Janet dragged her eyes up from People magazine and a serving of indigestible Healthy Choice turkey and dressing. It was 40 past noon in Room 217 of General Lafayette High School, where Janet allowed her students to work on their homework during lunch. The only person in the room besides herself today was Kim Cheever, a college-prep junior who had fallen behind due to basketball practice. “Yes, Kim?” Janet mumbled. She realized she was speaking with her mouth full of stuffing and gulped down a flavorless lump. “How goes it?”
“I’ve been trying to figure out these quadratic equations? The ones from section 15.8?” Kim explained, her soprano rising at the end of each sentence. “The ones where you have to, like, figure out where they, like, cross the x-axis?”
“Fifteen-eight,” Janet repeated, her mind still on the article she was skimming in People. A voice actress from the Lil’ Warriors line of Christian animated videos had been arrested for possession of narcotics at a La Quinta in Amarillo, Texas, it reported, and Janet was feeling the campfire warmth of schadenfreude. “Is that the section on rational functions? With, I want to say, fourth-degree polynomials?”
“Ummmm, like, the W shapes?”
“Yes, their graphs cross the x-axis at up to four points. What problem are you having?”
“Well, like, when I graph this weird function, the one in 42, it, like, loops back on itself? Well, like, not like a circle, but…it’s freaky. Am I just stupid, or—?”
“Let me look,” Janet interrupted, not overtly dismissing Kim’s theory. She popped the last bite of turkey in her mouth and dumped its microwaveable dish in the trash. Then she slid a pencil into her magazine as a bookmark, hoping to savor the public downfall of a cartoon squirrel after fifth-period geometry. Kim dragged her scuffed desk next to Janet’s larger one, the metal legs squawking obnoxiously over the hardwood floor. Janet cringed, readjusted her bifocals, and scanned Kim’s cluttered homework for exercise 42. “I see the graph. This is rather a mess, Kim. Where’s your work on this equation?”
“Oh, like, it’s here, on this page?”
“You really must take the time to work neatly,” Janet admonished, then muttered as she worked through the problem. “Okay, so let’s see now. You multiplied by the least common denominator, which is 30. You factored the polynomial by grouping, I can see that. Good. Exponents in descending order, right. That factors as…Good, Kim, you noticed the monomial factor and difference of cubes. Well done. This is a tough one, though, isn’t it? Did I assign this?”
“I guess for extra credit?” Kim replied.
“Well, that was cruel of me. I must’ve been angry at the world that day. Then the quadratic, good, which you’ve factored as…right…then you…Oh!” Janet exclaimed happily. “I see what the problem is! You got five as one of your solutions, right?”
“I guess so?” Kim leaned closer. “Five and negative five.”
“Exactly. But look at the original problem here. If you substitute five for x in this fraction, you get a denominator of zero. Remember, you can never divide by zero. Das ist verboten, meine Freundin.”
“Uh, riiiight,” Kim drawled, blank-eyed. Then she shook off her confusion. “Okay, except there’s not, like, a, what do you call it, a break in the function?”
“I don’t believe that’s the case,” Janet said quietly, smiling, and pushed Kim’s notes back in her direction. “You might want to look that one over again.”
“No, like, that’s my problem,” Kim argued, pushing them back. “I’ve been looking at it for, like, an hour or something. It’s making me crazy. I mean, look at the example. Like, the function there has empty circles at places like that.”
“Okay, disconta…whatever. Except those are places where the function, like, wants to jump to another place, right?”
“You got it.”
“Except this one doesn’t do that. I mean, I ran it on Google and everything, so like…It, like, touches the axis from both sides? Where they actually touch? So I don’t know why there’d be a disconta-whatever there. I don’t get this.”
“This one has to do with limits,” Janet demurred. “We haven’t gotten to those yet. Let me look at it again. We can talk about this tomorrow. I’m sure other students will have the same question.” Kim shrugged and took her notes back, then awkwardly hauled her desk back to its place.
Janet jotted the problem number in a notebook as her lunch period ended. That night, she returned to the problem while her husband and son watched a Transformers sequel on cable. Exercise 15.8.42 was complex at first, then deceptively simple, then confounding. It seemed to be a logical Möbius strip. She believed she knew exactly which solutions the textbook author intended, but she couldn’t wrap her head around its point of discontinuity at x equals five. She beat her head against it for an hour, then realized the only reason she couldn’t trust her answer was the mathematical prohibition against dividing any quantity by zero. It was viewed as illegal, undefined; but if she allowed it just this once, the graph of the function resolved into beautiful splendor, even tucking back into itself in a way that seemed almost organic.
She struggled to recall salient lessons from her decades-past degree in mathematics. Why couldn’t she divide by zero, again? She remembered it caused issues when she reversed the division into multiplication—except it didn’t in this case, because the ratio threw things off and her multiplication worked out fine. The longer she wrestled with this equation, the more she became convinced that she’d found a special exception to a time-hardened rule. To the best of her understanding, in this particular case, it was perfectly okay to divide by zero. In fact, if she commuted the terms a certain way, she could even make a case for dividing zero by itself. That was impossible. And yet, there it was.
She leaned back into her chair, dazed but bracingly excited. This was something. This was something! She’d discovered a mathematical singularity. This bizarre textbook exercise, apparently beyond the ken of its own author, a math professor at Columbia University for Pete’s sake, was a point of departure from everything human science believed to be true. She laughed at herself joyfully, ran the math again. Only then did she accept her results. From her desk in a corner of the dining nook, she considered how to share this discovery with her family. They had no idea. She’d found something wonderful. If she tried to explain it to her husband Bobby, God knew, he could never comprehend it in six lifetimes. He introduced her to coworkers as the brains in the family, and he chuckled, but frankly, he was right. As for her teenage son, even with expert help, Jason was all but incapable of adding two fractions together.
As she thought about this, she began to resent it. How had her life come to this? She had discovered a mathematical marvel! The very universe had bloomed like an orchid, revealing exotic depths at its heart! Why was she stuck here enjoying this by herself? Where was the justice or logic in that? How could her husband, a man who used to read actual books, have degenerated into an oaf who got excited about robots who fought with karate? How many years had it been since her intellect was respected and fulfilled? Instead, here she was, Mrs. Janet Allen Luntz, high school math teacher from Nowhere, USA, obliged to spend her Saturday afternoons parked on the couch, rooting for football teams she cared nothing about. She was weeks from her 45th birthday but had never set foot in another country. How she dreamed of touring Europe! She longed to visit Paris and weep over the grave of Blaise Pascal. She dreamt of attending an Elizabethan drama in London with a family who wouldn’t spend the tube ride home mocking the costumes. Life was meant to be savored, explored and conquered, not endured! Janet Luntz had seen the light, and there was no going back! She was meant for bigger things, higher truths, vast discoveries! What she had seen could not be unseen! Salvation was hers! She had been born anew!
A month later, Janet Luntz quit her job, packed her car, and drove west to spend a week at the Oregon Country Fair. There she tried psychedelic mushrooms for the first time, shed her clothes and danced with strangers beneath the Walking Tree of Life. She hitchhiked across Europe while penning an 800-page novel about the scourge of materialism. She left a trail of broken hearts and champagne. Inspired by her quest for self-fulfillment, Kim Cheever took up with a slim sociology professor from Manhattan and finagled her way into Princeton, then established herself as the world’s foremost travel writer for lesbian readers. Jason Luntz got his act together and finished his high-school football career with 163 rushing yards per game, enough for a scholarship to the state college down the road. As for Bobby, he was never the same. He remarried two years later, a real estate broker from Branson, Missouri, but he trudged to his grave with the sagging expression of one whose dinner arrived cold from the kitchen.
Janet never worked exercise 42 again, though she expounded on her discovery that night in a bedside diary. So it wasn’t till decades later, when her memoirs were posthumously published after she crashed a helicopter near Lake Geneva at age 78, that her legions of admirers read the equation that launched Janet Luntz on her course toward the stars. And even now, as her radiant smile graces the five-euro note and a marble statue of her looks out over the Seine from Pont Neuf, her biographers find it unseemly to admit she forgot to distribute a factor of negative one.