Uploaded March 2010 | Updated September 2026, 3 weeks ago
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 = pi to get e^i*pi = -1).
In this video I descibe 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.
See another video on how I choose to derive the number e in a calculus course and yet another video as to why this is the same number e that arises in the theory of compound interest. (Full details, of course, appear in Volume 5 of the THINKING MATHEMATICS book.)
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 = pi to get e^i*pi = -1).
In this video I descibe 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.
See another video on how I choose to derive the number e in a calculus course and yet another video as to why this is the same number e that arises in the theory of compound interest. (Full details, of course, appear in Volume 5 of the THINKING MATHEMATICS book.)
