While most numbers lead anonymous lives away from the mathematical spotlight, e^{iπ } occupies hallowed ground. Douglas Hofstadter writes that when he first saw the statement e^{iπ} = −1, “. . . perhaps at age 12 or so, it seemed truly magical, almost other-worldly.”

At the risk of deflating the celebrity status of e^{iπ}, what follows is an intuitive, dynamic geometry interpretation described in *The Book of Numbers *by John Conway and Richard Guy.

Written as a limit, e^{iπ} is equal to the following:

Let’s start small and evaluate (1 + *i*π/*n*)^{n} for *n* = 8. And rather than use algebra, we can instead approach the computation geometrically: given two complex numbers, *z* and *w,* the argument of *zw* is the sum of the arguments of *z* and *w, *and length of *zw* is the product of the lengths of *z* and *w*. (For more information and a proof, see this prior blog post.)

In the sketch below, point *B* represents (1 + *i*π/8) in the complex plane. By measuring ∠*AOB* (its argument) and *OB *(its length)*,* we can construct (1 + *i*π/8)^{2}, (1 + *i*π/8)^{3}, … all the way up to (1 + *i*π/8)^{8} by using desktop Sketchpad’s iteration feature. Point *P*, whose location is reported as –1.7554 + 0.262*i*, represents (1 + *i*π/8)^{8}.

Now drag slider point *n* to the right. When *n* = 130, we see that point *P, *which represents (1 + *i*π/130)^{130}, is equal to –1.0387 + 0.001*i*. Not bad for a rough approximation of e^{iπ}!

Note that for large *n, *∆*BOA* closely approximates a circle sector with arc length π/*n*. As there are *n* stacked triangles in the final product, it makes sense that point *P* should nearly lie on the negative real axis. The length of *OB* is (1 + π^{2}/*n*^{2})^{0.5}, and so *OP* is (1 + π^{2}/*n*^{2})^{0.5n}. Graphing this expression, entering large *n* values on a calculator, or using a little calculus suggest *OP* approaches 1 as *n* grows infinite. Thus, combining the angle and length observations leads to the conclusion that point *P,* our e^{iπ} approximation, nears –1 for large *n.*