IMAGINARY NUMBERS
Copyright 2011 by P.K. Gardner
Smashwords Edition, License Notes
This ebook is licensed for your personal enjoyment only. This ebook may not be re-sold or given away to other people. If you would like to share this book with another person, please purchase an additional copy for each recipient. If you're reading this book and did not purchase it, or it was not purchased for your use only, then please return to Smashwords.com and purchase your own copy. Thank you for respecting the hard work of this author.
Table of Contents
It's déjà vu if Daniel's ever had it: this split second where he's alone in his closet-sized dorm room before the scene doubles and he's somewhere else. It's the same size and shape as his room but the clutter has quadrupled. He doesn't have a roommate, but the simple fact seems irrelevant to the gawky, grinning freshman perched on the bunk inexplicably situated over his bed. The kid dangles his feet, kicking them up just far enough for Daniel to see the well-worn tread on battered sneakers.
"Danny-boy!" he crows, sliding down to land soundlessly on the rug.
The freshman's name is Kyle Neaves. Daniel knows it with crushing certainty, just like he knows that the room has rearranged itself to fit this impossible person. The scattered gym shorts on the fan and scraps of paper coating the floor are all symptoms of the process.
Daniel presses his eyes shut and grabs at the space behind him for the doorknob. He stumbles over a pair of discarded basketball sneakers in his haste to get out and refuses to open his eyes until he's pulled the door closed behind him. The only person in the hallway is a ghostly blond fellow in nothing but a towel and a pair of flip-flops. He leaves a thin trail of water glistening in his wake. Daniel pushes the door back open.
Only the room is empty, not a single item out of its meticulously organized place. Daniel collapses onto the bed, careful not to disturb the hospital corners. "All right," he mumbles to the oddly shaped pattern of stains on the ceiling. It looks vaguely like an old professor. "Start with the implication."
There is a statement that proceeds any proof: If p then q.
If Daniel has an imaginary roommate, then he is insane.
The statement seems obvious, but the obvious is often hardest to prove. He needs something that fits his language. Approximate the person, Kyle Neaves, as the imaginary number, n. But what good is an imaginary number?
The set of imaginary numbers is closed under addition. No matter what two things are summed together, the answer never becomes anything real.
Daniel wakes up the next morning in an empty room and thinks Kyle is an isolated incident until he spots him leaving the physics building. He catches Daniel's eye and calls, "Hey, Danny, can I still use your spare basketball ticket?"
A kid on a skateboard makes a parabolic curve around Kyle, and Daniel seizes on the moment as evidence. He nods his affirmative and keeps walking. But when he turns over his shoulder to check, Kyle has evaporated, swept away by the howling wind.
Daniel skips class and finds himself huddled in an empty corner of the library, fingers shaking as he dials his older sister's work number. "Do you remember back when we were kids, and I spent a few weeks wondering why you weren't dead?"
"What's this about?"
"Do you remember?"
"Yeah. I just assumed that you wanted me dead for a few weeks. Everyone feels that way about siblings."
"I'm hallucinating a roommate."
"Sounds like you're more lonely than crazy. Get wasted and pick up a girl or something. It'll help."
"Gina…"
The dial tone greets him. His next class starts in four minutes but he can't deal with introductory physics today. There's no room in his head for math that avoids fact. He's never understood how it's possible to approximate every parameter and still expect useful results.
The cloying scent of flowers stings at his nose even though it's mid-November and campus is a barren, windswept wasteland. Images of twisted sheet metal and shattered glass haunt his memories. A car crash, they'd told him at the wake. Gina's body is too damaged to show to the public. It was the defining moment of his childhood—or at least it would have been had it actually happened.
Daniel takes the stairs to the top floor, stares out the window, and calculates terminal velocity of a falling body.
That night he gets spectacularly wasted with his imaginary roommate, the two of them laughing over videogames and using the pauses in action to construct a pyramid of beer cans on their windowsill.
He wakes up the next morning with a splitting headache even though the cans have mysteriously disappeared. The bitter taste of last night's beer sticks to his tongue and three minutes of brushing his teeth fails to dislodge it. On the desk, his laptop sits open, his most recent proofs assignment staring out at him. Pinching the bridge of his nose, he wills the room to stop spinning.
At the very bottom of the document, there is an addendum to his work, a row of red type in 24-point font.
TWO PLUS TWO EQUALS FOUR ONLY WHEN ADDITION IS DEFINED BY CONVENTIONAL MEANS. EXPAND YOUR PARAMETERS.
Daniel's finger finds the backspace key and holds it down as the mountain of logic unwrites itself. He collapses onto his bed refusing to look at the professor in the stains on his ceiling. Proof by contrapositive, he decides.
If Daniel is not insane, he does not have an imaginary roommate. Assume the new hypothesis is correct. There are no beer cans on the windowsill, only his wilted green plant, slowly dying from exposure to the cold. A tinge of mint clings to his tongue. He steadies himself, breathes deep and assumes he is sane.
Three hours later, his sister calls to make sure he's all right. "When's the last time you had it this bad?"
For as long as he can remember, he's had fleeting flickers of a life where he's made different choices. Gina calls it perfect hindsight. Daniel doesn't agree because the two months he spent thinking she was dead were far from perfect.
"Not since you."
"Do you think things are better in the other version?"
"I feel like I'm someone else, but that's not always a bad thing."
"Sounds like two roads are diverging in the woods, little brother."
"I'm insulted you think I'd get lost in the woods."
"That's not my point. If you assume both of these things are real, why you can't pick the one you like better."
"How about schizophrenia?"
"You're not a schizophrenic," Gina assures him. "Mom had you tested."
"I don't think you exist," he tells Kyle that night.
"I don't think Arkansas exists," Kyle retorts.
"I'm being serious."
"Arkansas has no professional sports teams and as far as I know no colleges. University of Arkansas doesn't count unless you have the recruitment mail to prove it. Their capital city's probably a little rock in a field in real Kansas."
"That's ridiculous."
"Have you ever met anyone from Arkansas?"
"No, but—"
"Danny, have you met me?"
A thin haze of fumes engulfs Daniel's dorm room. The sharpie had been new when he started but now it leaves feeble streaking lines punctuating the futility of every aborted proof. He lies back against the floor, staring at the jumble of stains on his ceiling as the acrid chemical smell twists into the sweet scent of the funeral flowers coating the coffin Gina never had. "Come on, Danny," he mutters to himself. "What are you trying to prove?"
But Danny is an ill-defined object, an imaginary quantity like Kyle himself. He's never been Danny, only Daniel. The sharpie slips from his hand, slicing across the page without leaving a mark. Out of ink. He grabs the next pen from his desk and tosses the paper into the stack behind him. Proof by contradiction? Assume Kyle is real, Daniel is sane and find the flaw.
Only once he's made that assumption, he doesn't want to disprove it.
He spends the day watching football with Kyle and wakes up the next morning, his life back in place. The room is in perfect order. The textbooks on his shelf are arranged first by subject then semester. He calls his sister who answers the phone in a foul mood. "You do realize I'm not supposed to take personal calls at work, right, Danny?"
"Daniel," he corrects automatically.
"Since when does it even matter?"
She hangs up before he can answer.
Daniel's always been fascinated by the corridor that separates the math and physics building. Aesthetically it's flawless: a great stone archway that looks like something from the Middle Ages. But there's a quirk to the architecture that magnifies even the smallest breeze into hurricane force gusts. It's an illogical construct, a failure of engineering, and his favorite route on campus.
The wind screams past him, blistering the tips of his ears red, and he tucks his head into his chest, regretting his forgotten scarf. He's deaf to the people around him until a guy on a bicycle slams into his leg and they both go careening into the concrete walkway.
"Shit, man," the biker says, scrambling to his feet. The fall had knocked his jacket open enough for Daniel to make out his screen-printed t-shirt. "I'm sorry. I should be paying more attention. You came out of fucking nowhere."
"I squared," Daniel says, still dazed.
"Are you concussed or something?"
The biker offers him a hand. His skin is warm. "No, that's not it. It's on your shirt."
"Yeah," a sheepish grin spans the biker's face. "High school mathlete. I try to keep it on the DL but laundry day snuck up on me. It's a really geeky joke."
"Keeping it real."
"I'm sorry?"
"I squared," Daniel says. "It's a real number."
He's met with a wide grin and the offer of a handshake.
"Kyle Neaves."
"Danny Mendez."
Daniel's mother calls him the Thursday before Thanksgiving, her voice thick with grief. It's the six-year anniversary of the day Gina didn't fall asleep at the wheel and wrap her car around a tree. "Are you driving home alone?"
He'd been scribbling notes nonstop onto computer paper since he filled up his notebooks but the statement stops him cold. "I'm sorry?"
It's an impossible statement, a piece of reality he has rejected as false.
"Gina never would have crashed if she had someone with her."
Imaginary numbers, he thinks. An entire class of mathematics invented to solve a problem deemed impossible. But if means to manage a problem don't exist yet, it doesn't mean the problem is unsolvable, just that the current thought process is insufficient.
He reevaluates his data, makes a decision.
"I'm not driving home alone. I'm giving Kyle a lift."
"Kyle?"
"He's a friend of mine." Danny twirls the pen in his hand. "We're going to room together next semester."
"And what time will you be home?"
"Friday afternoon. Should beat Gina by a few hours."
"We can't wait to see you, Danny."
He drops the pen to paper and makes three quick marks: i2. He smiles. "I'll see you soon, mom."
He's got it now: Define quantities Danny Mendez and Kyle Neaves, m and n, as objects in the set of imaginary numbers. The set of imaginary numbers is open under multiplication. The two values are multiplied by conventional means. The original implication fails. Imaginary numbers become real. (Q.E.D.)