How does "e" (2.718) help apply to applications/implications in real life?
As for growth problems, imagine you went to a bank where you have 1 dollar, pound, or whatever type of money you have. The bank offers you
This is better, actually! Let's take it further.
Now, they give you
If we repeat the process, at the end of the year you will have
We can see a pattern! If we take a general case, say you get
So, we saw that it was advantageous to get a smaller interest over shorter intervals of time. Let's confirm this; let
Yes, it does increase, but it seems to be slowing down, converging to a value even. But what is this value?
Well, let's say your bank does the impossible and offers you an interest with
This is one of the definitions of
But this is not exactly practical, because real life banks don't work this way. However, it does offer us a pretty good image of how
I will continue this in another answer.
Another application of
Suppose you have a population with
So the number of people after 180 years, which we will denote as
Now, we wish to find the instantenous rate of growth of the population. If we find it, it will be helpful to maybe compare it to former rates and form a pretty good impression of what the future holds. This is where
The population after
Now, the instanteneous rate of change represents how much the population will have grown in an infinitesimal amount of time.
Basically, we ask what will
If we denote the infinitesimal interval of time to be
In mathematics, we usually just write
Appearances of e in Physics
In statistical mechanics, the Boltzmann distribution is a probability measure that gives the probability that a system will be in a certain state in terms of that state’s energy and the temperature of the system.
This is all pretty complicated stuff, especially for a precalculus student. To simplify, let's say the system can only have 2 different states. Then, the probability that it will be in one of the two states,
You can often find
However, for the example I'm going to give, we're going to talk about sticks, just to show how far from standard Math
Let's say we have a stick of lenght
The answer is, quite surprinsingly,
Unusually, this answer streak does have a conclusion. I wish to say that the fact that
This just makes you love Mathematics even more, doesn't it?