Buzz Mauro

Twenty-seven Uses for Imaginary Numbers

I was just a kid and I suppose that means there’s a lot I can’t really be blamed for, but can’t take any credit for either, which is an unfortunate consequence but the way it goes. And Artie Hillsweiger, being also only a kid at the time, therefore probably doesn’t deserve any of the not insignificant amount of credit or blame I might be inclined to throw his way either, but the temptation is great in me at times to lay all of it on him. On scrawny Artie rolling around on my bed in an ecstasy over Euler’s Formula or some stupid thing I had happened to say about Fourier series or some girl at school, in this half-gaga half-terrorizing way he had of zeroing in on me when I least expected it, or especially when I most expected it and was most dying for it secretly deep down.
            The Euler’s Formula thing went on the longest and should have gotten old a lot faster than it did, for both of us, but it was the kind of thing that sent us over the edge back then, when it was still possible for numbers to explode open like that into a whole new picture of the complexity we’re embedded in, when your mind can still feel blown by the perfection of a system, because you’re just sixteen and you haven’t yet seen Gödel’s proof that no system can be perfect and, besides, it’s not even so much the system but the numbers themselves and how they interlock that you’re in awe of. Because you haven’t yet graduated on to Bertrand Russell where you start to understand that numbers really have nothing to do with it because it’s all about the fundamental rules that govern objects and processes of any ilk whatsoever. That’s when you become a mathematician.
            But I was sixteen and unusually interested in such things and there it was, Euler’s Formula, in the manual that came with my new scientific calculator, which was a birthday present from my parents. Euler’s Formula had no business being in that manual, since, as it is, it’s about imaginary numbers—or, more correctly, complex numbers, as the set of all sums of the pure imaginaries and the reals is gorgeously called—and calculators being pretty much unable to handle imaginary numbers in those days. But there it was, probably because whoever wrote that manual simply loved and feared it too much to leave it out. I was standing at my desk in my room, one of those lanky kind of kids with too much energy to ever be sitting down for very long so I was as likely to be standing or pacing as sitting down or lying in bed on a Friday after school. I was standing with the manual, bolted to the floor by an idea from the eighteenth century, no genius myself but proficient enough to see implications of mathematical realms that no one had even hinted at to me before, and tied up in those smoky images of imaginary exponentiation were images of flying off into those realms with Artie at my side.
            When I showed it to him on Saturday morning he punched me in the arm.
            “No way,” he said. “Prove it.”
            He did that kind of thing a lot, bravado and stuff, always trying to find something to hold over me or prove me wrong about something because it was a kind of rivalry thing between us a lot of the time, but I have to say it was always pretty clearly affectionate underneath.
            But I had the upper hand here because he couldn’t come over until eleven so I’d made it to the library and back before he arrived and they just happened to have a book that gave a proof using power series. It was unlike any proof I’d ever seen in my life but wasn’t too hard and had convinced me beyond a doubt. And it convinced Artie too, standing in my room at my side, craning his head into it at 11:05, because I had checked the book out, knowing that to show the formula to Artie without a proof might be fun in a tantalizing kind of way but that the real fun couldn’t start until he had reason to believe that what this stupid calculator manual had to say was actual universal truth.
            “I see it, but I don’t believe it,” he said, and I felt a little tingle in my dick, which was hard, because he was quoting Georg Cantor when he proved there are as many points on one side of a square as there are in the entire interior, which meant Artie was understanding this as a major revelation of an almost once-in-a-lifetime sort. But the tingle in my dick was more because of how I knew I was the only person Artie knew who could have known that, so it was like he was speaking to me in a secret shared language that bound us to each other, mind to mind, and kind of dick to dick.
            We knew about Cantor because that was infinity and irresistible, but we were still mathematical babies in most ways. We were no strangers to trigonometry of a pretty abstract kind and knew our limits and could take a derivative, as long as the function was everywhere continuous. Complex numbers were our special friends already, and we spent blissful afternoons raising them to powers and taking their roots. But blew our freaking minds. What the hell could an imaginary exponent even mean?
            “That’s what it means,” Artie said.
            “What is?”
            “Exactly what it says.”
            And of course he was right. Sometimes he could really surprise you.

And it only gets worse if you let theta equal pi, which is what you would naturally do first. You get , of course, but two seconds later this becomes and you and your best friend find yourselves laughing like loons and falling down on the floor all over each other because what you have there are the five most important numbers in the whole world related by some voodoo in this one tiny simple equation like the world adds up. And we saw it, and we believed it. We were pretty freaking proud of ourselves.
            And we were rolling around together on the floor.
            I try to think back to what really mattered to me back then, and the first response of course is that nothing really mattered because my food and shelter, et cetera, were paid for and I was pretty much loved and protected in all the typical ways, so I wasn’t out scrounging for survival. I wasn’t wrestling Artie for the right to decide which direction the clan would migrate that winter, or to steal his venison or whatever. But inside all that comfort of my life other things had room to matter and to mean things. It was March and light from a white sun filled the room and we rolled around through that. He was very clean and smelled like laundry, but moist and activated, like the smells were right now combining chemically with our bodies somewhere in the humidity between and around us, and all of that seemed to matter a whole lot.
            We were squished between the bed and the desk at first but then Artie twisted around and got me kind of compromised and gave me a noogie and broke away to the other side of the bed. But I was a pretty quick kid in my lankiness and didn’t have much trouble tackling him back to the floor and leaning down hard on his chest, so our faces were real close and I could see actual stubble on his chin from where he shaved, and there was a mean little red patch under his eye by his nose and I knew he would have a pimple there by the end of the day.
            I struggled all my weight down on him through my upper arm and sometimes my elbow, and he was so scrawny I worried a little bit that I might crack him. But he was struggling his chest back up against my arm and we squirmed there in an almost equilibrium. We were laughing and a little bit of his spit landed on my cheek. Then his head was straining away from me and I could see all the ropes of his neck muscles. I rolled in and pressed my head into his neck and felt the hard cords on my forehead.
We must have started to sweat, because I noticed the smells intensifying. Artie’s neck, though, with my face smashed into it like that, wasn’t wet but hard and taut and also smooth and soft and I made sure I kept my crotch out of the whole thing, scissoring my legs out up against the dresser.
            But then Artie twisted again, too fast for me to tell how, and he was on top, digging his elbows down hard into me in the soft sensitive flesh to the inside of the shoulders and I screamed and I was glad my parents were out at garage sales. The scream was loud and should have been enough to make Artie back off, but he didn’t at all. He was on his knees over me, leaning all his weight down into those two points of pain, and I didn’t have the breath to scream again. I started to tense my legs to kick him off, but then I didn’t. The pain spread out in a rush and flowed away into a numbness that left me able to breathe just the tiniest bit and staring up at Artie’s face with my mouth stunned open.
            His whole face was as red as the pimple spot now and his eyes were wide wide open. He was breathing really hard and his breath was hot and had no smell. The eyes were intensely green and I couldn’t remember if I’d ever noticed that before. I’m sure I must have, but if you had asked me to tell you what color Artie’s eyes were I probably wouldn’t have been able to. Which seemed impossible to believe with those green green eyes so close now to my boring brown ones.
            And they were mad, too. I’d never seen Artie look that mad, like he was trying to burn me up with those eyes, like he might nail me right to the floor with those bony elbows, like he had lost his mind. And I felt no power to reject his anger or to understand it or to push him off me or to maintain myself as much of a separate consciousness from Artie Hillsweiger any longer.
            And that was when he dropped down on top of me, replacing the elbows in one swift move with the much less painful butts of his hands and stretching his whole body out on top of me, still looking into me hard and breathing hard and still looking angry with a certain scary confusion mixed in, and I felt his hard cock next to mine through our pants. We held there a moment and kind of breathed into each other’s mouths, then he did a pushup on top of me and said, “Hey, what if theta equals i ?” and rolled off to get a pencil.

Turns out that if you let theta equal i you’re really off and running. Because now you’re looking at trig functions of imaginary numbers even though triangles can’t have imaginary angle measures, and when you’re sixteen that’s somehow weirder than even an imaginary exponent, and it begs you to find values for the sine and cosine of i, specifically.
            “Shouldn’t the Pythagorean identity still hold?” I asked Artie.
            “I was just thinking the same thing,” he said.
            Artie did the substitution in his notebook and I did it in mine and we worked on the system of equations in parallel, the same two equations in the same two unknowns, and it was clear that a race was on. Whether this race would end in new parameters for our friendship was the real question and I saw that Artie felt it too, and that was why we scribbled so intensely there side by side on the bed, cross-legged and hell-bent. The bed moved underneath us a little every time one of us started a new line.
            But I couldn’t resist and I looked up briefly and saw pain and anger in his eyes still, staring down at his solution, and I wondered if he felt what I felt as often as I thought he did, or if maybe I was making our whole friendship up inside my head, and how pathetic that would make me. He didn’t talk much, so it was possible. Perhaps there was no race. Perhaps I should have abandoned childish thoughts like that by now, and perhaps Artie had. Perhaps Artie no longer thought in pathetic terms like races and rivalries and friendships and there were other, better things going on in him that I hadn’t even dreamed of yet. But then that thought, too, spun around in my head, because it meant I was still trying to know what Artie was thinking, and all I really seemed to care about was whether it was the same as what I was thinking or not.
            “The cosine is real!” Artie yelled.
            His eyes didn’t come off his paper and his pencil didn’t stop moving and I realized he was still breathing heavily, not as much as when our dicks were pressed together, but heavier than usual. I had noticed that he had been washing his hair every day and it was shiny now and most of it was hanging down on his neck and shoulders, but one long lock of it hung straight down his face and I wished he’d brush it away. I wanted to do it for him, but not gently. I was starting to get a little embarrassed, because my own hair was exactly the same length as his, because when he grew his long I grew mine long, too. And for the last couple of weeks I had been washing my hair every day, which my parents thought was weird and a waste of shampoo, and I was afraid now that maybe other people had been noticing that I was doing things because Artie was doing them and that made me feel like I’d lost myself somewhere along the way and had been subsumed too much by Artie and his likes and his ways, which maybe weren’t my own likes and ways at all and were never meant to be. But one thing I knew was that the math was mine first—I was sure of that—and my paper was showing me that the cosine was going to have to be real, as in not imaginary, which was freakish, and Artie discovering it before me, even a few seconds before me, made me think for a second that I might start crying.
            Artie stopped writing, and then I stopped, and there was a tension during which he wouldn’t look at me. We were sitting there with actual trig functions of imaginary numbers on our laps, and one of them had a real value for some ungodly reason, and I wanted him to talk to me about whatever he was freaking feeling about the whole situation we found ourselves in. He was wearing brown corduroys and I let my eyes drop for a second to see if his dick was still hard, but the way his notebook was positioned I couldn’t tell.
            At last he looked up at me and when he did the anger was gone from his face, replaced by something much more troubling that looked a little bit like fear. But I held the gaze of his green eyes and didn’t say anything. That long lock of hair was still hanging there and now I didn’t want to brush it away anymore, because I liked how it didn’t bother him. But there we were looking at each other, which we had never done before for this extreme length of time, and no matter how much I couldn’t tell what was really going on inside him, I knew we were both afraid of something. And I knew it was the same thing for both of us, even though I didn’t know what it was.
            You don’t get to this point of obsession with mathematical scenarios without realizing how different you are from other sixteen-year-old boys. I do remember that at sixteen pretty much every sixteen-year-old was very clearly different from every other one, but Artie and I just seemed an awful lot the same, and our way of being different was more different than most. My dream was to write a book, the first mathematical bestseller. I thought I saw in Artie’s eyes that he would write it with me. We’d spend our afternoons solving and simplifying, specializing now in complex functions no one else had yet encountered, finding practical applications, opening up worlds for the professionals and lay people alike, revealing mysteries the general public didn’t know were in store for them. It all seemed absolutely irrevocably new, like no one had ever thought or felt any of this before, and they were in for some eye-opening when the book came out.
            I didn’t see any other way to do it, so I grabbed Artie’s notebook and threw it on the floor, and my own too, then pushed him back on the bed. He popped right back up, but I sprang at him and pinned him back down and his hair splayed out on my pillow and his eyes kept their focus on me. He resisted, but less this time, only pretend. It was easier, and slower, and very scary because there was more time to think, but still we were full-out against each other again, only me on top now, and it hadn’t occurred to me before but I pressed my mouth down onto his and we both moaned and sank more into each other then, and our pants were still on and I knew, whether Artie did or not, that we were about to make an awful mess, but the time for turning back was past.