e to the Pie Eye

Not knowing enough about the math of complex numbers, I find the equation eπi = -1 intriguing. This is apparently the famous Euler’s Identity. I mean, how exactly do they get a real number out of an imaginary one without squaring? I had gotten some insight by playing with my calculator. I have a good one which is able to compute e^{\pi i} without choking on it. It helps that it understands complex math, like many new calculators do these days.

So, I tried something using an idea such as

*** QuickLaTeX cannot compile formula:
 \lim_{x \to e} x^{\pi i}

*** Error message:
Fatal Package fontspec Error: The fontspec package requires either XeTeX or
leading text: \msg_fatal:nn {fontspec} {cannot-use-pdftex}
Emergency stop.
leading text: \msg_fatal:nn {fontspec} {cannot-use-pdftex}

, where x was allowed to approach Euler’s constant. The way I “approached” e was a tad kludgy, but useful: I began with a “2”, computed the result, then “2.7”, adding the next successive decimal, computed that, and kept the process going for several more decimal places. My results could be tabulated as:

x Result
2 -.5702332490 + 0.8214828311 i
2.7 -.9997752846 + 0.02119859190 i
2.71 -.9999540533 + 0.009585998981 i
2.718 -.9999999469 + 0.0003257333283 i
2.7182 -.9999999955 + 0.00009457240125 i
2.71828 -1.000000000 + 0.000002112790593 i

So, as you can probably suss out, as we add successive digits to the base the base approaches e, the real part of the number becomes -1, while the imaginary part of the number slowly melts away to nothing. By the time we get to 2.718281828 (the value of e seen on most calculator displays), we get the number: -1.000000000 + 0.0000000001202932385 i. Like many things we associate with the understanding of Euler’s constant, it seems the limit (in my case, in the form of playing with my calculator) is one approach to understanding Euler’s identity. There is another, better approach.

Of couse, the next lesson I feel the need to proceed to, is, what the heck is all this about e^{\theta i} = \cos \theta + i \sin \theta?