What is e (2.718…) and why does it literally appear everywhere?
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A lot of these answers are missing pieces. Either answering what it is or some places it does up but not answering why. So here is my crack at it.
The number e is special because it is a constant that shows up in a lot of places, and it’s important for math in general much like the more familiar pi or lesser known but equally fun and more important i (square root of negative 1). What is it? It is the number such that the derivative of the function e^x is equal to itself. That seems interesting but not that important. So why do we care? Well the obvious application is in growth. The definition is an exponential function and exponential functions deal with growth and anything modeling growth deals with expectation functions… But that is only a tiny part of the power of e.
A lot of things in math are about finding relationships between other things. And then generalizing those relationships to find the underlying patterns. When you do this you find some freaky and eerie coincidences. These are not coincidences. They are a result of how math works. The numbers that we deal with are full of patterns that exist. The number e deals with a special relationship related to change in general. This is why it shows up when we are looking at growth functions. It also shows up in oscillation functions, because those involve change. So functions involving sine can be expressed as a combination of e to a real and e to an imaginary component. e^(πi)=-1 because that is a special case of an expression of rotation where the real part is 0 and the imaginary part is -1. This has to do with the 2D number plane as opposed to the more familiar number line or more realistic number space that we live in (kinda) but it does illustrate why this stuff shows up. We live in a mathematical world where we can describe things with these functions. As a result, the special numbers we build these functions from, the constants, show up in the universe because they build the universe.
So the tldr, e is defined as the number such a that it’s the derivative of itself in an exponential function and it shows up everywhere because it describes how change happens in math and math is all about change so it shows up everywhere.
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