Old Math expects you to accept a line as having area #7 TRUE CALCULUS; without the phony limit concept (textbook 1st ed.)

[Archimedes Plutonium]

Alright, this little textbook is moving along splendidly so far, and
even learned something new-- a formal definition of continuity when
successive numbers have holes or gaps of 10^-603 between them. And now
I am on the chapter of the integral and integration.

The first thing I did to prepare for this page tonight is to search
whether the "picketfence" concept had been traced in mathematics
history. So where is the first time that a mathematician weaved into
calculus the picketfence model? Was it Leibniz and Newton, in order
for them to discover the antiderivative concept? I do not know, and
perhaps someone who is expert on the history of Leibniz and Newton can
tell us if both knew of the picketfence model?

I personally recall when I first heard of the picketfence model was in
College at UC in 1968 in Freshman Calculus class using the textbook by
Fisher and Ziebur. Now I do not recall if the picketfence model was in
the text or whether it was the professor at the blackboard that used
the picketfence model. I do not recall but it was 1968 that I first
learned of this picketfence model and it is important, perhaps the
most important model in all of mathematics since the Calculus is one
of the most important branches of mathematics.

So far I have covered the derivative and am just starting with the
Integral having discussed the antiderivative in the last post. And we
can easily see that the tops of the picket fence with its angle
delivers to us the derivative for that angle can vary from 0 degrees
up to but not including 90 degrees
and here is another picture of a picket fence model:

|\
||
||
||

and here is the reverse angle:

/|
||
||
||

Now, here is the very most important fact about Integrals and
Integration that all the Calculus books before this book failed to
teach. Now I read many of Calculus books for information through the
years. I studied in school the Fisher and Ziebur and read some of
Strang and some of Stewart and some of Ellis & Gulick and I do not
recall coming upon the picketfence model in these texts except for
maybe Fisher and Ziebur. I think it is worthwhile to track down the
history of this picketfence model because the major error of all these
Calculus books and of the teaching of Calculus in classrooms across
the world is that with the limit concept, the student is expected to
understand that the integral is the summation of many lines, each line
containing some area. This is the result of mathematics not defining
the borderline between finite and infinity which then yields what the
smallest nonzero number is, which then tells us what the size of the
holes or gaps are between successive numbers.

So in Old Math their Integral and Integration based on limit concept
has students thinking that the integral of the function y=x or y=x^2
are the summations of these rectangles that have no width but are just
lines

|
|
|
|

Whereas in New Math, we find that the borderline of infinity is Floor-
pi*10^603 and its inverse is the size of the hole or gap between
successive numbers, I call it 10^-603 for short abbreviation and that
allows Calculus to have a true rectangle with a tiny triangle sitting
atop the thin rectangle whose width is no more than 10^-603

Do you see the difference between New Math and Old Math? In Old Math,
they want you to be a fool, not as foolish as believing the Moon is
made of cheese, but not far from such foolishness, for they expect you
to believe and accept that the integral is
a line segment yet it has area.

In New Math, we do not want students turned into fools. We want
students to think without contradiction, and that means area must be a
rectangle however thin that rectangle width is.
That is why the Calculus cannot exist unless successive numbers have
holes and gaps between them.

So in New Math, the integral and integration are long slender
rectangles of width 10^-603 and atop that rectangle sits a tiny
triangle with a hypotenuse that becomes the derivative.

In New Math, we do not force students to believe and accept a
contradiction that the limit concept makes integration the summing up
of line segments when line segments have no area.

Now I do recall the math professor at UC when I was a freshman in 1968
lecturing on integration and saying words to the effect that
integration made the rectangles as thin as possible by the limit
concept.

So, the Calculus taught in schools is nothing but a sham teaching of
contradictory nonsense.

As I said so many times before, if you do not know the borderline of
finite into infinite, you cannot have holes or gaps between successive
numbers and thus you cannot have Calculus.


--

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