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Topic: Fundamental Theorem of Calculus is superfluous in New Math #9 TRUE
CALCULUS; without the phony limit concept (textbook 1st ed.)

Replies: 3   Last Post: May 22, 2013 1:20 AM

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Registered: 3/31/08
Fundamental Theorem of Calculus is superfluous in New Math #9 TRUE
CALCULUS; without the phony limit concept (textbook 1st ed.)

Posted: May 21, 2013 4:38 PM
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Alright, I have come to some decisions at this stage of this tiny
textbook, that I shall make it no longer than 10 pages. I decided on
that because a bright Middle School student can handle 10 pages of
mathematics that he or she has never seen before and learn something.
However if 25 pages, they would likely be too discouraged. And also,
because True Calculus has no limit concept, that most of modern day
calculus of those 700 page texts, much of that gobbleygook phony
baloney or gibberish nattering nutter speak is about the limit. When
you have true math, you need just 10 pages to explain it. When you
have fake math, you need 700 pages of symbolism and abstractions to
hide and cover up.

So I have two pages remaining in this edition. And when I do the 2nd
edition I can do it all in two days posting 5 posts per day. So a
rapid fire book is this, because I can have the 100th edition within
one year.

Now since this text is only 10 pages long, I need no chapters to
organize because from start to finish, anyone can read it in one day
and no point in dividing 10 pages into chapters.

Now let me outline the education system of how calculus is taught in
Old Math. The first Calculus in the school system is called
"Precalculus" and it teaches about function, about area under graph
and about slope and tangents, but stops short of the limit concept.
Calculus taught in College is basically a year study of the "limit
concept". So in Old Math, calculus and limit concept were one and the

In New Math, we start first with the concept of finite moving into
infinity and have to find a borderline of where is the last finite
number and the start of infinity. I found that borderline to be Floor-
pi*10^603 where pi has three zero digits in a row and is evenly
divisible by 2,3,4,5 or 120. This divisibility is important for it
gives us the Euler regular polyhedra formula and it gives us where the
surface area of the pseudosphere equals the area of the corresponding
sphere. The inverse of that infinity borderline I denote as 1*10^-603
is the smallest nonzero number possible in mathematics and geometry.
It is the metric size of the hole or gap or empty space between 0 and
the next number which is 1*10^-603 and the next number after that is
2*10^-603 and there are no numbers in between those three numbers
listed. There is just empty space, however one can draw a line or line
segment in that empty space even though there are no number points.

So these holes and gaps make the limit concept as fictional, for there
is no need of a limit. The holes themselves serve as a limit. The
holes prevent pathological functions from forming such as the
Weierstrass function or the function y = sin(1/x).

The holes allow the derivative to form or come into existence because
the hole gives the derivative room to form a angle, an angle between 0
and 90 degrees. Without the hole or empty space, the neighboring
infinite points would obstruct as in the Weierstrass function,
obstruct the formation of the derivative. But since every point in the
Cartesian Coordinate System is surrounded by a hole of at least
10^-603, that every point of the function is differentiable.

The holes allow the integral to come into existence because with the
empty space the integral is a summation of very thin picketfence
rectangles with a triangle on top of the rectangle. The hypotenuse of
the triangle top is the derivative. The integral is the summation of
all these picketfences whose width is exactly 10^-603. In Old Math,
the integral was a summation of line segments, but even Middle School
children have learned that lines and line segments have no area, yet
calculus professors seem to have lost sight of the fact that line
segments have no area when they explain calculus. So the hole of
10^-603 allows the integral to form and exist.

Now in most Old Math calculus texts of those 700 page gibbering
nattering nutter symbolism of limits, once they cover derivative and
integral, they usually want to tie the two together in what is called
the "Fundamental Theorem of Calculus". And they make a big stir and
fuss about this. But in New Math, we not only throw out the limit as
fakery, but we have no need to show that the derivative is the inverse
of integral and vice versa. In mathematics, do we need to have a
Fundamental theorem of add subtract or a Fundamental theorem of
multiply divide and prove they are inverses? No, we need not go
through that silliness.

In New Math, in True Calculus we merely note that the derivative is
the angle of the hypotenuse atop the picketfence which determines a
unique area of the picketfence, so that the derivative is the inverse
of the integral. If I change the area of the picketfence, I change the
derivative proportional to the area. If I change the angle of the
hypotenuse, I proportionally change the area inside the picketfence.

So in True Calculus we throw out the phony baloney limit concept and
along with it we have no need for a hyped up exaggerated Fundamental

Now let me speak more about geometry, since I have just these 2 last
pages. It is important to know the relationship of geometry to numbers
and that should have been the Fundamental Theorem of Calculus. The
fundamental theorem should have embodied the idea that why the
Calculus exists at all is because in Euclidean Geometry when we have a
Cartesian Coordinate System of dots separated by 10^-603 holes, that
no matter what the size of the graph is, the relationships of where
those dots are to each other always forms the same angles. So that the
function y= x is always a 45 degree angle. So the Fundamental Theorem
of Calculus should have been a theorem that explores and proves why
Euclidean Geometry can yield a calculus but that Elliptic geometry or
Hyperbolic geometry cannot yield a calculus.

And another geometry feature I want to start to explore is a truncated
Cartesian Coordinate System.
Here I have just two points for the x-axis of 0 and 1*10^-603 and I
have all the points of the y-axis from 0 to 10^603 or 10^1206 points
in all. Now I call that a truncated Coordinate System of the 1st
quadrant. And you maybe surprized as to how much one can learn from
this truncated system. It has the functions of y=3, and y=x, and
y=x^2. It also has the functions of Weierstrass function and the
function y = sin(1/x).

So for the function y=3 we plot the point (0,3) and (1*10^-603, 3). It
has the function y=x and we plot the point (0,0) and (1*10^-603,

What is nice about the truncated-Coordinate System is that we can
instantly learn a lot about functions without being bogged down with
distractions of a lot of point plotting. We can home in on just the
derivative or integral in that truncated interval and we can see how
in New Math, all the points and numbers of mathematics should be
transparent and visible to the mind's eye all in one glance.

Now we can even extend that learning to asking a question of huge
importance. Not with a truncated x-axis only but say a truncated x and
y axis. Suppose we truncated the y-axis to be just 10 points in all
and the x-axis its 2 points in all. Now the question of huge
importance is "What are all the possible functions that exist in that
truncated coordinate system?"

Now in Old Math if ever such a question was asked
"how many functions can exist (continuous functions)" the math
professor would answer-- infinity number. In New Math, that question
has a more precise answer. Of course it is a number larger than Floor-
pi*10^603, but in New Math, we can compute precisely what the total
possible functions that can exist.

For example, if we truncated the axes to just 2 points, 0 and
1*10^-603 then the total number of functions that exists is 4 from
probability theory.

f1 = (0,0), (1*10^-603,0)
f2 = (0,0), (1*10^-603,1*10^-603)
f3 = (0,1*10^-603), (1*10^-603,0)
f4 = (0,1*10^-603), (1*10^-603,1*10^-603)

So, what is the huge number by probability theory for a nontruncated
1st quadrant of total possible functions of mathematics?

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:

Archimedes Plutonium
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies

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