Euler’s identity

2 min read

It’s 2004, and two elderly Russian brothers have just finished explaining how they famously they generated the 1 billionth digit of pi (and then later the 2 billionth, and later the 4 billionth).  The critical path issue was a pragmatic one:  How to cool that many homemade 1980s-era parallel computers in a cramped Brooklyn apartment?

“What’s your favorite equation?” someone asks in Q&A, baiting them into recitation of a credal answer the geeky audience expects:  e^i(pi)+1=0.    

“Why?”  Mathematicians adore the elegant and mysterious union of disparate concepts:  exponential growth, imaginary numbers, irrational numbers, and the defining ratio of the circle.  It contains all the key arithmetic operations: addition, equality, multiplication, exponentiation.  And a prime number.  And the powerful notion of zero.  Next question.

Leonhard Euler (pronounced “oiler”) ranks among the most prolific and important of mathematicians, and this 18th-century equation is his most popular legacy.  Euler’s identity has long greeted visitors on my doormat:

doormat

But, beyond aesthetics, what does it mean?  Conceptually, how is exponential growth related to the ratio of diameter to circumference of a circle?   As with so many questions, a satisfying explanation had to wait until the days when Internet content exploded.

At Burning Man 2013, an early morning quest for serendipity traps my decrepit cruiser bike in thick desert dust… right in front of a banner proclaming e^i(pi)+1=0.  The math-themed camp plies me with hot, heavily-liquored espresso while musing about mathematical beauty.  It’s only when an applied math professor dressed as a robot happens upon us, that someone is able to address my pressing question as to what it means:  Something to do with signals, he recalls from grad school.  Think of a coiled spring.  It’s a circle…that grows sideways.

signal diagram

In 2015, I get a super-cool job in telecom, working with radio frequency engineers.  Sinusoidal functions describe radio waves.  In polar coordinates, they’re circles.  It all becomes clear and (sort-of) applicable.  (Well, RF people may but vaguely remember the equation from circuit theory classes, but the understanding is widespread that this equation underlies everything in communications.)

Thus, my faded, old doormat acquires a new layer of meaning.  My scientific mind understands the new job to be a random, yet “fitting”, event.  But, after years of priming from my surrounding new-age culture, I prefer to enjoy it as a “fateful” sign of satisfying structure in the not-so-random walk of life.  Fittingly, this is the same, ineffably satisfying feeling — of things mysteriously interlocking — that Euler’s equation provokes among those who revere it without needing to know what it means.

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