictG.html

              This says:

              "googol

              a unit of quantity equal to 10^100 (1 followed by 100
zeroes). The
              googol was invented by the American mathematician Edward
Kasner (1878-
              1955) in 1938. According to the story, Kasner asked his
nephew Milton
              Sirotta, who was then 8 years old, what name he would give
to a really
              large number, and "googol" was Milton's response. Kasner
also defined
              the googolplex, equal to 10^googol, that is, 1 followed by
a googol of
              zeroes. These inventions caught the public's fancy and are
often
              mentioned in discussions of very large numbers."

              A slightly different version is in

              Googolplex
              http://www.fpx.de/fp/Fun/Googolplex/

              which says:

              "The American mathematician Edward Kasner once asked his
nine-year-old
              nephew to invent a name for a very large number, ten to the
power of
              one hundred; and the boy called it a googol. He thought
this was a
              number to overflow people's minds, being bigger than
anything that can
              ever be put into words. Another mathematician then shot
back with
              googolplex, and defined it to be 10 to the power of
googol."

              Here's a review of the 1940 book in which Kasner discussed
the googol:

              Edward Kasner and James Newman. Mathematics and the
Imagination
              http://www-users.cs.york.ac.uk/~susan/bib/nf/k/kasner.htm

              Also check out our Dr. Math FAQ on large numbers:

              http://mathforum.org/dr.math/faq/faq.large.numbers.html

              Summing up, the googol was named by Milton Sirotta, and the
googolplex
              by his uncle Edward Kasner, who I suspect had set Milton up
by asking
              for a name for the googol, just so he could name something
incredibly
              larger.

              Incidentally, you'll find that the googol can just as well
be called
              "10 duotrigintillion" following the (more or less) standard
              conventions for naming large numbers; googol is just the
fun name,
              which allows us to name "googolplex" easily; and if a
mathematician
              or scientist ever had occasion to use either number, they
would just
              call them 10^100 and 10^10^100 because numbers are much
easier to work
              with than names.

              - Doctor Peterson, The Math Forum
                http://mathforum.org/dr.math/



	 Corey...

Hello,
I'm new to the group.
I'm trying to get my brain wrapped around the concept of a
Googleplex. I *think* * understand a Google. It's one followed by
100 zeros, right?
The definition of a Googleplex is one followed by a google of zeros?
I don't get it. The definitions sound the same.
If anyone can help shed some light on this mysterious concept I'll
be very grateful. I've looked around on the web for a site that will
explain it but most searches turn up crap about the google search
engine.
Thanks in advance for any help!

> is there anything in my so called S.S. Code?

Hard to say. The arithmetic follows from known algebraic identities.
That part of your findings isn't new.

As for the historical/biblical connections, I wouldn't know if
there's anything new there. It all looks similar to other results
I've seen, but that doesn't mean much. It could be that you've
discovered something new.

I expect that it will be hard to find mathematicians who also
have the extra expertise necessary to competently judge your
work. I don't know of any. Good luck!

Walter

--- numberland2001 wrote:
> I was wondering.
>
> Yes, I was wondering this morning about some basic arithmetic, I have
> pompously titled, the S.S. Code, (the reason for such could be given
> in another post).
>
> And my wonderment requires an explanation from people who understand
> mathematics, for I simply know not why it works out.
>
> OK, I must admit I am expecting to make a fool of myself with this,
> but no matter.
>
> The exercise is this, and my key numbers are 1,473.230769 and 6,840
> and 14.
>
> a. 6,840 less 1,473.230769 and /1,473.230769 and x 14 is the answer
> 51.
>
> b. 6,840 less 1,473.230769 and less 1,473.230769 and /1,473.230769 x
> 14 is the answer 37.
>
> c. 6,840 less 1,473.230769 and less 1,473.230769 and less
> 1,473.230769 x 14 is the answer 23.
>
> d. 6,840 less 1,473.230769 x 4 and /1,473.230769 x 14 is the answer 9.
>
> Thus there is a difference of 14 between each item.
>
> OK, next to:
>
> e. 1,473.230769 x 5 lots less 6,840 and /1,473.230769 x 14 is the
> answer 5.
>
> f. 1,473.230769 x 6 lots less 6,840 and /1,473.230769 x 14 is the
> answer 19.
>
> g. And 1,473.230769 x 7 lots has the answer 33
>
> h. And 1,473.230769 x 8 lots has the answer 47
>
> i. And 1,473.230769 x 9 lots has the answer 61
>
> j. And 10 lots is 75 and 11 lots is 89 and 12 lots 103 and 13 lots is
> 117.
>
> So a difference of 14 between them.
>
> OK, now some of the above reference numbers I use when considering
> the architecture of Ancient Egypt, noting that the Seked of the
> Pyramid Age included the ratio 14/11.
>
> And the standard measuring unit for the Pyramid Age was 20.612 inches
> to the cubit, and the measuring rod consisted of 28 Ancient Egyptian
> gods, each with a glyph a finger width wide 1/28, but the powerful
> god Horus was indicated twice.
>
> So 1,473.230769 x 28 lots and less 6,840 and /1,473.230769 x 14 is
> 327.
>
> Now 327 is made up of thrice 109.
>
> And the reference number 109 is that for `Fire'.
>
> So when I place Fire at 109 into the Lamp at 17 (the seventh prime of
> course) the number is 126, that of the `Eye of Horus', and
> the spirit body inside all humanity released at natural death.
>
> And 7 x 109,000 + 6 x 17,000 is 865,000 miles the diameter of the Sun.
>
> OK, now the Lamp, that is the Serpent, at 17 x 1,473.230769 less 6840
> and /1,473.230769 x 14 is 173, and 1,730,000 is twice 865,000 miles.
>
> So the Lamp changes into twice the Sun using 1,473.230769 and 6,840
> and 14.
>
> And just to finish, I go to St. George's Hall, that is St.
> George's Chapel, Windsor Castle, was rebuilt in 1829 in Gothic
> style,
> at a length of 185 feet.
>
> And the number 185 is seen in the Bible's 185,000 dead.
>
> "And it came to pass that night, that the angel of Yahweh went
> out, and smote in the camp of the Assyrians an hundred fourscore and
> five thousand: and when they arose early in the morning, behold, they
> [were] all dead corpses. So Sennacherib king of Assyria departed, and
> went and returned, and dwelt at Nineveh." 2Kings 19:35-36
>
> And 6,840 less 1,473.230769 twice and /1,473.230769 x 70 is 185.
>
> And so I will stop here, and ask this board, (I have asked no others
> of any persuasion), is there anything in my so called S.S. Code?
>
> Thanks
>
> John D. Miller
> North London, UK.

I was wondering.

Yes, I was wondering this morning about some basic arithmetic, I have
pompously titled, the S.S. Code, (the reason for such could be given
in another post).

And my wonderment requires an explanation from people who understand
mathematics, for I simply know not why it works out.

OK, I must admit I am expecting to make a fool of myself with this,
but no matter.

The exercise is this, and my key numbers are 1,473.230769 and 6,840
and 14.

a. 6,840 less 1,473.230769 and /1,473.230769 and x 14 is the answer
51.

b. 6,840 less 1,473.230769 and less 1,473.230769 and /1,473.230769 x
14 is the answer 37.

c. 6,840 less 1,473.230769 and less 1,473.230769 and less
1,473.230769 x 14 is the answer 23.

d. 6,840 less 1,473.230769 x 4 and /1,473.230769 x 14 is the answer 9.

Thus there is a difference of 14 between each item.

OK, next to:

e. 1,473.230769 x 5 lots less 6,840 and /1,473.230769 x 14 is the
answer 5.

f. 1,473.230769 x 6 lots less 6,840 and /1,473.230769 x 14 is the
answer 19.

g. And 1,473.230769 x 7 lots has the answer 33

h. And 1,473.230769 x 8 lots has the answer 47

i. And 1,473.230769 x 9 lots has the answer 61

j. And 10 lots is 75 and 11 lots is 89 and 12 lots 103 and 13 lots is
117.

So a difference of 14 between them.

OK, now some of the above reference numbers I use when considering
the architecture of Ancient Egypt, noting that the Seked of the
Pyramid Age included the ratio 14/11.

And the standard measuring unit for the Pyramid Age was 20.612 inches
to the cubit, and the measuring rod consisted of 28 Ancient Egyptian
gods, each with a glyph a finger width wide 1/28, but the powerful
god Horus was indicated twice.

So 1,473.230769 x 28 lots and less 6,840 and /1,473.230769 x 14 is
327.

Now 327 is made up of thrice 109.

And the reference number 109 is that for `Fire'.

So when I place Fire at 109 into the Lamp at 17 (the seventh prime of
course) the number is 126, that of the `Eye of Horus', and
the spirit body inside all humanity released at natural death.

And 7 x 109,000 + 6 x 17,000 is 865,000 miles the diameter of the Sun.

OK, now the Lamp, that is the Serpent, at 17 x 1,473.230769 less 6840
and /1,473.230769 x 14 is 173, and 1,730,000 is twice 865,000 miles.

So the Lamp changes into twice the Sun using 1,473.230769 and 6,840
and 14.

And just to finish, I go to St. George's Hall, that is St.
George's Chapel, Windsor Castle, was rebuilt in 1829 in Gothic
style,
at a length of 185 feet.

And the number 185 is seen in the Bible's 185,000 dead.

"And it came to pass that night, that the angel of Yahweh went
out, and smote in the camp of the Assyrians an hundred fourscore and
five thousand: and when they arose early in the morning, behold, they
[were] all dead corpses. So Sennacherib king of Assyria departed, and
went and returned, and dwelt at Nineveh." 2Kings 19:35-36

And 6,840 less 1,473.230769 twice and /1,473.230769 x 70 is 185.

And so I will stop here, and ask this board, (I have asked no others
of any persuasion), is there anything in my so called S.S. Code?

Thanks

John D. Miller
North London, UK.

Hi Walter,

Wow!  Great stuff on 3*5=15.  You clearly put a lot of work into this.

I read your page on Mantis.  It sounds good, and I'll be sure to give
it a try sometime.  Thanks for pointing it out.

Take care,

Wayne


Thursday, January 1, 2004, 1:46:46 PM, you wrote:

WV> The identity 3*5=15 quickly leads to more mathematics:
WV> various algebraic identities, an infinite product,
WV> Fibonacci numbers, the Golden Ratio, Mersenne primes,
WV> sums of squares, complex numbers, and quaternions.

WV> I've posted an essay on this at
WV> <http://www.gbbservices.com/math/threefive.html>.

WV> Hope you enjoy it,

WV> Walter

WV> P.S. The programmers on the list might also be interested
WV> in "Personal Bug Tracking with Mantis on Windows" at
WV> <http://www.gbbservices.com/prog/mantis.html>





WV> Yahoo! Groups Links

WV> To visit your group on the web, go to:
WV>  http://groups.yahoo.com/group/understandingmath/

WV> To unsubscribe from this group, send an email to:
WV>  understandingmath-unsubscribe@yahoogroups.com

WV> Your use of Yahoo! Groups is subject to:
WV>  http://docs.yahoo.com/info/terms/

The identity 3*5=15 quickly leads to more mathematics:
various algebraic identities, an infinite product,
Fibonacci numbers, the Golden Ratio, Mersenne primes,
sums of squares, complex numbers, and quaternions.

I've posted an essay on this at
<http://www.gbbservices.com/math/threefive.html>.

Hope you enjoy it,

Walter

P.S. The programmers on the list might also be interested
in "Personal Bug Tracking with Mantis on Windows" at
<http://www.gbbservices.com/prog/mantis.html>

I've posted a new essay on "Large Numbers". It's at
<http://www.gbbservices.com/math/large.html>.

"This is about some ways I've come up with to get a gut
feeling for the behavior of numbers of the form 10^10^10^n,
where n is the range from 1 to 10. These numbers don't come
up in the physical sciences or economics, so you probably
haven't had a need to become familiar with them. Some of
their properties may surprise you."

Walter

I've posted a new essay on "Solving The Cubic".
It's at <http://www.gbbservices.com/math/cubic.html>

Hope you enjoy it,

Walter

P.S. I know that some of the list members are programmers,
so you might also be interested in "Using Visual Studio .NET"
at <http://www.gbbservices.com/prog/vsnet.html>

My book claims that,

|x| = x, whenever x => 0
|x| = -x, whenever x < 0

So, this just means that |x| breaks into two equations,

x => 0
-x < 0

	 But, I am not clear how my book jumps from this to saying that x <=
|x|.  Can someone explain how to make this leap?


	 Corey...

Hi Corey,

  >>The definition you're using for "symmetric difference"
  >>is of course crucial.
  > Well, the symmetric difference of A and B denoted (A D B)
  > Is all x in (A but not in B) union all x in (B but not in A).

This is where spelling things out really becomes useful.
Your approach may work, depending on a whole lot of context
which I haven't seen, but your definition seems to mix
set theory and logic unnecessarily. I'd be much more
comfortable with just using logic.

Firstly, a little bit of context:
When you're starting to prove theorems in some domain of
mathematics, call it A, what you want to do is to define
your terms in terms of some other, more fundamental,
domain of mathematics, call it B. And furthermore,
at the beginning, when proving theorems in domain A,
you'll be using only results from domain B in your proofs.
After you've built up a body of theorems in domain A
you can start using them to prove more theorems
in domain A. If you don't do things this way, you're in
great danger of engaging in circular reasoning, and
not really proving anything at all.

Some examples from a typical mathematics curriculum are:
Point Set Topology (A="topology" B="set theory")
Probability Theory (A="probability theory" B="measure theory")
Group Theory (A="group theory" B="set theory")
Complex Analysis (A="complex analysis" B="real analysis")

In your case (where A="set theory", B="logic") I'd expect
to see definitions of set theoretic operations like
union, intersection, symmetric difference, difference
in terms of logical operations like
inclusive or (aka or), and, exclusive or (aka xor),
negation (aka not).

So, in your case, I would be working with definitions like
x is an element of A intersect B iff
x is an element of A AND x is an element of B
and then using results from logic to establish results
in set theory. A nice example would be using the fact
that "(a xor b) xor b" is the same "a" (a result that's
heavily used in computer graphics) to establish
"(A D B) D B" is the same as "A".

Incidentally, the definition I'd use for A D B is
x is an element of A D B
iff x is an element of A XOR x is an element of B.
If you don't like "xor", restate it in terms of
"not", "or", "and"
(since a XOR b iff (a OR b) AND ( NOT (a AND b))
and work from there. If you prefer to use
a XOR b iff (a AND (NOT b)) OR (b AND (NOT a))
that's also okay :-)

  > And many years ago when I took Discrete Mathematics,
  > this stuff would have been easy.

Well, of course. Results from logic would have been
right at your fingertips!

  > And with nearly 300 math books in my room, I cannot
  > seem to find the one book I need. <chuckle>

That's a lot of books! One of the reasons that I
recommended seeking out a live mathematician is that
this kind of thing typically isn't written down.
Apprenticeship is alive and well in graduate
schools of mathematics.

  > Thanks for the reference material.

You're welcome.

Hope that helps,

Walter

Walter Vannini wrote:
>
> Hi Corey,
>
>  > I was not sure how to prove that ...
>
> Wow, this is hard to answer, especially via email!
>
> It seems to me that you really want to clarify just
> what you're given, and what you have to establish.
> Knowing what you're given includes knowing the
> definitions of the terms you're working with.
> The definition you're using for "symmetric difference"
> is of course crucial.


	 Well, the symmetric difference of A and B denoted (A D B)  Is all x in
(A but not in B) union all x in (B but not in A).

So, if A = {1, 2, 3} and B = {2, 3, 4}

Then A D B = {1, 4}


And many years ago when I took Discrete Mathematics, this stuff would
have been easy.  But, I forgot a lot of those important identities.  And
with nearly 300 math books in my room, I cannot seem to find the one
book I need. <chuckle>


>
> Probably your best bet is to look at a simpler but
> similar problem. One example that comes to mind is:
> show that A-(A-B) is the same as A intersection B.
> Whatever technique you use for proving this will
> most probably apply to your problem.


Hmmm!  I think that one is resolved by saying,

If x is an element of [A - (A - B)]

Then x is an element of A but x is not an element of (A - B)

Furthermore, x is an element of A and x is an element of (~a union B)

And this is logically equivelant to saying,

x is an element of (A intersect ~A) union x is an element of (A
intersect B)

	 Since A intersect ~A is the empty set, we are just left with x is an
element of the empty set union x is an element of (A intersect B)

Consequently, x is an element of (A intersect B).

	 So, we just start from this conclusion and prove that it is true in
reverse to demonstrate logical equivalance.

>
> If you're at a university and have access to mathematicians
> (in person), I would strongly suggest asking one of them
> to show you how to give a rigorous proof of
> A-(A-B) = A intersection B. Incidentally, the way I'd
> prove it is by showing that if x is an element of the
> set on one side, then it's an element of the set on the
> other side, and vice versa. For now, I'd recommend
> being really pedantic and being explicit about every
> step and assumption.
>
> You should keep in mind that there are usually many ways
> to prove a result, and you shouldn't stop looking once
> you've found a proof.
>
> Also, in case you haven't come across it, I'd suggest
> becoming familiar with Polya's "How to Solve It".


	 I'll have to see if I can find that.  Thanks for the reference
material.


	 Corey...

Hi Corey,

  > I was not sure how to prove that ...

Wow, this is hard to answer, especially via email!

It seems to me that you really want to clarify just
what you're given, and what you have to establish.
Knowing what you're given includes knowing the
definitions of the terms you're working with.
The definition you're using for "symmetric difference"
is of course crucial.

Probably your best bet is to look at a simpler but
similar problem. One example that comes to mind is:
show that A-(A-B) is the same as A intersection B.
Whatever technique you use for proving this will
most probably apply to your problem.

If you're at a university and have access to mathematicians
(in person), I would strongly suggest asking one of them
to show you how to give a rigorous proof of
A-(A-B) = A intersection B. Incidentally, the way I'd
prove it is by showing that if x is an element of the
set on one side, then it's an element of the set on the
other side, and vice versa. For now, I'd recommend
being really pedantic and being explicit about every
step and assumption.

You should keep in mind that there are usually many ways
to prove a result, and you shouldn't stop looking once
you've found a proof.

Also, in case you haven't come across it, I'd suggest
becoming familiar with Polya's "How to Solve It".

By the way, your distrust of accepting a proof by diagram
has a long tradition. The ancient greeks were known to
deliberately draw misleading geometric diagrams so that
they would not be fooled into assuming something they
had not logically established. For example, if they were
proving something about an isosceles triangle, they might
draw a triangle with all three sides being very different
in length. Apparently they weren't careful enough, and
Hilbert found some holes in some of Euclid's proofs
that were most probably due to relying on diagrams. Things
like assuming that a point was inside a polygon simply
because it looked like it should be in the polygon.

Good luck,

Walter

--- Corey Bray wrote:
> Hello Group,
>
>
>  Let D represent capital Delta and denote the symmetric difference of
> two sets A and B.  Prove that,
>
> (A D B) is logically equivelant to (A union B) - (A intersect B)
>
>  I understand to prove that two sets are logically equivelant means to
> show that P is a subset of Q and also Q is a subset of P.  But, I was
> not sure how to prove that the above was logically equivelant.  I can
> see that it is true using pictures, but I want to improve my
> understanding of how to do such proofs abstractly rather than relying on
> pictures.
>
>
>  Corey...
>  isomorphics@...

Hello Group,


	 Let D represent capital Delta and denote the symmetric difference of
two sets A and B.  Prove that,

(A D B) is logically equivelant to (A union B) - (A intersect B)

	 I understand to prove that two sets are logically equivelant means to
show that P is a subset of Q and also Q is a subset of P.  But, I was
not sure how to prove that the above was logically equivelant.  I can
see that it is true using pictures, but I want to improve my
understanding of how to do such proofs abstractly rather than relying on
pictures.


	 Corey...
	 isomorphics@...

It occurred to me that in all the discussion about
Pythagoras' identity, and its generalization to general
triangles, the actual generalization (called the law
of cosines) was never explicitly stated. Here it is:

Given a triangle with side lengths a,b,c,
and with angle theta between the sides of length a and b,
the following identity holds:
c^2 = a^2 + b^2 - 2ab cos(theta)

Pythagoras' identity is just the special case where
theta is ninety degrees, so that cos(theta) is zero.

The other two special cases are theta = 0 and theta = 180 degrees.

For theta = 0:
c = abs(a-b), so that
c^2 = (a-b)^2
      = a^2 + b^2 - 2ab
      = a^2 + b^2 - 2ab cos(theta), since cos(theta)=1

For theta = 180 degrees:
c = a+b, so that
c^2 = (a+b)^2
      = a^2 + b^2 + 2ab
      = a^2 + b^2 - 2ab cos(theta), since cos(theta)= -1.

The general identity actually follows quickly from
Pythagoras' identity, since it is easy to construct a
right angled triangle with sides  a - b cos(theta), b sin(theta), c
from the general a, b, c triangle.
It then follows that
c^2 = (a-bcos(theta))^2 + (b sin(theta))^2,
and then cos^2+sin^2=1 quickly gives us
c^2 = a^2 + b^2 - 2ab cos(theta).

Walter

Thanks for the spellings and the background info.  This helps me out a
lot.

	 Corey...

Walter Vannini wrote:
>
> Hi Corey,
>
> It's "Dedekind". He was a German mathematician:
> Julius Wilhelm Richard Dedekind.
>
> If you search on "Dedekind cut" you'll find lots of references.
> In short, Dedekind cuts are a way of defining the real numbers
> in terms of the rational numbers.
> Dedekind cuts do it via subsets of the rationals that are
> defined using the standard ordering.
> "Dedekind cut arithmetic" is probably a way of defining the
> arithmetic operations on the reals in terms of operations on
> subsets of the rationals.
>
> Another popular way to define the real numbers in terms of the
> rational numbers is via equivalence classes of Cauchy sequences.
> If you search on "Cauchy completion" you'll find out more.
> Incidentally, Cauchy was a French mathematician:
> Augustin Louis Cauchy.
>
> Walter
>
> --- Corey Bray wrote:
>  > Hello group,
>  >
>  >      I'm trying to learn about something called, and please correct my
>  > spelling, "Dedican Cut arithmetic".  Does anyone know what it is?  I'm
>  > reading a book on Advanced Calculus, well I am actually listening to it
>  > on tape, but that has caused me some difficulty here.  If anyone can
>  > explain what this type of arithmetic is and how it works, I would really
>  > appreciate it.
>  >
>  >      Corey...
>
>
> To unsubscribe from this group, send an email to:
> understandingmath-unsubscribe@yahoogroups.com
>
>
>
> Your use of Yahoo! Groups is subject to http://docs.yahoo.com/info/terms/

Hi Corey,

It's "Dedekind". He was a German mathematician:
Julius Wilhelm Richard Dedekind.

If you search on "Dedekind cut" you'll find lots of references.
In short, Dedekind cuts are a way of defining the real numbers
in terms of the rational numbers.
Dedekind cuts do it via subsets of the rationals that are
defined using the standard ordering.
"Dedekind cut arithmetic" is probably a way of defining the
arithmetic operations on the reals in terms of operations on
subsets of the rationals.

Another popular way to define the real numbers in terms of the
rational numbers is via equivalence classes of Cauchy sequences.
If you search on "Cauchy completion" you'll find out more.
Incidentally, Cauchy was a French mathematician:
Augustin Louis Cauchy.

Walter

--- Corey Bray wrote:
  > Hello group,
  >
  >  I'm trying to learn about something called, and please correct my
  > spelling, "Dedican Cut arithmetic".  Does anyone know what it is?  I'm
  > reading a book on Advanced Calculus, well I am actually listening to it
  > on tape, but that has caused me some difficulty here.  If anyone can
  > explain what this type of arithmetic is and how it works, I would really
  > appreciate it.
  >
  >  Corey...

> Do you have a good or favorite proof or way to see it intuitively???
  > ... But still I dont "Grok" (Heinlein) it.

I thought about it last night, and came up with something
that might help.

Given an a,b,h triangle, we know that h is a function of a and b,
i.e. h=F(a,b).

Take the partial derivative with respect to a. Looking at
a diagram, the triangle with sides "delta h" and "delta a"
is, in the limit, similar to the a,b,h triangle, so that

dh/da = a/h

Multiplying both sides by h, we have

h dh/da =a

Integrating gives us

(1/2) h^2 = (1/2) a^2 + f(b)

When a=0, we have that h=b , so that f(b) must be (1/2) b^2.
Multiplying by 2 gives the Pythagorean identity:
h^2 = a^2 + b^2.

So, Pythagoras' identity can be viewed as the integral version
of a differential relationship involving similar triangles.

Incidentally, after writing the above I did a search to
see if the above proof has been published. Of course it
has! It's essentially number 40 in the list of 43 proofs at
<http://www.cut-the-knot.org/pythagoras/index.shtml>
It was published in 1988 in an issue of The Mathematical
Intelligencer.

All the best,

Walter

--- monkeyEinstein wrote:
> Hi Walter,
> This is Dave C. of the StonyBrook Aquatic team.
> I was looking at you connections on Pythagoras.
> Although I know how to "prove" a^2 + b^2= h^2 for a,b,h=hypoteneuse
> the sides of a rt. triangle, I still have a hard time seeing why in
> an intuitive way.
> Do you have a good or favorite proof or way to see it intuitively???
> I saw a cut and paste proof I liked in long ago in a book The Ascent
> of Man. But still I dont "Grok" (Heinlein) it.
>
> Maybe an algebraic proof would be better. Like...
> "Certaintly a+b=h (n=1) can't be true, so how about a^n + b^n = h^n.
> For n large, it certaintly cant be true.
> Now there is a right angle, which divides the plane into 4 equal
> regions...." I give up.
>
> And what about the generalization to non-right angles triangles?
> (Law of cosines I think)
> Have you "connected" that to anything?
>
> Got to go,
> Take Care,
> Dave

Hello group,


	 I'm trying to learn about something called, and please correct my
spelling, "Dedican Cut arithmetic".  Does anyone know what it is?  I'm
reading a book on Advanced Calculus, well I am actually listening to it
on tape, but that has caused me some difficulty here.  If anyone can
explain what this type of arithmetic is and how it works, I would really
appreciate it.


	 Corey...

Hi Dave,

  > Do you have a good or favorite proof or way to see it intuitively???

There's a multitude of proofs of Pythagoras' theorem! Even
President Garfield got into the act and provided a proof in 1876.

To answer your question, I do have a favorite, and it does help
me "see" the result. It uses several ideas that I like.
There's the idea seen in Combinatorics of establishing an
identity by counting the same thing in two different ways.
There's also the idea seen in Probability of calculating a
probability indirectly by directly calculating the probability
of everything else that could happen. And then there's the
seemingly unhelpful idea of adding clutter to the mix by
working with several copies of the triangle instead of just
the one copy in the statement of the theorem. To me, that last
idea is the creative leap. Everything else is just details,
in comparison.

I think that's enough talk about ideas used in the proof.
Here's the proof:

The basic plan is to look at the area left over when four
copies of the triangle are used to cover up some of the area
of a large square. One way of arranging the four triangles
results in the uncovered area being h^2, while another way
results in the uncovered area being a^2+b^2.

Here are the details:

For the first arrangement, take four copies of the a,b,h
triangle and attach them via the hypotenuse to each side of
a square with side length h. This results in a larger square
of side length a+b, with a "hole" of size h^2.

For the second arrangement, take the four triangles and combine
pairs of triangles to form two rectangles with sides of length
a and b. Now take these two rectangles and arrange them so
that one is rotated 90 degrees relative to the other, and
they share a vertex. They're inside a square with side length
a+b, with two "holes", one of size a^2, the other of size b^2.

As for the generalization to general triangles:
I don't have a way to intuitively understand it, other than
as a perturbation of the right angle version of the theorem.
If you come across an approach that sheds some light on
the result, please let me know.

It's great to hear from you Dave! It's been a long time since
the SB Aquatic team got together! I'd almost forgotten it once
existed.

All the best,

Walter

--- monkeyEinstein wrote:
> Hi Walter,
> This is Dave C. of the StonyBrook Aquatic team.
> I was looking at you connections on Pythagoras.
> Although I know how to "prove" a^2 + b^2= h^2 for a,b,h=hypoteneuse
> the sides of a rt. triangle, I still have a hard time seeing why in
> an intuitive way.
> Do you have a good or favorite proof or way to see it intuitively???
> I saw a cut and paste proof I liked in long ago in a book The Ascent
> of Man. But still I dont "Grok" (Heinlein) it.
>
> Maybe an algebraic proof would be better. Like...
> "Certaintly a+b=h (n=1) can't be true, so how about a^n + b^n = h^n.
> For n large, it certaintly cant be true.
> Now there is a right angle, which divides the plane into 4 equal
> regions...." I give up.
>
> And what about the generalization to non-right angles triangles?
> (Law of cosines I think)
> Have you "connected" that to anything?
>
> Got to go,
> Take Care,
> Dave

Hi Walter,
This is Dave C. of the StonyBrook Aquatic team.
I was looking at you connections on Pythagoras.
Although I know how to "prove" a^2 + b^2= h^2 for a,b,h=hypoteneuse
the sides of a rt. triangle, I still have a hard time seeing why in
an intuitive way.
Do you have a good or favorite proof or way to see it intuitively???
I saw a cut and paste proof I liked in long ago in a book The Ascent
of Man. But still I dont "Grok" (Heinlein) it.

Maybe an algebraic proof would be better. Like...
"Certaintly a+b=h (n=1) can't be true, so how about a^n + b^n = h^n.
For n large, it certaintly cant be true.
Now there is a right angle, which divides the plane into 4 equal
regions...." I give up.

And what about the generalization to non-right angles triangles?
(Law of cosines I think)
Have you "connected" that to anything?

Got to go,
Take Care,
Dave

Thanks Walter!  That was very helpful...


	 Corey...

Walter Vannini wrote:
>
>  > Can someone here demonstrate this?
>
> Sure, I'll give it a go. There's lots of ways to do it.
> Here's the most direct one I can think of:
>
> Assume the conclusion is false.
> That means, assume it is false that a > 0.
> Well, that  means that a <= 0.
> If that's so, and we're given that -a < 0,
> then a+(-a) < 0+0, (since w<=x and y<z implies w+y<x+z).
> Simplifying the two sides, this says that "0<0"
> is true. But, this isn't true, and so the initial
> assumption that the conclusion is false was wrong.
> End of Proof by Contradiction.
>
> Of course, if you already know that w<=x and y<z implies w+y<x+z,
> why not substitute w=a, x=a, y=-a, z=0 and immediately get
> that a<=a (always true) and -a<0 (given) implies that a+(-a)<a+0,
> i.e. 0<a, which is a>0 (the desired conclusion).
> This bypasses proof by contradiction altogether.
> Then again, who says you can't use a knife as a screwdriver.
> It's clumsy, but it works, as I've proved many times.
>
> Hope that helps,
>
> Walter
>
> --- Corey Bray wrote:
>  > Hello Everyone,
>  >
>  > Given that -a < 0, prove that a > 0 by assuming the conclusion
>  > is false and prove that it must be the case by contradiction.
>  > I was not exactly certain how to show this to be the case using
>  > a proof by contradiction method.
>  > Can someone here demonstrate this?
>  >
>  > Thanks in advance...
>  >      Corey...
>
>
> To unsubscribe from this group, send an email to:
> understandingmath-unsubscribe@yahoogroups.com
>
>
>
> Your use of Yahoo! Groups is subject to http://docs.yahoo.com/info/terms/

> Can someone here demonstrate this?

Sure, I'll give it a go. There's lots of ways to do it.
Here's the most direct one I can think of:

Assume the conclusion is false.
That means, assume it is false that a > 0.
Well, that  means that a <= 0.
If that's so, and we're given that -a < 0,
then a+(-a) < 0+0, (since w<=x and y<z implies w+y<x+z).
Simplifying the two sides, this says that "0<0"
is true. But, this isn't true, and so the initial
assumption that the conclusion is false was wrong.
End of Proof by Contradiction.

Of course, if you already know that w<=x and y<z implies w+y<x+z,
why not substitute w=a, x=a, y=-a, z=0 and immediately get
that a<=a (always true) and -a<0 (given) implies that a+(-a)<a+0,
i.e. 0<a, which is a>0 (the desired conclusion).
This bypasses proof by contradiction altogether.
Then again, who says you can't use a knife as a screwdriver.
It's clumsy, but it works, as I've proved many times.

Hope that helps,

Walter

--- Corey Bray wrote:
  > Hello Everyone,
  >
  > Given that -a < 0, prove that a > 0 by assuming the conclusion
  > is false and prove that it must be the case by contradiction.
  > I was not exactly certain how to show this to be the case using
  > a proof by contradiction method.
  > Can someone here demonstrate this?
  >
  > Thanks in advance...
  >  Corey...

Hello Everyone,


	 Given that -a < 0, prove that a > 0 by assuming the conclusion is false
and prove that it must be the case by contradiction.  I was not exactly
certain how to show this to be the case using a proof by contradiction
method.  Can someone here demonstrate this?

	 Thanks in advance...


	 Corey...

a^2-b^2=(a-b)(a+b)
Read about it by following the link at
http://www.gbbservices.com/mathematics.html

Enjoy,

Walter

Conservation of Energy gives yet another interpretation
of the trigonometric identity cos^2(t)+sin^2(t)=1.

I've updated the essay on this identity. Just follow the
link at http://www.gbbservices.com/mathematics.html
to see the details. Also, I've started to add some
images to the text. They've been generated via the
Python Imaging Library.

Walter

Why do 3d graphics programmers know about quaternions?
If you want to know, follow the link from
http://www.gbbservices.com/mathematics.html

Walter

My first essay is now online.
You'll find a link at http://www.gbbservices.com/mathematics.html
The essay revisits the trigonometric identity cos^2 + sin^2 = 1
using different mathematical contexts.

I'm always delighted when a "basic" result from the early stages
of the mathematics curriculum reappears in a later stage, with a
new meaning in a new context. The result I'm writing about in
this essay, which first appears in the context of trigonometry
and right angled triangles, reappears in different contexts again
and again (and again). I've catalogued a few of the
reinterpretations that I'm aware of. If you become aware of others,
please let me know.

Enjoy the essay!

Walter Vannini

Hi,

This is a test.

Walter