I read a book recently called "A Tour of the Calculus" by David Berlinski, written to describe the calculus and what it does, rather than teach it. There's a couple of pages of analysis in most of the chapters, which is to say, just slightly more than I can take in. He thinks through the ideas of integration and derivatives, and includes things like the relationship between mathematical objects, physics and "la vie vecue". I like best that he calls it the calculus, the way Sir Isaac did.
But Euler's e, I still don't think I get it. It's a transcendental number, yup, it a logorhythmic function, it can be used as a base same as we commonly use base 10 or computers use base 2, right, and...it's the number whose...log is itself...no, but it loops back on it self somehow...Nope. Dunno. It has something to do with ants on a Moebius strip somehow.
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Good stuff. I will get back to you on it.
A function y=f(x), right?
An exponential function is a function where the variable (x) is in the exponent, i.e.
y = 2 to the (x) power. In this case, f(2) = 4, f(3) = 8, etc.
The 2 in the original function is the 'root' of the exponential function.
The derivative of a function is another function that describes the slope of the line of the function at any point. We call the derivative of f(x), f-prime(x), right?
If you take the function y=2 (no matter what x you put in, the answer comes out as 2, a constant function. If you graph this, it will be a horizontal line. It has a slope of zero at every point, the derivative is zero. f-prime(x)=0. You see that the derivative is different from the function. There is one function, which is an exponential function, for which the derivative is the same as the function. The slope of the curve at any point is the same as the y-position of the curve at this point. Get it? This special function is not just any exponential function. It is not f(x) = 2 to the (x) power, it is not 3 to the (x) power, it is Euler's (e) to the (x) power. We call this the one true exponential function and we write f(x) = exp(x).
Once you have this tool, there are a whole lot of things you can do. The first of these is to create the 'inverse function', i.e. the function that goes backwards from the exponential function. We call this function the logarithmic function. log(exp(x)) = x. Get it?
Another example of an inverse function is the square root function, which is the inverse of the square function.
SquareRoot(Square(x)) = x, right?
The inverse function of the (e) exponential function is called the natural logarithm, but it is the only true logarithm. All other logarithms (which are the inversesn of other exponential functions), are only understandable in terms of the natural logarithm.
So now you know about functions, exponential functions, Euler's e, inverse functions, and logarithms. Once you have the natural logarithm, you have a kind of secret key into math that allows you to define and understand all kinds of things that seemed impossible before, including complex numbers. Certain leaps of faith are often required during the study of such things, but as it turns out, the airplanes do fly, etc, etc. So it all appears to be kosher, if unorthodox (ha, ha).
In the world of numbers, zero is the first key. Zero was invented in Mesopotamia (Iraq) a long time ago. Without zero, you really couldn't do anything, because you wouldn't move to a different power. You know, the one's place, the ten's place, etc. You would have to have a separate name for each possible quantity of things. That would not work very well.
After zero, the next most important number is Euler's e. As Zero unlocks arithmetic, e unlocks calculus, which is sometimes called 'the arithmetic of infinitesimals'. Less prosaically, it is also called 'the last of the trivialities', since it is still only a tool of applied math, and nothing like the crazy stuff that 'real' mathematicians do.
After Zero and (e), Pi is a very distant third in importance, if it is even to be considered in the same kind of category. Probably best to think of them in three separate categories. Zero makes it possible for us to do any math at all. Pi shows us that there are some pretty cool relationships out there that can be expressed mathematically. (e) unlocks the arithmetic of infinitesimals to allow us to use math to model the physical universe.
Have you heard about the palimpsest of Archimedes?
And a very good morning to you too.
Torture demo planned for later this week to help derail the new attorney general nominee. Could be fun.
bring your own waterboard?
natch.
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