What is e (2.718…) and why does it literally appear everywhere?
In: 756
Most things in physics come from differential equations: nature doesnt directly tell you what something IS, it tells you how it changes. Mathematically this means the laws of physics are equations written in terms of the *rates of change* of the quantity you care about. You then solve these equations to get an expression for that quantity.
For example, newtons 2nd law of motion give you an equation that says the 2nd order rate of change of positoon (acceleration) is proportional to force. Solving that gives you an expression for position.
e is special because the function e^x has the property that it’s rate of change with respect to x is e^x . This is very relevant to differential equations because many equations are written such that the rate of change of your quantity is proportional to your quantity, therefore e^x is a solution.
An example of this is a mass on a spring: the 2nd order rate of change of displacement is proportional to displacement.
There are many others such as the heat equation, diffusion equation, shroedinger equation (ok these are all the same equation), radioactive decay, exponential growth, etc. that are written similarly, and therefore have some form of e^x in their solution.
In conclusion, e appears so much because it is the solution of many laws of physics due to its special rate of change property.
Latest Answers