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 same.
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 theorem.
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, 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.
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: