The square root of negative one is a nasty bit of business.
The square root of 25 gets invited to parties, has followers on Instagram. Even the square root of 2 is asked to dance occasionally. But the square root of negative one? The square root of negative one is hard to wrap your head around.
It seems like the square root of negative one could stay outside in the rain and no one would know or care. Can the square root of negative one even be said to exist? Even mathematicians had no use for the square root of negative one for centuries. The square root of negative one can be likened to the Ghost of Christmas Past: Scrooge doesn't actually believe in him and is therefore especially displeased when he shows up.
The square root of 25 is five. Fine. Nothing to see here. Move along. Five times five is 25. That’s it. I get it. Rejoicing throughout the land. The square root of two is something, something bigger than one. I don’t know exactly what, but I know the square root of two is there somewhere. Because 1.4 times 1.4 is 1.96; Not so bad. And 1.414 times 1.414 is 1.999396. Close enough to two. And 1.4142135 times 1.4142135 is 1.999999823. Let’s say 1.999999823 is the same as two and go get a drink. Who cares if 1.999999823 is off by a piddle from two? This is a column about parenting. Nobody is building a bridge across a river or cramming for a trig exam. Maybe the square root of two is a decimal that goes on forever. (It is.) You thought holiday dinners at your in-laws were interminable? Turns out your spouse’s parents have nothing on the truly infinite decimal that is the square root of two.
But the square root of negative one? What number multiplied by itself gives you negative one? Positive one times positive one is positive one. Duh. But negative one times negative one is also positive one. No help there either. So, we designate the square root of negative one by the letter “i.” And we just say that i^2 = -1; i (pronounced “eye”) squared is negative one.
Before a couple hundred years ago, nobody knew about the square root of negative one. And why would they? The square root of negative one isn’t a distance. The square root of negative one could not meet any girls in a women’s prison holding a fistful of pardons. The Parthenon was constructed without the square root of negative one. So were Hamlet, Eine kleine Nachtmusik, Sojourner Truth’s “Ain’t I a Woman?” speech, and the Magna Carta.
But was the square root of negative one there all along?
Turns out the square root of negative one is ubiquitous. The square root of negative one is necessary for Erwin Schrodinger’s wave equation. Wave mechanics don’t work, can’t be explained, won’t fill up a black board without the square root of negative one. Freeman Dyson (astrophysicist, 1923-2020) pointed out that “it was one of the most profound jokes of nature” that the square root of negative one that mathematicians never trusted or believed in, is necessary for matter to exist. Mathematicians thought they had invented or discovered the square root of negative one. “Nature, Dyson famously said, got there first.”*
And what about your relationship with your children? Who are they? Do they exist independent of your perception of them? Or were they—like the square root of negative one—there all along? Are your children who you think they are? Is your relationship with your kids what you think it is?
If there isn’t something real about the square root of negative one—indeed the square root of negative one is termed an “imaginary number”—it sure does show up everywhere. You can't explain the square root of negative one; you can't internalize the square root of negative one. But the square root of negative one is fundamental to the reality of everything in the world around you. Could your relationship with your children be similar? Maybe you can’t define, understand, or articulate your connection to your kids, but it’s there just the same. Could it be time to stop worrying, defining, thinking and just accept your kids for who they are? Because without our beloved children—like the square root of negative one—the entire universe would not exist.
You don't have to understand the square root of negative one for the atoms in your body to remain bonded to one another. You don't have to articulate your connection with your kids for it to exist. Sometimes the most powerful forces in the cosmos are the most difficult to put into words.
* Owen O’Shea, The Call of the Primes: Surprising Patterns, Peculiar Puzzles, and Other Marvels of Mathematics, Prometheus Books, 2016, page 141.