What is e (2.718…) and why does it literally appear everywhere?
In: 756
There’s a lot of stuff in this thread but want to add a way of explaining what e^x is in calculus. In maths there is a concept called an identity element for different kind of functions.
For addition and subtraction, the identity element is 0. If I add or subtract 0 I get the same as I started. 5 + 0 = 5
For multiplication and division, the identity element is 1. 5 / 1 = 5
For a similar function, x^y, 1 is again an identity element for x and for y. 5^0.1 gets smaller, 5^2 gets bigger, but 5^1 stays the same. 1^y = 1 no matter what y is.
e^x is an identity element for differentiation and integration. Differentiation is a way of telling how fast something is changing. For example, If you have an equation for the speed of the car, differentiation gives us the acceleration. Integration goes the other way: if you had an equation for the rate of acceleration you could calculate the speed of the car. (Or to be precise, how many units it has increased by).
If your car is travelling at e^x m/s and x is measured in seconds, then it is also accelerating at e^x. The rate of change of acceleration is also e^x, and the rate of change of that rate of change is also e^x. You get the same as you started. The same is true of integration.
(fun bonus maths fact: logarithms were known and used for hundreds of years before e was discovered. It was at least 80 years after e was discovered that it is some kind of opposite for logs.)
(Even weirder fun bonus fact: Euler’s Identity gives us relations for e that appear to have nothing to do with growth. Why?)
(Yes, I forgot c. Just give me half marks.)
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