Euler's Formula

By Susam Pal on 04 Jun 2021

I know that Euler's identity is widely regarded as the most beautiful theorem in mathematics. In my opinion, the truly beautiful concept involved here is Euler's formula: $e^{ix} = \cos x + i \sin x.$ It unifies algebra, trigonometry, complex numbers, and calculus. Euler's identity is only a special case of Euler's formula, i.e., Euler's formula with $$x = \pi$$ gives us Euler's identity: $e^{i \pi} = -1.$ This is cute but Euler's formula is truly beautiful. In fact with $$x = \tau = 2\pi,$$ we get another cute result: $e^{i \tau} = 1.$ Quoting an excerpt from Chapter 22 of The Feynman Lectures on Physics, Volume I:

We summarize with this, the most remarkable formula in mathematics: $e^{i \theta} = \cos \theta + i \sin \theta.$ This is our jewel.

We may relate the geometry to the algebra by representing complex numbers in a plane; the horizontal position of a point is $$x,$$ the vertical position of a point is $$y.$$ We represent every complex number, $$x + iy.$$ Then if the radial distance to this point is called $$r$$ and the angle is called $$\theta,$$ the algebraic law is that $$x + iy$$ is written in the form $$r, e^{i \theta}$$ where the geometrical relationships between $$x$$ $$y,$$ $$r,$$ and $$\theta$$ are as shown. This, then, is the unification of algebra and geometry.

See the bottom of the page at https://www.feynmanlectures.caltech.edu/I_22.html for the above excerpt.