Beauty is in the i (and the e, and the pi) of the beholder.
Just a couple of days left to preorder a signed copy of my new book.
I’m signing them in just 48 hours!
Also, Twin Cities friends, I’m doing launch events at Magers & Quinn (Mon. 9/9 at 7pm) and The Thinking Spot (Sat 9/14 at 2pm). It’d be fabulous to see you there!
I recently posted this cartoon on Facebook, with bad handwriting to match my signature art style:
There’s a reason people love this equation. Euler’s identity is a kind of mathematical “to be or not to be,” combining several of the language’s simplest words into one of its most examined and celebrated phrases. It has the concision and symmetry of an aphorism from G.K. Chesterton (“There are no uninteresting things, only uninterested people”), except if G.K. Chesterton had known complex analysis.
On Facebook, my pal Duane Skaggs replied with a story, vindicating the idea that pure math can have surprising applications in everyday life:
I was going in for an eye surgery once, and the surgeon knew I was worried. He has an undergraduate degree in mathematical economics, and we had discussed maths topics before.
He comes in the prep area, confidently writes something on a napkin, hands it to me, and walks out.
This equation completely calmed me and the surgery went well.
But on Twitter/X, my pal Andrew Stacey replied with a compelling argument to the contrary.
(Please do not take this as an allegory about the differences between social media platforms, even though it totally is.)
As Andrew points out, once you understand the symbolism in the equation (what does a complex power mean?), the new content in the statement itself isn’t much to write home about:
Thus, I submit, a more honest description of eiπ=−1 would be:
“This equation relates the fact that if you go halfway around a circle then you end up at a point diametrically opposite where you started.”
That’s a little less likely to end up on anyone’s top ten favourite equations.
Who to believe? Duane’s thumping heart or Andrew’s analytical eye?
I am a middle child, not only in birth order, but in spirit. I’m always looking to reconcile seemingly opposed camps. Now, there’s no need for Duane and Andrew to have the same aesthetic reaction to this equation, any more than they need give Greta Gerwig’s Little Women the same score on a ten-point scale (the correct score, by the way, is 8.9).
But I think there’s a way to square their impressions. Andrew is regarding the equation as a self-contained statement; Duane, as a summary of a longer chain of thought.
By analogy, consider “to be or not to be.” What makes it a great line? Certainly not the content (which is literally just “should I keep living or go off and die?”).
But it’s not just the form, either. It’s the context.
Those six words begin the most memorable monologue in the finest play by the greatest writer in the history of the literature. When we quote “to be or not to be,” we invoke that context, in all its rich meanings.
So too with Euler’s identity. When we quote it, we’re not saying: “Given your favorite definition of a complex exponential, here’s a surprising new statement.” We’re invoking the context.
You can exponentiate an imaginary number. In fact, that number is just an angle of rotation in the complex plane. There is a fundamental unity between the algebra of exponentiation and the geometry of rotation, and thus, a link between the irrational constant at the heart of growth and the irrational constant at the heart of circles.
Why do we quote “to be or not to be”? Not because it’s the most surprising or information-dense line in the play. Not because we’re trying to separate it from the rest of Hamlet. We quote it because it gives us so much of the essence of Hamlet, distilled to just a few syllables.
And that’s why we quote Euler’s identity. Not because it is a novel statement within the study of complex exponentiation, but because it conjures in the mind the whole beauty of complex exponentiation.
Because it’s a terse combination of symbols, and that’s the only kind of profundity we know.
Anyway, I have a newly vested interest in the matter. Even though I’ve often thought the same as Andrew (in particular, why not cite the general form e^ix = cos(x) + isin(x)?), I nevertheless made the equation the final note in my new book, Math for English Majors.
It’s not about the formula itself. It’s about the context.
