What is e (2.718…) and why does it literally appear everywhere?
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You probably know that π is found everywhere in math because of circles. *e* is similar to that but has to do with growth and speed of growth.
Suppose you are driving a car. Now suppose we graph the total distance it travels with relation to total time traveled, that is, we have a graph in which the total distance traveled is on the vertical (y) axis and the time its been traveling is the horizontal (x) axis.
Now, you may know that we can write an equation, in terms of time, to express the distance traveled with respect to time. Lets call this function f(t).
Now, this car you are driving travels at either a constant speed, or is accelerating/daeccelerating. Now, if we were to graph the speed of the car, a flat line would indicate a constant speed, a line with a positive slope would indicate an accelerating car, and a line with negative slope would indicate the car is slowing down. Lets call the function for the speed of the car f'(t)
Now, what if I were to tell you that the speed of the car is equivalent to its distance traveled? That is, when it travels x miles, it is also traveling at x speed?
This is called the exponential function. Its defined to be the function for which its speed of growth is equivalent to itself, namely f'(t) = f(t).
Now, in physics, and the rest of math, all things are measured as they change. The rate of change and speed of things and their relation to each other (in this case speed and displacement) are the fundamental backbone of physics, because we measure how natural phenomena change with regards to each other. Due to this, the exponential function is the absolute *backbone* of physics and other areas of math and science.
Now, how does e tie into this? Well, if exp(x) is the exponential function, exp(1) is e.
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