The meaning of common functions.

Ok so i feel a little silly for never having learned this.

So i understand what lots of common functions do. For example, i understand that e^1 is somewhere around 2.7-something, sin90=0, natural log of 2 is about .7 and so on and so on.

I guess what I'm wanting to know is how exactly these functions are what they are. WHY does e=2.7? Where does that number come from? Same thing for log and natural log functions. What do the trig functions represent (note: I'm NOT looking for the answer "They represent an oscillating curve", or "It's the opposite over hypotenuse on a right triangle". I want to know more than that. Think about it like, when you enter "tan(67)" on a calculator, what exactly is it calculating?)

At least in Calc 1,2, and 3 there were explanations of what exactly everything meant and their theories. But i never understood these basic functions that I use almost every day.

Are you ready for limits? Just for starters, take this expression

Now, use your calculator and substitute for 'n'.

n = 1 gives 2

n = 2 gives 2.25

n = 4 gives 2.44140625

n = 8 gives 2.565784514

n = 16 gives 2.6379284974

Care to guess where this is headed?

That's an nice parlor trick, but it doesn't say where it comes from. There are many stories about such discoveries. Just trying to solve a problem and it turns up. You left of .

much of this behavior can be explained by the series:

, which as luck would have it, converges for all complex numbers z (and therefore, for all real numbers as well).

in particular, e is the number you get when z = 1.

all of the functions you mention, such as log, sin, cos, etc. all derive from this basic series, the complex exponential.

in the x-direction, it expresses the notion of "self-similarity in growth".

in the y-direction, it expresses the notion of "periodicity".

because of the rapid convergence of this series (factorials grow much faster than powers), it is often used to calculate values of functions like sin and cos to high degrees of precision, with a minimum of memory usage.

the calculation of such functions to many decimals places by hand, is a time-consuming process, and the people who tabulated these values for sailors in the 17-th century, are well deserving of respect. nowadays, computers are capable of performing such calculations rather trivially, the architecture of a math-processing chip doesn't have to be very sophisticated to be able to get 14 decimals places of any desired function within a fraction of a second.

one of the great advances of calculus, was its utility in getting results involving infinite sums, which allowed for tremendous advances in accuracy to be made in using actual values of functions (including n-th roots, logarithms, and inverse trigonometric functions, not to mention the ordinary trigonometric functions and power functions).

in fact, numerical integration allowed calculation of functions that we didn't even have names for, such as the standard probability distribution, or elliptic integrals.