Alright, sorry about that detour there of y = 1/x. I suffer, as everyone else suffers from the pollution of Old Math with their continuums and their inability to ever properly define the borderline of finite with infinite. The Old Math pollution comes to a large confrontation with y = 1/x.
I, like everyone else has been polluted and brainwashed as to how to think about the number "e" and the derivative and integral and function graphs. In that detour, I finally see the clear picture and in this post, we will do and see the number "e" like it has never been seen before. So let us do that exciting and beautiful math with no further delay.
When we mark out infinity borderline such as in the 10 grid where 10 is the largest finite number and thus 0.1 is the smallest nonzero finite number and then, 10.1 becomes the first infinity number and the inverse of 0. In the 10^603 Grid, we do the same thing, only the first infinity number is 1*10^603 + 1*10^-603. In these Grids of infinity borderlines, they are successive numbers with empty space in between. They are filled with gaps and holes. That is Euclidean Geometry, and there are "no curves" in Euclidean Geometry, but only a straightline segment connecting one point to another point. These connected segments appear to be curved but whenever we focus in on the points connected themselves do we see they are all straightline connections.
Now many functions in Euclidean Geometry are full straightlines overall such as y = 3 or y = 10 or y =x. So the function graph overall is a straightline itself. But many functions are overall not straight lines but rather the collective sum of smaller straightline segments and these I have defined as straightlinecurves.
The function y = x is a Overall-Straightline The inverse function of y = x is y = 1/x and it is overall a straightlinecurve wherein all its connections from x=1 to x = macroinfinity are successor points such that the dy/dx stays smooth as a curve.
So, here is the deal. In Euclidean Geometry there is one and only one function that is the Identity function and it is y = x. That function has an inverse y = 1/x. Since the Identity function is special, its inverse is special. The derivative of Identity is dy/dx = 1, for that is why it is identity.
The derivative of inverse identity is not 1 but rather is "e" in that it involves e^x.
What is this special number "e"? Well, it is what pi is to Elliptic geometry, for "e" is the pi of Hyperbolic geometry.
The trigonometry is about pi and how pi generates the trig functions in a right triangle. Trigonometry is the connecting of Euclidean with Elliptic geometry of connecting circles with right triangles.
The number "e" is the connecting of Euclidean geometry with Hyperbolic geometry. Of connecting the straightline of y = x with a straightlinecurve of y = 1/x. The function y = 1/x is a hyperbola.
The derivative of y = x is always 1. Never changes, but always 1.
The derivative of y = 1/x is always e^x produces e^x and never changes but always e^x.
In the graph of the function y = 1/x, we use only the portion of the function from x=1 to x = macroinfinity. We do not use, nor need the portion from x=0 to x =1, and the reason should be obvious, in that both portions are mirror images of one another and the one portion along the y axis asymptote is microinfinity, whereas the portion along the x-axis from x=1 to x= macroinfinity is a repeat of the other asymptote.
So, what was the pollution of Old Math, that prevented me and everyone else from understanding of what "e" is?
Well, if we take the function y =x and graph it along with y = 1/x, and now, throw away all of the graphed functions in the interval x=0 to x=1. What we have remaining is a hybrid angle. I call it a hybrid angle because one side of the angle is the straightline of y =x and the other side of the angle is the straightline curve of y = 1/x. To keep the derivative the same dy/dx for y =x is simple and easy for it is always 1. To keep the derivative the same for y = 1/x, the function has to be a low lying curve looking with such very tiny changes that the derivative stays the same value.
So here we begin to learn what "e" truly is. It is the counterpart to the derivative of Euclidean geometry that is always the same y = x has y' = 1. The derivative is always 1. So that causes the inverse of y = x the function y = 1/x to have a special number so that its derivative is always the same, always "e". It cannot be pi, for pi forms a closed straightlinecurve and is not even a function as a closed curve. It has to be a open curve and thus a new and special number, that of "e".
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: