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How many functions are there and zooming in on every number point of a function #2 Textbook 2nd ed. : TRUE CALCULUS; without the phony limit concept
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Archimedes Plutonium
May 23, 2013, 1:39:23 AM5/23/13
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Alright, I could spend the whole 10 pages just on continuity and
discontinuous functions, but this textbook is focused on the prime
essentials of Calculus, the derivative and integral.
Now because in New Math there is always a 10^-603 hole or gap between
successive numbers and number-points we could say that all functions
are discontinuous according to Old Math. In Old Math a line segment
looked like this:
____________>
whereas in New Math that same line segment is
really this:
........................>
where each number point has a 10^-603 hole between it and the
neighboring point leftwards and rightwards. So that would be
discontinuous in Old Math and in Old Math, they dwelled upon "a
continuum" for to them, between any two Reals or points of geometry
was an infinite number of more points or Reals. In New Math, once you
get to the smallness of 10^-603, there are no Reals or points of
geometry. Mathematics stops at infinity, whether the large infinity of
beyond Floor-pi*10^603 or the small infinity of less than 10^-603.
Mathematics is a science of precision that lies within finite numbers
and finiteness.
Now one of the problems is that of perception and this problem is made
more aware of in our modern times by the computer, for the computer is
much like the dots with holes for a line segment, only in computer
speak it is called bytes and pixels. So that if we zoom in on a line
segment in New Math to the size of 10^-603 we see the line segment as
this
................> but if we zoomed out we perceive that line segment
to be this _________> because of the sheer density of so many dots.
Now I defined continuity as being a dx along the x-axis wherein the dx
is no larger than 1*10^-603 so that in the case of y = 1/x where
division by 0 is undefined means the function is discontinous at 0 for
the dx is 2*10^-603 since the interval of -1*10^-603 to +1*10^-603,
but then a step function such as F(x) = 1 for x from 0 to 1 and F(x) =
2 for x from 1+1*10^-603 to 10^603 is a step function that looks like
this not drawn to scale:
...........................> 2
........> 1
......................................> x-axis
So that step function looks like two horizontal line segments, two
steps, with a break at x=1. It is not a discontinuous function for all
the dx are intact and no larger than 1*10^-603.
And amazingly the step function is continuous for the successive dx
are all 10^-603. But there is a break from one step to the other.
Now in graphing these functions we maybe compelled to connect the dots
by a straight line or some curve line, but in truth it is just dots
and the density of dots that compose the graph.
So how would I reconcile the idea that a step function is continuous,
yet it has that break?
I reconcile it with what David Bergamini writes in Life Science
Library titled MATHEMATICS, 1963 on page 108. Quoting: "There is one
major difficulty in the process of integration-- a difficulty so
enormous and so recurrent that most of our largest computers today
have been built specifically to cope with it. This is the problem of
so-called "boundary conditions." When the area of a picket fence is
measured, the boundary conditions are established by the two pickets
that mark the two ends of the fence. But there are no ends to many of
the curves that represent equations" end quoting.
What Bergamini is discussing is the boundary of the integral and when
you have a step function that is continuous everywhere but has "breaks
in the graph" that it is the job of "boundary conditions integration"
to overcome those breaks.
Grid system of all the points in geometry
being transparent and able to zoom in on a neighborhood of number-
points
The Cartesian Coordinate System grid in New Math of 1st quadrant looks
like this:
.......................................>
.......................................>
.......................................>
.......................................>
.......................................>
.......................................>
.......................................>
.......................................>
Where there are 10^1206 dots in the x-axis and 10^1206 dots in the y-
axis and in the entire grid there are 10^2412 dots in all.
Now I want to discuss in this chapter the unique ability of New Math
to see fully a function and to zoom in on a function of its smallest
interval. In Old Math we never had such abilities, except for perhaps
real simple functions like y =3 or y =x. In Old Math we have
complicated functions and the way mathematicians had worked them was
to plot various 100 points and hope to begin to see a pattern for
which he/she then guesses where all the other points are in the
graph. And Old Math never had the ability to zoom in on the smallest
interval for there were no smallest intervals because there was
always another infinity of points to reckon with.
In New Math all the points follow the same sequence:
0, 1*10^-603, 2*10^-603, 3*10^-603, 4*10^-603, 5*10^-603, . .
So that in the first quadrant we have a total of 10^1206 points
along the x-axis and the same number along the y-axis and thus for
the first quadrant we have a total of 10^2412 points in all, and can
zoom in on any region.
So that as we plot the Weierstrass function, we merely plot what
every one of the 10^1206 x values gives of 10^1206 y values. And
because there are holes or gaps of no numbers in between successive
numbers, a gap has a metric of 1*10^-603, which guarantees the
Weierstrass function is well behave and no longer a pathological
function. In fact the Weierstrass function is now continuous
everywhere where we define continuous function whose hole size width
dx is no larger than 1*10^-603. So as in a Weierstrass type function
if 0 has a y value of 0 and the next successor number is 1*10^-603
whose y value is 10^603 which would be the steepest angle in Calculus
derivative but would be continuous. And would be differentiable
everywhere.
So here we have a new concept in mathematics of zooming in on graphs
of functions. We have the entire universe of points of the Cartesian
Coordinate Grid and we have the possibility of defining every
function that can exist. It is a huge number of possibilities to be
sure for example just the first two points of 0 and 1*10^-603 has
each 10^1206 possible y values. So we have a probability combinations
and permutations.
So in New Math, we have graphs of functions where all is open to view
and scrutiny. We have every point or number layed out in a huge grid
and we can zoom in on any point and its neighboring points with holes
in between points.
In Old Math, zooming in never was satisfactory because once you zoom
in, there is yet another infinity of points. And in this manner, a
false function like the Weierstrass function is borne and nurtured
even though it is a fakery.
So as we plot the Weierstrass function in New Math of its 10^-603
holes between successive points, we do not end up with a pipedream of
a function of fractal jagged mountain inclines that is alleged to be
nowhere differentiable as seen in Wikipedia on Weierstrass function.
In New Math, since we have all the points of the grid in front of our
eyes and can zoom in to a specific point, that we no longer need a
concept of limit. The limit concept was to sweep the small infinities
under the carpet or rug of shame. But since we have those holes,
there is no need of a limit concept since we can instantly find the y
value of a point of interest and its neighbor point to the left and
right and thus look for the derivative. If we had that blizzard of
smaller infinities, we need some sort of help and that is why the
limit came about. The limit in mathematics is akin to the Aether in
physics, and then finally when it was realized that in the Maxwell
Equations the moving bar magnet in a stationary closed loop is the
same as a moving closed loop and stationary bar magnet, that the
Aether was phony. It is interesting to note that the Luminiferous
Aether dates to the 19th century and the limit concept of mathematics
dates to the 19th century with Cauchy. However, physics wised-up and
reasoned the Aether was nonexistent, but the mathematicians never
wisened up over the phony limit concept. The stumbling block was the
inability to reason that a borderline must exist between finite
ending and infinity starting which is the number Floor-pi*10^603 and
its inverse would be the holes and gaps in successive numbers.
So is there a better name for this concept of being able to list all
the numbers of the Cartesian Coordinate Grid with holes of 10^-603
separating the successive numbers on the axes? Is there a name for
the concept of zooming in to the smallest interval of numbers and
seeing exactly what the function plots? In politics there is a name
called "transparent and open government". But apparently in
mathematics, they are back in the stone age or cave dwelling when it
comes to Calculus and points of a graph.
How many functions altogether?
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,
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:
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
discontinuous functions, but this textbook is focused on the prime
essentials of Calculus, the derivative and integral.
Now because in New Math there is always a 10^-603 hole or gap between
successive numbers and number-points we could say that all functions
are discontinuous according to Old Math. In Old Math a line segment
looked like this:
____________>
whereas in New Math that same line segment is
really this:
........................>
where each number point has a 10^-603 hole between it and the
neighboring point leftwards and rightwards. So that would be
discontinuous in Old Math and in Old Math, they dwelled upon "a
continuum" for to them, between any two Reals or points of geometry
was an infinite number of more points or Reals. In New Math, once you
get to the smallness of 10^-603, there are no Reals or points of
geometry. Mathematics stops at infinity, whether the large infinity of
beyond Floor-pi*10^603 or the small infinity of less than 10^-603.
Mathematics is a science of precision that lies within finite numbers
and finiteness.
Now one of the problems is that of perception and this problem is made
more aware of in our modern times by the computer, for the computer is
much like the dots with holes for a line segment, only in computer
speak it is called bytes and pixels. So that if we zoom in on a line
segment in New Math to the size of 10^-603 we see the line segment as
this
................> but if we zoomed out we perceive that line segment
to be this _________> because of the sheer density of so many dots.
Now I defined continuity as being a dx along the x-axis wherein the dx
is no larger than 1*10^-603 so that in the case of y = 1/x where
division by 0 is undefined means the function is discontinous at 0 for
the dx is 2*10^-603 since the interval of -1*10^-603 to +1*10^-603,
but then a step function such as F(x) = 1 for x from 0 to 1 and F(x) =
2 for x from 1+1*10^-603 to 10^603 is a step function that looks like
this not drawn to scale:
...........................> 2
........> 1
......................................> x-axis
So that step function looks like two horizontal line segments, two
steps, with a break at x=1. It is not a discontinuous function for all
the dx are intact and no larger than 1*10^-603.
And amazingly the step function is continuous for the successive dx
are all 10^-603. But there is a break from one step to the other.
Now in graphing these functions we maybe compelled to connect the dots
by a straight line or some curve line, but in truth it is just dots
and the density of dots that compose the graph.
So how would I reconcile the idea that a step function is continuous,
yet it has that break?
I reconcile it with what David Bergamini writes in Life Science
Library titled MATHEMATICS, 1963 on page 108. Quoting: "There is one
major difficulty in the process of integration-- a difficulty so
enormous and so recurrent that most of our largest computers today
have been built specifically to cope with it. This is the problem of
so-called "boundary conditions." When the area of a picket fence is
measured, the boundary conditions are established by the two pickets
that mark the two ends of the fence. But there are no ends to many of
the curves that represent equations" end quoting.
What Bergamini is discussing is the boundary of the integral and when
you have a step function that is continuous everywhere but has "breaks
in the graph" that it is the job of "boundary conditions integration"
to overcome those breaks.
Grid system of all the points in geometry
being transparent and able to zoom in on a neighborhood of number-
points
The Cartesian Coordinate System grid in New Math of 1st quadrant looks
like this:
.......................................>
.......................................>
.......................................>
.......................................>
.......................................>
.......................................>
.......................................>
.......................................>
Where there are 10^1206 dots in the x-axis and 10^1206 dots in the y-
axis and in the entire grid there are 10^2412 dots in all.
Now I want to discuss in this chapter the unique ability of New Math
to see fully a function and to zoom in on a function of its smallest
interval. In Old Math we never had such abilities, except for perhaps
real simple functions like y =3 or y =x. In Old Math we have
complicated functions and the way mathematicians had worked them was
to plot various 100 points and hope to begin to see a pattern for
which he/she then guesses where all the other points are in the
graph. And Old Math never had the ability to zoom in on the smallest
interval for there were no smallest intervals because there was
always another infinity of points to reckon with.
In New Math all the points follow the same sequence:
0, 1*10^-603, 2*10^-603, 3*10^-603, 4*10^-603, 5*10^-603, . .
So that in the first quadrant we have a total of 10^1206 points
along the x-axis and the same number along the y-axis and thus for
the first quadrant we have a total of 10^2412 points in all, and can
zoom in on any region.
So that as we plot the Weierstrass function, we merely plot what
every one of the 10^1206 x values gives of 10^1206 y values. And
because there are holes or gaps of no numbers in between successive
numbers, a gap has a metric of 1*10^-603, which guarantees the
Weierstrass function is well behave and no longer a pathological
function. In fact the Weierstrass function is now continuous
everywhere where we define continuous function whose hole size width
dx is no larger than 1*10^-603. So as in a Weierstrass type function
if 0 has a y value of 0 and the next successor number is 1*10^-603
whose y value is 10^603 which would be the steepest angle in Calculus
derivative but would be continuous. And would be differentiable
everywhere.
So here we have a new concept in mathematics of zooming in on graphs
of functions. We have the entire universe of points of the Cartesian
Coordinate Grid and we have the possibility of defining every
function that can exist. It is a huge number of possibilities to be
sure for example just the first two points of 0 and 1*10^-603 has
each 10^1206 possible y values. So we have a probability combinations
and permutations.
So in New Math, we have graphs of functions where all is open to view
and scrutiny. We have every point or number layed out in a huge grid
and we can zoom in on any point and its neighboring points with holes
in between points.
In Old Math, zooming in never was satisfactory because once you zoom
in, there is yet another infinity of points. And in this manner, a
false function like the Weierstrass function is borne and nurtured
even though it is a fakery.
So as we plot the Weierstrass function in New Math of its 10^-603
holes between successive points, we do not end up with a pipedream of
a function of fractal jagged mountain inclines that is alleged to be
nowhere differentiable as seen in Wikipedia on Weierstrass function.
In New Math, since we have all the points of the grid in front of our
eyes and can zoom in to a specific point, that we no longer need a
concept of limit. The limit concept was to sweep the small infinities
under the carpet or rug of shame. But since we have those holes,
there is no need of a limit concept since we can instantly find the y
value of a point of interest and its neighbor point to the left and
right and thus look for the derivative. If we had that blizzard of
smaller infinities, we need some sort of help and that is why the
limit came about. The limit in mathematics is akin to the Aether in
physics, and then finally when it was realized that in the Maxwell
Equations the moving bar magnet in a stationary closed loop is the
same as a moving closed loop and stationary bar magnet, that the
Aether was phony. It is interesting to note that the Luminiferous
Aether dates to the 19th century and the limit concept of mathematics
dates to the 19th century with Cauchy. However, physics wised-up and
reasoned the Aether was nonexistent, but the mathematicians never
wisened up over the phony limit concept. The stumbling block was the
inability to reason that a borderline must exist between finite
ending and infinity starting which is the number Floor-pi*10^603 and
its inverse would be the holes and gaps in successive numbers.
So is there a better name for this concept of being able to list all
the numbers of the Cartesian Coordinate Grid with holes of 10^-603
separating the successive numbers on the axes? Is there a name for
the concept of zooming in to the smallest interval of numbers and
seeing exactly what the function plots? In politics there is a name
called "transparent and open government". But apparently in
mathematics, they are back in the stone age or cave dwelling when it
comes to Calculus and points of a graph.
How many functions altogether?
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,
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:
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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