Archive of Our Own

"So, hey, Mr. Minami. I was in class the other day, messing with a calculator and I got an error for dividing by zero? I wanted to ask, like, why can't you do that? I mean, like, we can multiply by zero, right? And that gets us nada! So what's the deal? What, is the universe gonna implode or something?"

Mr. Minami glanced up from his supersized superhero soda, eyes narrowing in some kinda mixture of annoyance, incredulity, or disdain, either from the question or from having his hourly sugar injection interrupted. "Calculator code's got limits. Math doesn't. What, you using an arithmetic set?" Immediately he popped his mouth back onto the straw. Fret couldn't tell whether his intense voice—loud even without the megaphone—or his intense-r sucking was more disruptive to all the poor burgerlovers at Justice Burger. Not very justice of Fret to have treated Mr. Minami during their busiest time of day.

Fret blinked a few times, freedom fry in hand, half-way to making his belly feel liberated. His mouth opened to speak though, not eat. "Uh, arith-what now?"

"Are you such an ignorant isosceles you haven't even learned arithmetic? Ha!" Mr. Minanmi's smirk widened. "If you want to gain a term or two, we could start from the basic axioms."

Shooting Mr. Minami one of his token kinda-lost, loosey-goosey grins, Fret bit down and tasted a little bit of freedom. And the extra salt he piled on, too. "We're talkin' math here, right? So I'm guessing arithmetic has something to do with numbers, am I right? Uh... But what does this have to do with dividing by zero?"

"Heh. I'll show you." Reaching across the table, Mr. Minami snatched the remaining half-fry right out of Fret's fingers and held it up. "We'll start with a pop quiz. How many fries is this, zeptogram?"

Fret's eyes darted away and back, like he was uncertain at the really obvious answer. "...One?"

"Correct!" Mr. Minami slammed the fry directly onto the table itself hard enough to make the seats shake, then grabbed a second one fresh from the bag and put it down next to the first. On the bare table. "Now, what'd I just do?"

"You...have two fries. Oh, you added one?" Fret smiled, the gears in his head working not-so-overtime. More like average-time.

"Naturally. Next question: what's a real number?" As he spoke, Mr. Minami filched another few fries from the bag, breaking them into halves and arranging them on the table. One fry here, two fries there, another two fries there, another two fries but at a right angle...

Fret reached up, after wiping the grease off his fingers, and scratched at the back of his head. A real number? What, was a fake number any different? Fret thought of a bunch of numbers. 100 was a real cool number, in his opinion. Or was it fake because he was making it up in his head? He looked down at the fries Mr. Minami was arranged in a bit of a loss. "Um, a real number...is one that is real! Like, you have—one, two, three—seven fries there! That's a reeeeaaaal number. And a fake number is one we make up in our heads! You know, like, 24!"

Mr. Minami had been grinning—up until Fret said the number 24. He glowered like Fret'd just personally thrown one of his weirdo trash piles away. "You're not just an ignorant isosceles. I miscalibrated. You're an entire stupid scalene, unbalanced as unintelligent." A flurry of furious fry-fingering, and Mr. Minami had drawn...a letter?

ℝ, the bottom facing Fret.

"Imagine a distance between two points, say, me and you. Any number you could use to define a distance along that line is real. 1, 2, 3, 7, 24, 7/2, π, the square root of 3, -1618033988749894... They're all real, whether counted in fries or represented symbolically by electrochemical impulses in that yoctolitre of usable tissue in your cranium." At least he got his grin back as soon as he started spouting math shit again.

24. Fret struggled to hold in a laugh, but actually tried to listen, too. "Okay, okay, so if I said the distance between you and me was...10! That would be a real number?"

"Sure." Mr. Minami set the superhero soda back down. "Heh, you might be a stupid scalene, but at least the rise-over-run of your hypotenuse's not so zetta slow."

Fret beamed. He sorta got what Mr. Minami said, but he could tell it was kind of a compliment? It tickled his gut in a different way than the Justice Burger did. "Okay, okay, so I can just say a number means something and it's real. But, what's a fake number?"

"More commonly defined as imaginary numbers. Irrelevant to the discussion of zero division in an arithmetic set." Mr. Minami's eyebrows arched up, and he suddenly grinned a very different grin, all teeth, more like a lion about to pounce. "Unless you want to take that tangent? Heh heh heh. Riemann spheres are zetta fun times..."

"H-hah, I think we should..." Fret thought for a second. "...stick to the, uh, s-sine? Or, the cosine! Because there's two of us?" N-nailed it?

"Ha ha ha ha ha!" Mr. Minami threw his head back in laughter so loud and rowdy that Fret could feel the burger-nation's collective gazes burning into their table. "Golden!"

Didn't just nail it, he hammered it home! Fret grinned.

"So, with any set of real numbers, we have operations that we can apply to the set, forming a group. In this case, you already enumerated one: addition." Mr. Minami somehow bent the original half-fry and the whole-fry into loops without breaking them. He pointed to each in turn. "Call this 𝑎, and this 𝑏. These represent any real numbers. Doesn't matter what. With any two real numbers 𝑎 and 𝑏, you can add them together to form a third real number." Mr. Minami slid two fries laid across each other between the loop-de-loops. In a way he'd lived up to the freedom fry name, having liberated them from their fry-shape. "𝑎+𝑏. The other operation..."

"Subtraction?"

"Are you trying to be obtuse?" Mr. Minami snapped. "Subtraction's congruent to addition! Nothing but the addition of a negative number!"

Fret jumped at the snap-crackle-pop of Mr. Minami's words. "Kweh? But you're taking away stuff instead of putting them in, right?"

It was like Mr. Minami was looking at walking garbage. "The difference between 𝑎−𝑏 and 𝑎+(−𝑏) equals what?"

"Uh..."

"Substitute any real numbers. The difference between 3−1 and 3+(−1) equals what?"

"Oh, well, 3-1 is 2. And 3+(-1) is...2! Hold on a second..."

Mr. Minami smirked.

"They're the same?"

"There's no difference between addition and subtraction." Mr. Minami hmphed. "You really didn't know arithmetic field axioms. I'd already calculated academia as approaching the limits of irrelevant trash, but what the factor are those yapping yoctograms even teaching you in school?"

Fret laughed, kind of embarrassed. All this stuff really did go right over his head in the classroom. "Hah, nothing good, I guess. Not like you, Mr. Minami!"

"That's ninety degrees." Only Mr. Minami managed to make the act of tilting his hat up radiate that degree of smug.

"Okay, so if you didn't mean subtraction, then..." Fret scratched the corner of his lip before reaching over the table. Only one other idea occurred to him! He fingered the fries from a + to an × symbol. "You gotta mean multiplication!"

Mr. Minami stretched his arm out over the table—and ruffled Fret's hair. "That's my Zeptogram No. 1!"

Fret "oop"ed and blushed at the surprise headpat. His gaze quickly danced around the restaurant, just to see how many of those borgorlovers had been looking their way. The weird looks had focused on Mr. Minami, sure, but he still felt the invisible pressure of their sight's weight. Fret looked down at the table instead, trying to ignore that awkward mix of adoration and unease settling in his brain. "S-so, what's the deal with multiplication?"

With a final ruffle and a gentle pat, Mr. Minami withdrew his arm. "You can multiply any two real numbers 𝑎 and 𝑏 and output a real number 𝑎𝑏." He paused, studying Fret for a moment, then sighed. "Which is another way of symbolizing 𝑎×𝑏."

"Ooh! I got it now." Fret smirked, feeling that sudden pressure in his head ooze away as soon as Mr. Minami backed off. And so did the warmth of his hand. "𝑎×𝑏, so like, 2×2 is 4! And 6×4 is, heh, 24!"

"Correct!" Mr. Minami fully smiled now, almost encouragingly leaning forward with his elbows on the table. "With these two operations, we can define field axioms used in the arithmetic we're constructing. Zeroth: the difference between 𝑎+𝑏 and 𝑏+𝑎 is equal to what?" He shuffled fries this way and that, as if demonstrating his point as he asked. "And between 𝑎×𝑏 and 𝑏×𝑎?"

Fret squeezed his brain a little harder this time, cranking the gears a little closer to overtime. This, he was pretty sure he learned in school. "𝑎+𝑏 and 𝑏+𝑎 are, um, equal to each other? And same with 𝑎×𝑏 and 𝑏×𝑎?"

Mr. Minami nodded approvingly. "We'll define that as the commutative property. Who gives a digit what order you add or multiply terms together in, as long as you're using the same operation? Let's add parentheses to the mix." Just as he'd begun to bend fries into little curvy shapes, he stopped and glanced up at Fret. "You know what those are, right?"

"Uh, yeah, totally!" In spite of how sus that sounded, Fret meant what he said.

Mr. Minami arranged more fries on the table. At least he looked like he was having fun. He held up an itty-bitty little fry bit squeezed into a bend. "𝑐's another real number. First: the difference between (𝑎+𝑏)+𝑐 and 𝑎+(𝑏+𝑐) is equal to what?" With how quickly he rattled off the number-y bits, Fret didn't always keep up, but the freedom fry visual aids that Mr. Minami put together just as lightning-fast helped keep it in his head. "And between (𝑎×𝑏)×𝑐 and 𝑎×(𝑏×𝑐)?"

The gears upstairs were oiled up with the fry grease. He squinted at the visual and quirked an eyebrow. (𝑎+𝑏)+𝑐... It was just 𝑎+𝑏+𝑐, wasn't it? And (𝑎×𝑏)×𝑐 was 𝑎×𝑏×𝑐 too, right? Fret was a little unsure how Mr. Minami would feel about his mental removal of freedom from the table, but it made sense in his head! "Okay, okay, stay with me for a sec, Mr. Minami. (𝑎+𝑏)+𝑐 and 𝑎+(𝑏+𝑐), it's all just addition! And (𝑎×𝑏)×𝑐 and 𝑎×(𝑏×𝑐) is all multiplication! It doesn't matter what order we add or multiply as long as we use the same operation, like you said! So we can just...get rid of these, right?" Fret reached over and grabbed two of the parentheses fries and chowed down on them. Meh, they were starting to get cold.

"Naturally! The associative property. The one I don't follow." Mr. Minami hehed to himself at his own joke. "Now, if we mix the operations together in the same equation, which one do we perform before the other?"

"Oh, uh." Come on, Fret, this was definitely something he learned in school! That expecting glare in Mr. Minami's eyes and his combo of right answers really piled on the tension. Didn't want to break that Super Smokin' Smart rank. "Um, PEMDAS, right? So, multiplication...?"

"Ah, PEMDAS. Correct again, zeptogram." Man, Mr. Minami sounded so proud. "For the next axiom..." He arranged the freedom fries in swift succession: 𝑎×(𝑏+𝑐). "You just said you wanted to eradicate the parentheses from your spatial coordinates." He gestured. "Go ahead. While keeping it equal."

Fret reached over, a couple of tiny sweat beads at the back of his neck. If he kept this up, he could hit quadruple 𝑆 and more! Just a little taunt— "This'll be a piece of π."—and then he'd show Mr. Minami what's what.

"Hmph. But what fraction of π will it be?"

Fret focused a little too hard and the first thing that came to mind came out his mouth. "Uh, the fraction you put in your mouth." He heard Mr. Minami's ha! as his fingers tickled the parentheses fries. If a number was next to a parentheses, it was multiplication—Fret easily remembered that from how much it confused him on his homework assignments.

"Ha! Inexpressible as a fraction, except through the only identity that matters! I already ate the entirety of ln(−1)/𝑖!" Mr. Minami gestured to himself. "That's how I reverse-engineered the solution for my greatest masterpiece: Myself."

While Mr. Minami ranted about pie and art or whatever, Fret picked up the parentheses fries. He shuffled the 𝑏+𝑐 closer to the 𝑎 and put the two fries in his hands down on the table, too cold now for his liking. "There! 𝑎×𝑏+𝑐!"

Mr. Minami stared down at the table, then clapped his hand over his face. "Zeptogram...substitute some real numbers in. Does 𝑎×(𝑏+𝑐)=𝑎×𝑏+𝑐?"

Oof... With that expression, he was knocked down from 𝑆𝑆𝑆 to a 𝐵. Fret frowned and kicked himself into overtime. "Okay, um, real numbers! So, 2×3+6. We start with multiplication: 2×3=6! And then we add 'em up, 6+6=12! So how about 𝑎×(𝑏+𝑐)...? 2×(3+6). With the parentheses, we add first, so 9? And then multiply, so 2×(9)=18. Oh..."

"Hmph. Iterate over your solution again." Mr. Minami swapped the fries around to form the original expression: 𝑎×(𝑏+𝑐).

"My solution? What's up with my solution, I followed PEMDAS and everything."

"No, your calculations for why your solution was wrong—" Mr. Minami emphasized the word by pounding his superhero soda onto the table. "—all added up. I'm talking about your original solution. Fine, I'll repeat the pop quiz. Eradicate the parentheses from the expression 𝑎×(𝑏+𝑐) while keeping it equal. You can sum as many fries as you need to and integrate whatever terms you want. The final expression should have zero parentheses."

"Oh!" Fret clapped. "Alright! Now this will be an actual piece of π." He quickly sprung into action with his hands, picking up the parenthese fries before taking a second look at the equation. It...was actually kinda hard to change it up the way he thought. Fret gestured with one of the fries in his hand, pointing at the equation and working it out in his head again. "Hold on... If I multiply 2×3 and 2×6, I get 6 and 12. And then when I add those together, I get 18... Ohh!"

Not unlike Mr. Minami, Fret's hands went into a short frenzy with rearranging the greasy digits and operators. He separated 𝑏 and 𝑐, curled a new 𝑎, and arranged both sides with the + in the middle that kept it nice and symmetrical.

Fret grinned. Boom. "𝑎×𝑏+𝑎×𝑐!"

Mr. Minami reflected that grin, angling his visor enough for Fret to get a rare full look at his face. His excitement made Fret think of a kid in a candy store. Then again, Mr. Minami would go ballistic in a candy store too, no kiddyness required. "Correct! Enough ninety degrees that you form a square! Ha ha ha ha ha! You've worked out the distributive property: perfectly balanced."

Wow! Look at that, perfectly balanced. Fret hummed in satisfaction as he sat back, absentmindedly grabbing a fry and tossing it in his mouth. He let out a very unsatisfied "hmm" at the room temperature diagram piece, but swallowed it down anyways.

"Heh, work out the other axioms and I'll add joules to those fries to your satisfaction," Mr. Minami offered in an unusually warm tone.

"Other axioms?" Fret asked, looking down at the fries. Adding joules, or whatever that meant, sounded pretty cool, but he didn't realize he'd attended a math class now.

"Can't derive much with the three we've got so far, but you've already more or less defined one of the others we need, and two more are easy as π." Fret could hear Mr. Minami tapping his boot under the table. "Not backing out now, are you, zeptogram? The series hasn't converged yet."

Fret tapped his foot against the floor, imitating Mr. Minami. He was starting to turn into a math-head too! A real universe-brain, like Mr. Minami was! One day, he'll be the one making all the math metaphors and hearing Mr. Minami's hoarse laugh over and over. He could take on a couple more, if it meant getting to see dividing by zero and making Mr. Minami proud. Fret shook his head. "No backing out here!"

"Good. Didn't take you for some cowardly cardinoid." Mr. Minami twisted a fry into a circle: a 0. He set the 0 next to another fry, this one—for once—not bent or changed in anyway, just a boring straight fry. "Let's define two more properties about our operations. There should be a way to apply both operations without changing the original value. In other words..." Mr. Minami whisked a few fries together. "Third: 𝑎+𝑥=𝑎. Find 𝑥."

Fret looked down at the equation, uncertain. Was this just another "oh, the answer is so obvious that it makes you feel like it isn't right" situation? Well, it worked out for him last time. "𝑥 is 0, right? Like, y'know, 𝑎 is still 𝑎 at the end, so it has to be added to...nothing!"

"Correct. The additive identity—identity meaning that 𝑎 doesn't transform—means 0 has to exist in our set." Mr. Minami swapped out the + for a ×. "Fourth: define the multiplicative identity."

𝑎×𝑥=𝑎. Fret touched his chin and tilted his head. He knew a few things about identity and making his identity his own, so he just had to do the same with the number! Keep it the same. "Then 𝑥=1! To keep 𝑎's identity!" Fret felt a little silly feeling kinda happy for the inanimate fry shaped like an 𝑎 for keeping its identity at the end of the equation. 𝑎 got to stay 𝑎! Not be forced to turn into a 𝑏 or whatever other not-math symbol they forced into math.

"Naturally. Your integration of novel information's accelerating. Let's see you keep these desired outputs constant."

Keep it up?! Back to 𝑆𝑆𝑆—Super Smokin' Smart Fret!

"The fifth axiom's one you've nearly already defined. Suppose I added 𝑎 and 𝑏 to get 𝑎+𝑏. Using only the operations we've defined," Mr. Minami said, his boot-tap coinciding with the word only, "how do I go from 𝑎+𝑏 back to 𝑎?"

"Uh, you subtract—I mean, add -𝑏 to 𝑎+𝑏?" Fret answered.

Mr. Minami had grimaced the moment Fret had said subtract, but then he'd grinned even more widely when Fret'd managed to save himself at the last second with the word add. Score 1 for his galaxy brain. "That adds up, zeptogram. Heh. We define that as the additive inverse. Every 𝑎 has its −𝑎, and 𝑎+−𝑎=0. So, we've established the pattern. Define the sixth axiom."

"K-kweh?" Fret looked at Mr. Minami with a real shocked and uncertain grin. "You want me to define number six?"

"Try it. You've got all the patterns you need. Think about the previous six axioms we've gone over." Mr. Minami smiled like he was looking at walking garbage, but the kind of walking garbage he could add to his latest...'art piece'. "If you don't believe in your probability calculation for your capacitance, believe in mine."

Fret's leg started bouncing suddenly, like he hadn't noticed. It was a different feeling, having Mr. Minami believe in him so much. He bounced his leg, thumbed the fries in his hand hard, squishing the potato innards onto his hands.

Phew. Okay. Calm down, Fret. Mr. Minami did the numbers in his head and believes in him. That had to mean the chances of getting it right were above 50%, right? Had to be. That means he was more likely to get it than not. Yeah. Yeah, he could do this. No need to worry.

Fret stamped his palm down onto his leg, keeping it still. Mr. Minami was still smiling at him, surprisingly not snarling in impatience for once the way he did whenever they had to wait in line for food. More like the way Fret had caught him a couple of times in the MIYAHISTA underpass: just kinda standing there, watching people go by, grinning at nothing.

Well, no need to make him wait any longer. With one more breath, Fret started to speak his thoughts aloud: "The last six axioms we've gone over... The commutative property, associative property, the distributive property, and the identity..." It was like a light flashed over Mr. Minami's eyes as Fret correctly recalled them, his smile growing warmer with every word. "And now we've got the inverse..."

"How many axioms did we use for the identities?"

"Two... One for addition, one for multiplication."

Mr. Minami nodded: the Math Man's blessing.

Fret squinted his eyes at Mr. Minami—not at him, more like through him, in thought. "Wait, there are two identities, but we've only gone over one of the inverses!" Oh, yeah, now the Mina-man was grinning ear to ear. "We went over the additive inverse, so we gotta go over the multiply—multiplee—multiplicative inverse!" Took him a couple tries, but he got there.

"You're exponential." Whatever that meant, it sounded like one of the biggest compliments Mr. Minami had ever given him. Especially with Mr. Minami leaning over to pat his head again, palm warm and heavy and downright cozy on Fret's hair, even more affectionately this time. "Go ahead. Define it, golden zeptogram."

The wires in Fret's brain were crossing from the warmth radiating from his Mina-man's words and hand, but Fret blinked his head clear. Wasn't gonna fail him now. He'd pass this class with an 𝐴+. "The additive inverse was, uh, adding a number and its negative buddy, to get 0. 𝑎+-𝑎=0, right?" Another nod from Mr. Minami in response. Fret's gaze glanced over the table, looking to the identities. 𝑎+0=𝑎. It looked kinda like the inverse equation... He just had to switch the 𝑎 around. "Then I need to switch 𝑎×1=𝑎 around too! I gotta look for 1!"

Fret's hands popped down to the fries, flicking them around to assemble an equation with a determined expression, and Mr. Minami's gaze followed suit, watching what Fret was doing so intensely, paying such close attention that Fret couldn't help but giggle. It made his hands shaky, nervous to impress his Mina-man. He worked it out in his head: he had to divide 𝑎 on both sides of 𝑎×1=𝑎 to get 1=𝑎/𝑎. But if there were only two operators, addition and multiplication, then...he had to take the inverse? And if the inverse of addition was 'subtraction,' then the inverse of multiplication would be—?

"Hey, Mr. Minami..."

Mr. Minami smirked knowingly. "Hey, zeptogram."

"I think I saw this in class once. A whole number can be written like 𝑎 and 𝑎/1, right?"

"Ninety degrees."

"Then..." Fret finished mixing around his fries.

𝑎×1/𝑎=1.

"Does this look right?"

Mr. Minami's palm settled on Fret's head. He tapped his thumb against Fret's brow almost tenderly. "I can practically detect the neuronal connections multiplying in here. Heh heh heh. Not so hollow-skulled of a hectopascal after all. As full and potentially endless as a fractal." Lifting his hand up, he jabbed a finger towards the fries Fret had remixed, his other arm sweeping dramatically over the table like he were boasting about Fret's accomplishments for him, like some dad embarrassing his kid at a family gathering. "Defined it all on your own. Heh. Feel that, zeptogram?"

Fret smiled. He beamed. He was lit up like the sun. Did he feel it? Hell yeah, he felt it! He could hardly even think about the rest of the restaurant—right here, right now, he was as much of a math man as his Mina-man.

"Your radiance is up to yottacandela."

"Huh?" Fret said, suddenly aware of the glow that had been shining from his eyes, his grease-covered fingertips, and his brow that contained his galaxy-brain.

Mr. Minami just kept grinning at him. "Any sun-sized star could output a couple octillion candela. My zeptogram's got the potential to output more than the factoring Milky Way. Heh...just look at you. An opus in progress."

Fret sputtered out a couple of sounds that sort of resembled Nagi's. His face burned bright red and—had his Mina-man just said his brain was beyond galaxy level? "Haha... You—you don't really mean that, right? I mean, I'm just regular ol' Fret, haha—!"

"You're not some random radian." Mr. Minami brushed his thumb against Fret's brow. ...Reassuringly? Or Mr. Minami's equivalent of it, or something. "You think I called you Zeptogram No. 1 without checking my work?"

Well, shit.

Fret had no more words. No more words but the quiet thoughts of "Boy, I hope this moment literally never ends."

Actually, a few words occurred to him, after a couple of minutes of Mr. Minami ruffling and puffing up his perfect hair into a mess—and Fret loved it—and after looking down at the multiplicative inverse he'd calculated out. Just one lingering question. "Mr. Minami, you haven't explained why I can't divide by zero."

"Heh heh heh. Correct. The series hasn't converged." Mr. Minami's smirk bordered on dangerous. Uh oh. "There are two possible final axioms for this field. One of them—the trivial solution—does allow division by zero. The other doesn't. The other one's the one we use in arithmetic. What's the only difference between the additive inverse and the multiplicative inverse, in terms of which numbers it applies to? If you can solve that...then you'll answer your own question." He gave Fret's head another pat. "And if you need a nudge to the right vector, I'm adjacent, zeptogram."

The weird brand of affection in Mr. Minami's voice did something to Fret's heart, he swore. Made the ba-bumps turn into BA-BUMPS.

Phew. Deep breath. And if he needed a push, Mr. Minami would be there. Fret had this one too.

Fret tried to speak through the overwhelming physical and mental heat all around him. "...Which numbers it applies to? Well, the additive inverse is 𝑎 to -𝑎. And the multiplicative inverse is 𝑎 to 1/𝑎. So...it applies to whatever number 𝑎 is, right?"

"Correct. So what numbers can 𝑎 be? Let's call 𝑎 any number that the additive inverse axiom applies to, and 𝑏 any number that the multiplicative inverse axiom applies to. The cardinality of the set satisfying 𝑏 is exactly one less than the cardinality of the set satisfying 𝑎." Mr. Minami met Fret's gaze. "Why?"

Fret closed his eyes, in thought. Well, adding and 'subtracting' could be done with any number, couldn't it? A real mix-and-match, just like with fashion. But with multiplication and 'division,' there was a number that didn't count? A number that wasn't included in 'all real numbers'? For some reason, his first thoughts bounced around to really high and really low numbers, but they could be multiplied too, right? So maybe it was a number that was closer to home... What number couldn't fit in 1/𝑎?

Fret opened up his eyes and examined the equation he wrote. 1/𝑎.

Hmmmmmmmmmmmmmmm.

Weren't fractions also, like, division signs—

"Zero!" Fret blurted out, his legs making him hop up a little in his seat. His eyes were wide-open and shone with realization. "The number that can't fit 𝑎—I mean, I—𝑏! The set for 𝑏! It can't be zero!"

Mr. Minami laughed this really bizarre laugh. Like he was just...having fun? "Ninety degrees. Why not?"

"Zero..." Fret cleared his throat, suddenly self-conscious about his reasoning. "Zero, uh, can't be on the bottom of a fraction? And...a division sign is a fraction, or something like that?"

"Those are all true statements," Mr. Minami observed approvingly, voice tight not with annoyance but with sincere interest, his superhero soda long since abandoned with the attention he'd riveted to Fret's face, "but you haven't proven anything. Why can't it be on the bottom of a fraction?"

Fret dug around in his brain for those remaining brain cells.

"Think about the axioms we just worked out."

Fret just needed to pull some real numbers from out of his head. No better way of finding out what's wrong than to plug in the wrong number.

0×1/0=1.

Oh. Well, now it was super obvious. "0×1/0... It can't equal 1, because it's multiplied by 0. Is that why?"

Fret was pretty sure he'd never seen Mr. Minami beam like that before. "Another ninety-degree answer, zeptogram. Don't stop your enharmonic progression. How does multiplying by 0 and any other real number differ, and what are the consequences of that?"

"It always equals 0 when you multiply by 0?"

Inclining his head—a stamp of endorsement—Mr. Minami made a circle with his hand, motioning for Fret to continue. After the ensuing few moments of Fret wracking his brain and coughing up a whole load of nothing, Mr. Minami spoke up again, his tone still even and patient. "Want that nudge, zeptogram? If you want to factor it out yourself in silence, I've got all the 𝑡-interval on the number-line. You're angling this vector, not me."

"Wait, wait." Fret motioned 'stop' with his hand. "I-I got this, Mr. Minami. What...does 0 mean in multiplication? It always equals 0... And that goes for, like, 𝑎×𝑏 and...𝑎×(𝑏+𝑐) too, doesn't it? If 0 is 𝑎, then both of those equations will be 0..."

Mr. Minami nodded along to Fret's words like he was headbanging at some thrash concert, like Fret's thoughts were the music.

"Mr. Minami..." Fret looked up at him, hesitant. "I think I, uh, I'm gonna need that nudge."

"Heh. Sure!" Mr. Minami didn't sound disappointed, at least, just weirdly excited. "As you said, the expression always evaluates to 0 when you multiply by 0, unlike for any other real number. With any other real number, if you wanted to get back to your original value, what would you do? You've written out that axiom already. So the fact that you can't do that with 0 means that 0 doesn't have...a what? It's as obvious as it seems, zeptogram. You're asymptotically there. I'm just pushing you right to the limit. Might even be so obvious and intuitive to your yottacandela sequencing you don't realize you haven't said it." After a moment, he picked up one of the remaining room-temperature fries. "Want a visual?"

Fret considered the nudge in the right direction. Ugh, with all this, Fret really wished to multiply his brain cells. Well, nothing better than the only thing you've got in your head, right? "When you multiply a number by 0, you get 0. And with any other number, you can divide to get back to your original numbers. But with 0 as the—the product, you can't divide 0 to get back to whatever number you multiplied with 0. And that means 0 doesn't have a—a—a value? Like, 0 can't be divided because there's nothing there! Can't split up a pizza if the pizza doesn't exist!"

Mr. Minami held up his hand, pinching his forefinger and thumb together—no, not quite, the tiniest of spaces shining through. "You're a fraction of a femtometre from the solution. Ninety degrees: you can't divide 0 to get back to whatever number you multiplied 0 by. 0 does have a defined value: don't forget the additive identity. But what can't you do? What doesn't have a defined value?" He smirked. "Well, there's nothing you can't do with enough time given your potential. But what can't you do arithmetically?"

"Heh, well, you can't divide by 0." Fret could almost chuckle. "So, if 0 has a value, then...does the multipla—multiplicative inverse of 0 have no value?"

"Ha ha ha ha ha!" Suddenly Mr. Minami wasn't there. And then suddenly he was, like, right there, next to Fret on his side of the table, having apparently blink-stepped right in the middle of Justice Burger just to sling his arm around Fret's shoulders, the hem of his coat draped around Fret's side, his voice as loud in Fret's ear as his breath was warm on it. "Exactly correct! The multiplicative inverse of 0 is undefined! And since division is just multiplying by the multiplicative inverse—"

"—then multiplying with the m—multiplicative inverse of 0 means you're multiplying with something that's undefined?"

Giving Fret's shoulder a squeeze, Mr. Minami beamed brightly enough to go supernova. "So, answer your own question. Why can't you divide by 0 in this arithmetic we've defined from these field axioms?"

Fret grinned. "I can't divide by 0 because division is just multiplication with the multiplicative inverse, and since the multiplicative inverse of 0 is undefined, it has no value! Can't use it!"

Mr. Minami grinned back. "Zeptogram No. 1! Radiamn straight!" With one arm around Fret's shoulders, he ruffled Fret's hair with the other, a two-prong attack right on his heart. "Your matrix's got one helix of a determinant. What your eigenvalues lack in erudition they more than make up for in enthusiasm. Zetta persistent sum of a digit." He laughed as he spoke, fluffing up Fret's hair. "Didn't even need me. Worked out that solution on your own. All I did was point you to the right coordinates. But you factored it out yourself. Heh...well on your way to making yourself a masterpiece with flawless calculations like those. Exponential? No—you're beyond that. Tetration, zeptogram. I want to see how far you can heap that pile of yours."

Make himself a masterpiece? Flawless calculations? Tetration, whatever that meant but the tone behind it made it seem even more powerful than exponential?

Fret's brain broke. Gears? Stopped. Wires? Mixed and tangled. Neurons? Firing no longer.

He dove into his Mina-man's chest, tucking his face close to him. Fret reached up with his noodly-arms, jiggly with no brain to guide them, and gripped onto the edge of Mr. Minami's coat, feeling the hard buttons with his fingers to regain any sense and brain power and—

Ugh, he felt like his brain cells got multiplied by the multiplicative inverse of 0.

In other words, his brain cells were undefined.

Mr. Minami's arm wrapped around him, holding him up in a tight, warm embrace, his Mina-man's appreciative, affectionate laughter so invitingly warm in his ear.

"Mmmrpph mmph," Fret said against Mr. Minami's inner shirt.

"That's my golden zeptogram." Mr. Minami hugged him. Behind his back, Mr. Minami's other arm shifted against Fret's shoulder: a series of soft sounds that reminded Fret of a real math class, chalk on board and all. Then something crackled, accompanied by the scent of fresh freedom fries. "I've balanced my side of the equation. But you haven't technically completed your proof yet. The series hasn't converged until you say it." He ran his palm over Fret's head with a surprisingly gentle, nearly doting hand. "The three letters that signal class is dismissed." His other arm circled Fret's waist, holding him warm, and close, and safe. "Make me proud, zeptogram."

"Heh..." Fret lifted his head up and looked Mr. Minami right in the eyes, Mr. Minami gazing back with the proudest, fondest smile Fret had ever seen on his face. "QED."