The beauty of future editions of a textbook is that errors can be removed and improvement of teaching can be improved. Already I made a major error, in that suggesting that we do not connect points of the graph of the function when we have holes between points, for we certainly do connect them since the hypotenuse atop the picketfence is a straight line segment that is connected to a "number point of the graph". In fact, that is what the Fundamental Theorem of Calculus is going to be in New Math. In Old Math, they thought their important Fundamental theorem was going to be the union of the derivative as inverse to the integral. In New Math we build the derivative as inverse and why need to prove it when we built it as such. So the Fundamental Theorem of Calculus in New Math concerns itself with why and how is it that Euclidean geometry with straight lines and straight line segments is the only geometry that can build a Calculus, while Elliptic geometry and Hyperbolic geometry are impossible geometries to build a Calculus therein?
Now I realized after post-page #3 yesterday (almost 1/3 done with the book) that it is too complicated for a High School student. So this post, which should be #4 is now numbered to be #0 so as to make simple and inviting to High School students. This should be the first page so that I can constantly refer to this simplified model.
Exercise on Graph Paper
Now I do not know if I can get 100 dots per line in the Usenet sci.math newsgroup. Let me try.
The above is a grid of 100 dots for the x-axis and 100 dots for the y- axis and 100x100=10,000 dots altogether. I doubt the post will show it in a square 100 dots wide and 100 dots long.
So I need the student or reader to get a graph pad. I have an Engineering graph pad where each sheet is 35 dots wide and 50 dots long so I need to take 6 sheets and cut and paste, or cut and tape them together to form a large sheet that is 100 dots wide and 100 dots long for a total of 10,000 dots altogether. So if a High School student, it is wise to do this exercise for you want to refer to it and to constantly make pencil drawings of functions and graphs to know what is going on.
Now we pretend, pretend that 10 is infinity borderline, the borderline where 10 is the last finite number and the largest finite number. So that has a deep and powerful implication. For it means that 1/10 is also a borderline of the small.
Now mathematics is the science of precision and that means that mathematics can handle precision and accuracy only with finite entities, finite numbers, finite points of geometry. Mathematics starts deteriorating once it reaches infinity, for it loses precision and accuracy. When mathematics is in infinity territory, it no longer is mathematics. The planet Earth is a good analogy here. The planet Earth has a borderline of solid matter Earth which is the surface of Earth as a borderline, including water of oceans. Another borderline is the atmosphere, but beyond the atmosphere we start getting into Outer Space. Now the magnetosphere of Earth that protects Earth from Solar rays that are harmful can be considered a borderline for Earth but beyond that, it is hard to say that Earth has any physical parameters. Now that is an analogy to mathematics. Mathematics is the science of precision and can be precise about finite numbers and finite points of geometry but once it reaches the borderline of infinity numbers or infinity points of geometry, we no longer have mathematics of total precision, and math begins to fall apart into imprecision. Now in the next post, I will talk about the precise borderline of infinity, but in this post, we pretend the number 10 is that infinity borderline. For High School students, they can handle the number 0, 1/10, 2/10, 3/10, .. on up to 10. And they can easily graph functions with only these numbers in play. They can draw picketfence structures on this large graph paper.
And as I write the next 9 pages of this 10 page textbook, I will constantly refer to this 100 x 100 = 10,000 points of the Cartesian Coordinate System.
-- 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: