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Topic: Cartesian Coordinate System, the union of geometry with numbers #0
Textbook 2nd ed. : TRUE CALCULUS; without the phony limit concept

Replies: 4   Last Post: May 26, 2013 3:40 AM

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plutonium.archimedes@gmail.com

Posts: 10,041
Registered: 3/31/08
Cartesian Coordinate System, the union of geometry with numbers #0
Textbook 2nd ed. : TRUE CALCULUS; without the phony limit concept

Posted: May 23, 2013 2:32 PM
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The beauty of future editions of a textbook is that errors can be
removed and improvement of teaching can be improved. Already I made a
major error, in that suggesting that we do not connect points of the
graph of the function when we have holes between points, for we
certainly do connect them since the hypotenuse atop the picketfence is
a straight line segment that is connected to a "number point of the
graph". In fact, that is what the Fundamental Theorem of Calculus is
going to be in New Math. In Old Math, they thought their important
Fundamental theorem was going to be the union of the derivative as
inverse to the integral. In New Math we build the derivative as
inverse and why need to prove it when we built it as such. So the
Fundamental Theorem of Calculus in New Math concerns itself with why
and how is it that Euclidean geometry with straight lines and straight
line segments is the only geometry that can build a Calculus, while
Elliptic geometry and Hyperbolic geometry are impossible geometries to
build a Calculus therein?

Now I realized after post-page #3 yesterday (almost 1/3 done with the
book) that it is too complicated for a High School student. So this
post, which should be #4 is now numbered to be #0 so as to make simple
and inviting to High School students. This should be the first page so
that I can constantly refer to this simplified model.

Exercise on Graph Paper

Now I do not know if I can get 100 dots per line in the Usenet
sci.math newsgroup. Let me try.

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The above is a grid of 100 dots for the x-axis and 100 dots for the y-
axis and 100x100=10,000 dots altogether. I doubt the post will show it
in a square 100 dots wide and 100 dots long.

So I need the student or reader to get a graph pad. I have an
Engineering graph pad where each sheet is 35 dots wide and 50 dots
long so I need to take 6 sheets and cut and paste, or cut and tape
them together to form a large sheet that is 100 dots wide and 100 dots
long for a total of 10,000 dots altogether. So if a High School
student, it is wise to do this exercise for you want to refer to it
and to constantly make pencil drawings of functions and graphs to know
what is going on.

Now we pretend, pretend that 10 is infinity borderline, the borderline
where 10 is the last finite number and the largest finite number. So
that has a deep and powerful implication. For it means that 1/10 is
also a borderline of the small.

Now mathematics is the science of precision and that means that
mathematics can handle precision and accuracy only with finite
entities, finite numbers, finite points of geometry. Mathematics
starts deteriorating once it reaches infinity, for it loses precision
and accuracy. When mathematics is in infinity territory, it no longer
is mathematics. The planet Earth is a good analogy here. The planet
Earth has a borderline of solid matter Earth which is the surface of
Earth as a borderline, including water of oceans. Another borderline
is the atmosphere, but beyond the atmosphere we start getting into
Outer Space. Now the magnetosphere of Earth that protects Earth from
Solar rays that are harmful can be considered a borderline for Earth
but beyond that, it is hard to say that Earth has any physical
parameters. Now that is an analogy to mathematics. Mathematics is the
science of precision and can be precise about finite numbers and
finite points of geometry but once it reaches the borderline of
infinity numbers or infinity points of geometry, we no longer have
mathematics of total precision, and math begins to fall apart into
imprecision. Now in the next post, I will talk about the precise
borderline of infinity, but in this post, we pretend the number 10 is
that infinity borderline. For High School students, they can handle
the number 0, 1/10, 2/10, 3/10, .. on up to 10. And they can easily
graph functions with only these numbers in play. They can draw
picketfence structures on this large graph paper.

And as I write the next 9 pages of this 10 page textbook, I will
constantly refer to this 100 x 100 = 10,000 points of the Cartesian
Coordinate System.

--
More than 90 percent of AP's posts are missing in the Google
newsgroups author search archive from May 2012 to May 2013. Drexel
University's Math Forum has done a far better job and many of those
missing Google posts can be seen here:

http://mathforum.org/kb/profile.jspa?userID=499986

Archimedes Plutonium
http://www.iw.net/~a_plutonium
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies




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