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Chapt5 improved Galois Algebra theory in New Math #1178 Correcting Math 3rd ed
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Archimedes Plutonium
Nov 25, 2011, 3:17:10 PM11/25/11
to
This is what I used to have for chapter 5
Chapter 5 Making an improved and more precise Galois Algebra theory by
recognizing a second successor number of 1*10^-603 along with the 1
But now I realize I have to change that to read like this:
Chapter 5 Improved Galois Algebra theory in New Math
That allows me to make 1*10^-603 as a topic of that chapter.
But what I want to stress for that chapter is the fact that with the
Infinity borderline at
10^603, automatically creates an inverse borderline for the micro-
scale infinity at
10^-603.
So in New Math, there are two infinities, a macro-infinity and a micro-
infinity.
In Old Math, there is only a macro-infinity. In Old Math, no matter
how close and
nearby is a number to 0, whether we have 10^-700 or 10^-9086, no
matter how close, those numbers are finite numbers in Old Math.
Old Math is one dimensional in one direction as regards to infinity.
Now that shortsightedness of Old Math causes grave problems for Galois
theory in that without a inverse, the numbers in Old Math cannot form
a Field, perhaps they form a Group but not a Field.
And as for Geometry, with only a macro-infinity, we already seen that
the Calculus and angles in NonEuclidean geometry are nonexistent.
Now sure, in Old Math we applied a Sequence to tiny numbers as they
approach 0
such as the Sequence 1/2, 1/3, 1/4, 1/5, . . . and we said there are
an infinity of
numbers in that sequence and the limit is 0. But that is not really a
Micro-Infinity
that New Math insists must exist.
And, in Old Math, infinity can be represented by a line ray such as
this:
_________________>
Where the arrow indicates going to Macro-infinity, but that Old Math
cannot do a
line ray going to Micro-infinity. New Math can do that for given any
point such as
the point 0 we can have a Micro-Infinity line ray as this:
-1*10^-603 <__ 0
or in the opposite direction:
0 ___> 1*10^-603
That Micro-Infinity line ray goes through what Old Math called numbers
like that of
10^-604 beyond 10^-9997 beyond 10^-557733 etc etc
So we see immediately that Old Math is highly deficient, not only in
incapable of building a Calculus or a Topology, but not even capable
of building a Geometry.
But let me get the theme of this post that without a Micro-Infinity in
Old Math, that
Galois Algebra theory falls apart also.
The reason it falls apart is because in Old Math, a Field had to have
inverses, otherwise it can never graduate beyond a Group in Old Math.
So what is the inverse of Infinity oo in Old Math? It has no inverse.
Infinity in Old Math is only a Macro-infinity. And so, in Old Math,
they cannot have Galois theory of Field or Ring, but they are stuck
with only a Group. At least I think they can have a Group, but perhaps
they cannot even have a Group without inverse to infinity.
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
Chapter 5 Making an improved and more precise Galois Algebra theory by
recognizing a second successor number of 1*10^-603 along with the 1
But now I realize I have to change that to read like this:
Chapter 5 Improved Galois Algebra theory in New Math
That allows me to make 1*10^-603 as a topic of that chapter.
But what I want to stress for that chapter is the fact that with the
Infinity borderline at
10^603, automatically creates an inverse borderline for the micro-
scale infinity at
10^-603.
So in New Math, there are two infinities, a macro-infinity and a micro-
infinity.
In Old Math, there is only a macro-infinity. In Old Math, no matter
how close and
nearby is a number to 0, whether we have 10^-700 or 10^-9086, no
matter how close, those numbers are finite numbers in Old Math.
Old Math is one dimensional in one direction as regards to infinity.
Now that shortsightedness of Old Math causes grave problems for Galois
theory in that without a inverse, the numbers in Old Math cannot form
a Field, perhaps they form a Group but not a Field.
And as for Geometry, with only a macro-infinity, we already seen that
the Calculus and angles in NonEuclidean geometry are nonexistent.
Now sure, in Old Math we applied a Sequence to tiny numbers as they
approach 0
such as the Sequence 1/2, 1/3, 1/4, 1/5, . . . and we said there are
an infinity of
numbers in that sequence and the limit is 0. But that is not really a
Micro-Infinity
that New Math insists must exist.
And, in Old Math, infinity can be represented by a line ray such as
this:
_________________>
Where the arrow indicates going to Macro-infinity, but that Old Math
cannot do a
line ray going to Micro-infinity. New Math can do that for given any
point such as
the point 0 we can have a Micro-Infinity line ray as this:
-1*10^-603 <__ 0
or in the opposite direction:
0 ___> 1*10^-603
That Micro-Infinity line ray goes through what Old Math called numbers
like that of
10^-604 beyond 10^-9997 beyond 10^-557733 etc etc
So we see immediately that Old Math is highly deficient, not only in
incapable of building a Calculus or a Topology, but not even capable
of building a Geometry.
But let me get the theme of this post that without a Micro-Infinity in
Old Math, that
Galois Algebra theory falls apart also.
The reason it falls apart is because in Old Math, a Field had to have
inverses, otherwise it can never graduate beyond a Group in Old Math.
So what is the inverse of Infinity oo in Old Math? It has no inverse.
Infinity in Old Math is only a Macro-infinity. And so, in Old Math,
they cannot have Galois theory of Field or Ring, but they are stuck
with only a Group. At least I think they can have a Group, but perhaps
they cannot even have a Group without inverse to infinity.
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
Archimedes Plutonium
Nov 26, 2011, 2:34:54 AM11/26/11
to
Chapter 26: precision definitions of Sequence, Series, Function,
Continuity, Topology, Algebra
I am happy I put this chapter near the end of the text since
sharpening the definitions
is an ongoing process.
Recently I had this pretty insight as to precision defining
"continuity" in mathematics
since numbers have 10^-603 holes between consecutive Reals or
consecutive points.
So that made it quite unsightly to define continuity when a line in
mathematics
looks like this .............. and not like this _______________
Continuity as defined in New Math is that an equation is continuous if
it is a function without a break such as a step function. A break is
defined as not having
a derivative.
A torus is not a function , nor is a circle a function and thus they
are not continuous.
An equation of y=sin(1/x) or y = x , or y=10^603 x are continuous
because they are functions.
Now Continuity defined as such also clarifies what the subject of
Topology involves and what it does not involve. Topology is a subset-
subject of Geometry and topology focuses on geometries that have holes
larger than 10^-603. Topology is a subject that does not deal nor
handles functions. All functions are continuous, except for step-
functions.
So this leads to intriguing insights. If you can plot a function and
it has no breaks in it, then it is continuous. You can plot a circle
and you can plot a torus but neither are functions and thus they
immediately qualify as having a torus topology.
As mentioned previously, what this does, is demonstrate that the
Calculus is a Projection, a Projective Geometry upon a Coordinate
System. In effect, I am connecting the entire subject of Calculus as a
Projective Geometry upon a Coordinate System, and the exercise of
finding a derivative or integral is an exercise of keeping all the
numbers fixed relative to one another and finding the angles with
respect to any two given coordinate-numbers.
Topology, likewise, is a subject that is also a Projective Geometry
upon a Coordinate System, however the difference with Topology versus
Calculus, is
all the plotting in Calculus are functions, and because they are
functions (other than step-functions) they are continuous and thus
differentiable and integrable. In Topology, the plotting may or may
not be a function, and if a function then it is uninteresting for it
does not have any topology tears like a torus. So Topology is a
plotting of nonfunctions with tears larger than 10^-603.
Now I am sure I have some flaws and errors in the above, but given
some time to
think over these items, I am sure to make more clear and error free,
for now, it is the basic outline of major ideas that I want to put
front and center. I think I can eliminate step functions, because to
me they seem to be obeying the rules of what forms a function. For
example when someone says the function y=1 then we clearly see it as a
continuous function, but when someone proffers a function of
y=1 for [0,4) and then y=2 for [4,oo) seems to me to be cheating in
what is a function for somehow it looks like a compound function of
stitching part of the function y=1 onto y=2. So I think I maybe able
to eliminate all step functions out of
mathematics as compound-functions which are not really functions at
all.
So that if I defined functions as meaning elemental functions versus
compound functions and eliminate compound functions altogether, would
result in higher clarity.
Continuity, Topology, Algebra
I am happy I put this chapter near the end of the text since
sharpening the definitions
is an ongoing process.
Recently I had this pretty insight as to precision defining
"continuity" in mathematics
since numbers have 10^-603 holes between consecutive Reals or
consecutive points.
So that made it quite unsightly to define continuity when a line in
mathematics
looks like this .............. and not like this _______________
Continuity as defined in New Math is that an equation is continuous if
it is a function without a break such as a step function. A break is
defined as not having
a derivative.
A torus is not a function , nor is a circle a function and thus they
are not continuous.
An equation of y=sin(1/x) or y = x , or y=10^603 x are continuous
because they are functions.
Now Continuity defined as such also clarifies what the subject of
Topology involves and what it does not involve. Topology is a subset-
subject of Geometry and topology focuses on geometries that have holes
larger than 10^-603. Topology is a subject that does not deal nor
handles functions. All functions are continuous, except for step-
functions.
So this leads to intriguing insights. If you can plot a function and
it has no breaks in it, then it is continuous. You can plot a circle
and you can plot a torus but neither are functions and thus they
immediately qualify as having a torus topology.
As mentioned previously, what this does, is demonstrate that the
Calculus is a Projection, a Projective Geometry upon a Coordinate
System. In effect, I am connecting the entire subject of Calculus as a
Projective Geometry upon a Coordinate System, and the exercise of
finding a derivative or integral is an exercise of keeping all the
numbers fixed relative to one another and finding the angles with
respect to any two given coordinate-numbers.
Topology, likewise, is a subject that is also a Projective Geometry
upon a Coordinate System, however the difference with Topology versus
Calculus, is
all the plotting in Calculus are functions, and because they are
functions (other than step-functions) they are continuous and thus
differentiable and integrable. In Topology, the plotting may or may
not be a function, and if a function then it is uninteresting for it
does not have any topology tears like a torus. So Topology is a
plotting of nonfunctions with tears larger than 10^-603.
Now I am sure I have some flaws and errors in the above, but given
some time to
think over these items, I am sure to make more clear and error free,
for now, it is the basic outline of major ideas that I want to put
front and center. I think I can eliminate step functions, because to
me they seem to be obeying the rules of what forms a function. For
example when someone says the function y=1 then we clearly see it as a
continuous function, but when someone proffers a function of
y=1 for [0,4) and then y=2 for [4,oo) seems to me to be cheating in
what is a function for somehow it looks like a compound function of
stitching part of the function y=1 onto y=2. So I think I maybe able
to eliminate all step functions out of
mathematics as compound-functions which are not really functions at
all.
So that if I defined functions as meaning elemental functions versus
compound functions and eliminate compound functions altogether, would
result in higher clarity.
Richard Shaft
Nov 26, 2011, 1:08:24 PM11/26/11
to
Worng newsfroup, you fucking imbeciale.
"Archimedes Plutonium" wrote in message
news:ba132993-ec9d-4fee...@h5g2000yqk.googlegroups.com...
"I have wasted my entire life."
"Archimedes Plutonium" wrote in message
news:ba132993-ec9d-4fee...@h5g2000yqk.googlegroups.com...
"I have wasted my entire life."
Archimedes Plutonium
Nov 26, 2011, 3:41:12 PM11/26/11
to
On Nov 26, 1:34 am, Archimedes Plutonium
What I am working towards is a definition of continuous for functions
that eliminates
all of Old Math discontinuities.
We can call three types of discontinuity in Old Math.
(1) point discontinuity such as f(0)=1 and f(x)=x
(2) break discontinuity such as a step function U=1 and U=0
(3) branch discontinuity such as f(x)=1/x^2
Now I used Strang's CALCULUS,1991, page 86 for these above three
examples.
Now I get rid of those three discontinuities of Old Math by several
methods.
First we get rid of all functions that are compound functions and in
the definition
of function we allow only elemental-functions. The function y=x^-2 is
a branch function
and is compound. So that we cannot have any compound functions. We can
consider one branch
of that as a function.
In Old Math, we eliminate all functions that have a F and G such as
(1) and (2) of Strang's
page 86, those are also compound-functions.
Now I am not sure of this statement or claim, that I am about to make,
but I think that every
step function is a compound function and thus in New Math, no function
is a step function.
So that in New Math, every function is elemental-function.
In New Math the function y=x^-2 cannot be both branches.
So in the definition of function in New Math, all functions are
continuous as per the definition of continuity is that the x-axis
points have a y-axis component, the purpose of which is that
continuity
exists even though there are tiny holes of 10^-603 in between
consecutive-points.
Now as for the function shown as y=sin(1/x) in Strang's page 86, it is
continuous in New Math because
in New Math we define 1/0 as being equal to 0. Division by zero in New
Math is defined as equalling 0.
Hopefully, all the bugs are worked out by the above of elemental-
function and compound-function, but
if there are more bugs, I suspect it is easy to get rid of them.
<plutonium.archime...@gmail.com> wrote:
> Chapter 26: precision definitions of Sequence, Series, Function,
> Continuity, Topology, Algebra
(snipped)
> Chapter 26: precision definitions of Sequence, Series, Function,
> Continuity, Topology, Algebra
What I am working towards is a definition of continuous for functions
that eliminates
all of Old Math discontinuities.
We can call three types of discontinuity in Old Math.
(1) point discontinuity such as f(0)=1 and f(x)=x
(2) break discontinuity such as a step function U=1 and U=0
(3) branch discontinuity such as f(x)=1/x^2
Now I used Strang's CALCULUS,1991, page 86 for these above three
examples.
Now I get rid of those three discontinuities of Old Math by several
methods.
First we get rid of all functions that are compound functions and in
the definition
of function we allow only elemental-functions. The function y=x^-2 is
a branch function
and is compound. So that we cannot have any compound functions. We can
consider one branch
of that as a function.
In Old Math, we eliminate all functions that have a F and G such as
(1) and (2) of Strang's
page 86, those are also compound-functions.
Now I am not sure of this statement or claim, that I am about to make,
but I think that every
step function is a compound function and thus in New Math, no function
is a step function.
So that in New Math, every function is elemental-function.
In New Math the function y=x^-2 cannot be both branches.
So in the definition of function in New Math, all functions are
continuous as per the definition of continuity is that the x-axis
points have a y-axis component, the purpose of which is that
continuity
exists even though there are tiny holes of 10^-603 in between
consecutive-points.
Now as for the function shown as y=sin(1/x) in Strang's page 86, it is
continuous in New Math because
in New Math we define 1/0 as being equal to 0. Division by zero in New
Math is defined as equalling 0.
Hopefully, all the bugs are worked out by the above of elemental-
function and compound-function, but
if there are more bugs, I suspect it is easy to get rid of them.
Archimedes Plutonium
Nov 26, 2011, 4:32:46 PM11/26/11
to
On Nov 26, 2:41 pm, Archimedes Plutonium
Now there maybe a real nice easy method of reform. We simply say that
Calculus takes place only in the first quadrant of the Cartesian
Coordinate
System and that 0 is not included and so the Calculus is from the open
set
(0,10^603)
Calculus takes place only in the first quadrant of the Cartesian
Coordinate
System and that 0 is not included and so the Calculus is from the open
set
(0,10^603)
Archimedes Plutonium
Nov 26, 2011, 9:49:05 PM11/26/11
to
On Nov 26, 3:32 pm, Archimedes Plutonium
Now I am not sure if a point-wise discontinuity is all that bad for
Calculus. A break as in a
step function is bad, but I think that Calculus gets by, always, if
the discontinuity is a pointwise
discontinuity. So that a function of y=x^-2 in 3rd dimension is easily
handled by Calculus at x=0.
What I want to reform or resolve is that functions have holes between
consecutive points and that those holes do not harm Calculus or
Topology or Geometry but in fact strengthen those subjects.
And it may take some time before all the bugs are worked out, but it
will eventually all work out.
I do not think I have the time to pause here to try to work out all
the bugs.
Calculus. A break as in a
step function is bad, but I think that Calculus gets by, always, if
the discontinuity is a pointwise
discontinuity. So that a function of y=x^-2 in 3rd dimension is easily
handled by Calculus at x=0.
What I want to reform or resolve is that functions have holes between
consecutive points and that those holes do not harm Calculus or
Topology or Geometry but in fact strengthen those subjects.
And it may take some time before all the bugs are worked out, but it
will eventually all work out.
I do not think I have the time to pause here to try to work out all
the bugs.
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