This is not a standard simplifience article. It provides a brief view of the nature of $\pi$ as is necessary to understand Euler’s identity. Both $\pi$ and Euler’s identity are contrives, which are normally avoided.

You’ve probably been taught that $\pi ≈ 3.14$. You might know it has something to do with circles. More specifically, $\pi$ is the ratio of a circle’s circumference to it’s diameter. If you walk $\pi$ diameters around a circle you’ll come all the way around.

As it turns out, circles with radius $1$ are far more useful than circles with diameter $1$. Both reality and mathematics (Euler’s identity included) are littered with circles where the radius is the natural unit. In radius-based circles $\pi$ isn’t an important number. What we actually care about is the number of radius lengths in the circumference of a circle.

$\pi$ is one of the most popular contrives in mainstream mathematics. $\pi$ is the number of radius-lengths in a half-turn, not a full turn. This is a silly unit to pick as your measuring stick. A quarter turn (right angle) and a full turn ($\tau$) are the natural units when working with circles.

Ignore all that for now. In order to understand Euler’s identity, all you need to remember is this: