Infinity borderline and MicroInfinity versus MacroInfinity #9 HS-Textbook 7th ed.: TRUE CALCULUS; without the phony limit concept
Now I am proud of any student that has stayed with the plotting of graphs of functions in the 10-Grid system with its fractions and decimal points. I am proud of them because now comes the time to make that Grid the importance it deserves. In Fake Calculus, they have you plot points of the usual of 0, 1, 2, 3, etc. They never have you plot points of the smallest nonzero number points, because they do not have a smallest nonzero number points. Their calculus is fakery because they do not have a borderline between finite versus infinite. And their fake calculus is bolstered up by another fake concept of limit (when you make a liaring, you usually need more liaring to cover up the first liaring). When you do not rise to reasoning that if you have a concept of finite and a concept of infinity, that a borderline must separate them, otherwise, everything is finite and no infinite exists.
So the moment that any person agrees there is a finite and there is a infinite, then, there is a borderline between them.
In the 10 Grid system that High School students have used in this 10 page textbook, the 10 is a pretend infinity borderline that 10 is the last finite Macro or large number, and the inverse of 10 is 1/10 or 0.1 and it becomes the smallest nonzero number in 10 Grid, the Micro number. And these two borderline numbers of finite to infinite I call Macroinfinity and Microinfinity, for they are the largest and smallest (nonzero) finite numbers. There are no numbers between 0 and .1 but rather there is empty space. We need that empty space to give the integral internal area, we do not fake it with a so called limit concept, and then go on and add up line segments that have no internal area for the integral. And we do not smother the derivative by endless strings of more infinites to get a slope line.
Now in mathematics of algebra, numbers and geometry, there are a lot of truths and theorems of those truths. So what is a large number where all those truths and theorems of mathematics are true that complies with a macroinfinity borderline? The number 10 is too small to be macroinfinity. In another textbook of mine-- Correcting Math, I found this large number that complies with all the known truths and theorems of mathematics that would serve as the borderline of MacroInfinity to be Floor-pi*10^603. You know pi as 3.14159... but this number Floor-pi*10^603 has 604 digits and is a large integer number of 314159..32000. That number is a integer with 604 digits in all. It ends in the digits of 32000 as an integer. That number is able to allow the Euler formulas of regular polyhedra and regular polygons of their requirement that pi is evenly divisible by the number 120 which is equal to 1*2*3*4*5, or 5 factorial. And that number ends in three zero digits in a row to make for three dimensions.
So I started the High School students learning True Calculus from the 10 Grid, and the student can then go up to the 100 Grid and then the 1000 Grid. And the student begins to realize that as they go to higher and higher Grids, that circles made by tiny straight line segments actually look like a "smooth curve only they are really tiny straightline segments." And the student begins to realize that a circle or spiral or trigonometry curve as seen on a computer screen is not a curve at all but tiny straightline segments.
But now we must go to the full and true Grid of mathematics which is the 10^603 Grid. And calculus itself lends a hand in determining the finite to infinity borderline. Calculus cannot have the infinity borderline be that of the huge integer 314159..32000 but rather, Calculus demands the borderline be that of 10000..000 where it is a 1 followed by 603 digits of 0.
So the science of Calculus has a major hand and influence on what the borderline of infinity must be. It must be in the region of Floor-pi*10^603 but it cannot be the digits of pi made into an integer, but rather 1 followed by 603 digits of zero.
Why must the infinity borderline be 1*10^603 and not 3.14159 *10^603? Because of a concept of Complementary Function. Fake Calculus does not have a Complementary Function concept because they never had a infinity borderline to allow construction of the Complementary Function.
The Complementarity function of a given function is the reason that I need to keep and use the function y= 1/x, the logarithmic function.
Of course we use the 10-Grid system where .1 is the smallest nonzero number in the 1st quadrant only.
Here is a picture of the function y = x and we want to derive the Complementary function denoted as Comp-y=x.
Now I leave as an exercise for the student to plot the function y= 1/x and its Complementary function of Comp-y= 1/x.
If we put the two functions, y=x and its Comp, together in one graph, they form a large letter X pattern and intersect at the coordinate point (5,5). And the Comp-y =1/x and y = 1/x form another pattern, somewhat similar to y = x
Now the MacroInfinity is 1*10^603 and the inverse is the MicroInfinity of 1*10^-603. If we kept the Macroinfinity as 314159..32000 then our Microinfinity would be messy. For example, in the 10 Grid, if we pretended that 13 was macroinfinity then, microinfinity would be 1/13 = 0.0769.. and very messy, but when we pretend 10 is macroinfinity we have the clean number of 0.1 as microinfinity.
So here is a definition of the Complementary Function of a given function. The Complementary Function is y = Macroinfinity subtract the y value of the given function.
So in the example of y = x in 10Grid, we kept subtracting the y value of y =x from that of 10, so that for x=1, y = 1 the Complementary function would be x=1, y =9, and for x=2, y=2 the Complementary would be x=2, y=8.
So what major role does Calculus provide in determining that the Macroinfinity must be 1 with 603 zeros following rather than pi digits made into an integer?
Well, if you look at the function y =1/x, you cannot subtract the pi digits and hope to have a Complementary function. If infinity of macroinfinity were 1*10^603 and microinfinity was 1*10^-603, then subtraction of y =1/x is all clean numbers that are numbers of the 1*10^603 Grid.
So, here, the Calculus itself has a major role in what the Infinity borderline must be for the Calculus to exist.