Number e Explained

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What is the number e?
There are essentially two ways to construct the number e: one through the study of compound interest and one through the study of basic derivatives in calclulus. (And it is not at all obvious these two numbers that result are the same number e!!) In this video I describe the calculus approach and briefly show why "e" is the most natural and eeeeasy number to use in calculus. (That it connects to compound interest is for another video.) The number e and Compound Interest
The number e arises as the number that makes calculus "easy," but it is often presented to students first in a pre-calculus class as a number arising from the study of compound interest. Very few folk seem to ask whether or not these two e's really are the same. They are! This video - which assumes familiarity with calculus - proves this. e^ix: Deriving Euler's Formula
Perhaps the most surprising and beautiful result in all of mathematics, Euler's formula, e^ix = cos(x) + i sin(x), turns the theory of trigonometry into a simple study of exponents. (And put in x = πi to get e^πi = -1).
In this video I describe a way to get to this result at the beginning of a calculus course (once one knows some basic derivatives) rather then leaving it to the end of a course using Taylor series.