#three integer

[chococrepes-art]

Yo! Y'all remember that theory that the last digit of a number is an important part of how they're like?

Like Five and Fifteen for example: both of them get easily annoyed by things, scenarios, and other people, or Four and Fourteen: both of them are crazy, eerie and dangerous, or the Zeros and Ten: they're all being and trying to be helpful to other people when they're in need.

(Example: Ten saving Nine's and Six's cats from the tree as well as saving Seven from that tree, Zro (I really doubt it was any other Zero because it was exactly Zro's voice and the Zeros including Zro do have small difference between eachother's voices) saving Pie from Pi, and and the other four Zeros saving all those puppies and cats from that burning building)

Well I was thinking ... or wondering, does that mean that Two Thousand Seven Hundred and Sixty-Three's personality could be somehow similar to Three's personality in some way? (Since 2763 ends in 3) The thing is, we don't really know anything/enough about Three's or Two Thousand Seven Hundred and Sixty-Three's personalities yet. And as we all know a new Bfdi:Tpot episode is coming out soon, and it's possible we might see Three there finally, and that could tell us about how Two Thousand Seven Hundred and Sixty-Three's personality might be like by finding out more about Three, but it's also possible we might not see Three yet, and if the continuation of the subscriber special comes out before Three's more official reveal, then perhaps Two Thousand Seven Hundred and Sixty-Three's personality could be telling us about Three's personality instead and what to expect or not expect from them.

This theory could still be wrong and just a coincidence of course, but I still think that this theory that the last digit that's a part of a number influences their personality more, is more accurate than the theory that the first digit influences the number's personality more, since we all know how 1 and 2 are like and none of the algebralien's we know seem to be that much like them (10, 14, 15, 2763) it's more likely that the bigger the number the more popular and scary they are among smaller numbers.

Anyways as a sidenote 2763 probably just has lots of layers to their being and personality because of all those digits that make this number up.

But basically what I wanted to say is that this could be an opportunity to learn more about both 2763 and 3, depending on who we're gonna see more of first.

Also completely off topic, but do you guys think that maybe Fourteen also has Thirteen trapped inside them, and it's like "tradition" for 4 ending numbers to trap 3 ending numbers inside them (not actually tradition but y'know what I mean), or do you think even Fourteen would find that really creepy and weird and terrible, like a sort of "professionals have standards" sort of thing (since they are a cannibal) or do you think Fourteen wouldn't do it but wouldn't care if others did it either? Because the only time we've seen Thirteen was when one of the people animating for the show doodled Thirteen on a Livestream and it was non-cannon. Or do you think that Thirteen actually literally does bring so much bad luck that they're exiled and have to live alone far away from algebralien kind, or maybe they've been actually sentenced to death even? Unless Thirteen hasn't been born yet, since we don't really even know how algebraliens are made (but considering the Zro scene I think it's obvious more than just romance exists, or at the very least heavily implied, I do think algebraliens would be hermaphrodites because they don't have gender AND they are aliens, but I talked about that in another post, but anyways I'm gonna stop thinking about that scene now)

[oneinteger-daily]

i am fashionably late to DAY THREE!

[gridgamesgalore]

im thirty-one and .. I love calendars ❤️

[asknine9]

[sigmaleph]

You are invited to participate in the following game: a panel of three judges sits before you, while you count up starting from 1. Every time you reach an integer power of ten (1, 10, 100, 1000, etc), that amount of USD is set as the prize. You must count out loud, in English; if you ever go more than two seconds without saying a number, say the wrong number, or fail to speak intelligibly (as determined by majority vote among the judges), the game ends and you win the highest prize you've reached. Different people will rotate in and out as judges every once in a while, as needed, but you do not get a break to eat or sleep or anything else; you must keep reciting numbers at least once every two seconds to stay in the game.

[charl0ttan]

its crazy how when u graph like x+y=10 thats a graph of all the two integer sums of 10. and when you graph x+y+z=10 thats all the three integer sums. and so on. and if you lived in infinitely dimensional space you could see a graph of every integer partition of every integer. idk i just thought about this. i love math. hi

[takes1]

needy bokuto x reader drabble

hmmmm this format is diff than what i usually do, but i liked it, lmk if it's acceptable. i wanna do a 'real' bokuto x tutor!reader but that's also been done 19328 times before

warnings. nsfw themes. impatience? or incompetence? minors dni.

info. nsfw themes / inexperienced!reader / bokuto begging / impatient!bokuto / airhead!bokuto / bokuto with raging hormones / dumb dick bokuto / 380 words

haikyuu collection. more here.

links. masterlist / my ao3 / request box closed for now

bokuto being brought to tears when you tell him you need to head back home in the middle of your makeout session. making every excuse in the damn book to keep going, begging, groveling, underneath you.

please don't go. i need you, please, god-- you don't get it.

it's true that you don't. you don't understand why he's never satisfied with how far you've gotten with him. for you, your progress is reason for celebration. for him, it's never enough.

you have to be careful around him. not showing too much skin is obvious, but even the things you say set him off and he spirals into a needy, whiny mess begging you for just a little kiss.

but you know it's never just anything. it's always as far as he can 'pretty please?' you to.

there were already enough 'could you please just--,' requests that you found yourself yielding to. there was always a pressure of from buzz in your head, his tongue against your neck, that you second-guessed every time he asked to see you now. he always got you there no matter the excuse.

he was failing three of his classes. you had to compromise. sitting side-by-side, a spread of textbooks on the table and assignments on his laptop from weeks ago he never opened.

even now he wasn't paying attention to them.

his shameless pinky prodding under the seam of your shorts was pushed back down to your mid thigh. you watched how his face tangled up, pain and frustration around his eyes and in the corners of his mouth. it tightened his broad shoulders, made his ears a cute tinge of pink.

it was endearing, at least. you liked the way he sat in silence sometimes, and suffered for you. if you could manage it, sometimes you would spare a glance to the near-permanent bulge in his lap.

a forced sigh.

you felt the pad of his thumb prod into your supple flesh and when your wince didn't tell him enough, you pulled it off and returned the hand to its owner.

maybe if he wasn't so eager.

taglist (love ya'll)

@yuchacco

@integers

[bignaz8]

Around 1770 BC, Babylonian scholars in ancient Mesopotamia inscribed a clay tablet (IM 67118), showcasing the Pythagorean theorem (a² + b² = c²) to calculate a rectangle’s diagonal, over 1,000 years before Greek mathematician Pythagoras. Discovered in modern-day Iraq, this tablet employs a base-60 numerical system, reflected today in our 60-minute hours, 60-second minutes, and 360° circles. The meticulous cuneiform inscriptions reveal step-by-step calculations, demonstrating sophisticated mathematical understanding. Even earlier, the Plimpton 322 tablet (c. 1800 BCE) records Pythagorean triples (e.g., 3-4-5), sets of three integers satisfying the theorem, written in elegant cuneiform. Likely used as a teaching tool for advanced students, it suggests a structured educational system for mathematical training. These artifacts, blending geometry and algebra, underscore Mesopotamia’s pioneering role in STEM, influencing modern mathematics and timekeeping systems millennia later.

[bfdiredux]

Algebraliens

Algebraliens are funny things but I suppose it's no shock that they exist! If we can have living objects, we can have living math.

They are all eccentric and on some level, quite chaotic. Some algebraliens are a dim sort of chaotic and others thrive off of chaos.

Algebraliens come in many shapes, sizes, and colors. They also have skin. They prooobably have blood and muscle too. I considered bones, but...

These things have no bones.

Gender is NOT a thing to Algebraliens!! They simply don't have it. They vibe with pronouns, though. (Remember: pronouns ≠ gender) You'll find that Algebraliens may have preferred pronouns but they are generally chill-- you may get gently told about the preferences but that's as far as most go.

Some confirmed pronouns--

Zero - He/they (the one with the hat!)

One - She/they

Two - They/them

Four - He/they

Five - He/they

Six - He/she/they

Seven - They/them

Eight - They/he

Nine - Any/all

Ten - They/It

Fourteen - He/him

Fifteen - She/her

Two Thousand, Seven Hundred, Sixty-Three(2763 from here on out) - It/its

X - They/any

-

ADAPTATIONS!

Heat Resistance - An algebralien is able to withstand incredibly high temperatures and lava. Somehow. X can be seen during the challenge to find their emeralds drifting through lava(likely unharmed due to their power) and Two, during the 2763000 subscriber special, merely had their skin burnt by lava.

Durability & Invulnerability - Algebraliens have skin that is difficult to penetrate. They're quite malleable. You are going to need tremendous force and effort to pierce their skin and make them bleed. They're also just... really difficult to kill.

I don't know what this is called - Your biggest concern as an algebralien would be dehydration although you could go an extended period of time without water and even longer without food. An algebralien not eating or drinking enough will eventually start to feel the side effects.

-

If an algebralien does not take care of themself, there are consequences.

The aforementioned side effects include worse physical health but also weakened power.

-

POWERS!

You could say the adaptations mentioned above are powers but! By algebralien standards, they aren't.

It's not known where powers originate from, nor is it known why/how it exists-- but it does. The current power scale between algabraliens looks something like this:

2763 > Two > One > Four and X > Literally everyone else

2763 and Two would be on the same level if it weren't for 2763's natural ability to suppress algebralien power. The further away an algebralien is from 2763, the more their power returns.

One is just below Two. She plans to change that, though.

Four and X are the embodiment of algebralien chaos.

-

EVERY algebralien has their own little pocket dimension.

They can project or transport themselves into their pocket dimension and powerful algebraliens like Four can transport items into their dimension. They also have total control over their domain-- and I mean total. They can shape it to their will, break the laws of physics, and bend its reality. This does mean an algebralien can trap another in their domain if they play their cards right.

The algebraliens who live primarily at the Equation Playground have doorways to their pocket dimensions that connect to the Playground in case of emergency. These doorways can be used, destroyed, materialized and dematerialized, ecetera, all depending on what the user wants.

-

FUSION!!!

By adding or multiplying, integers can fuse together or unfuse. It takes a lot of willpower from one side to keep from unfusing if the other is against it.

By subtracting, you can whittle a number down to smaller numbers. They have to be manually added together again to refuse and act as a hive mind.

You can kill an integer by division if you use a zero. However, trying to divide powerful integers by zero will instead leave their soul/spirit inside of the zero instead.

Any symbol or object can be used for this process as long as at least one integer is involved.

[tinydefector]

Marine Centre 2- merformers Au

Word count: 1.7k

Warnings: mention of wounds.

Masterlist and rules

Prev

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PHYSICAL EXAM / TREATMENT RECORD: Big Blue

Species: Oceanides. Sex: Hermaphrodite/ Pod Bull

Age: 50+yrs

Recent illness/injury: lacerations to back/ dorsal/ shoulder. Injuries received from altercations with The Meg ( Aggressor), believed to be due to feeding. (Wounds are healing, no sign of infection)

Behaviour: Gentle, Social.

PHYSICAL EXAM Examined by: Dr Bayley Quin

time:12:42pm

Date: 23/05/XXXX

Wt 253.592 Kg. Temp 21.8°c

GA:

Blue and red Oceanide with swirling patterns across the body, fins and gills are red fading into blue. Larger than any of His Pod. Eyes: blue

INTEG: covered in many scars from, Fighting, feeding and Mating.

External Parasites: lamprey

RAVS Animal Condition Score: Good

Reason for RAC Score assigned: For Release out of Pool 4

Addtl Comments: big blue has recovered remarkable from his injuries and is ready to be set back out to the Lagoons with his pod.

________________________________________________________________

PHYSICAL EXAM / TREATMENT RECORD: The Meg

Species: Oceanides. Sex: Hermaphrodite/ Pod Bull

Age: 50+yrs

Recent illness/injury: lacerations to head and stomach/ bite wounds to arm. Injuries received from altercations with Big blue ( The Meg was the Aggressor), believed to be due to feeding. Wounds still have not headled due to scratching.

Behaviour: aggressive, Terratioral, temperamental.

PHYSICAL EXAM Examined by: Dr Bayley Quin

time:2:54pm

Date: 21/05/XXXX

Wt 269.52 Kg. Temp 24.8°c

GA:

Dark grey, light grey with red markings, fins and gills are red . Larger than any of the Oceanides in the centre. Eyes: red

INTEG: covered in many scars from territory fighting, feeding, and mating.

External Parasites: none

RAVS Animal Condition Score: bad

Reason for RAC Score assigned: request to keep Meg in Pool 3

Addtl Comments: the Meg still hasn't fully healed due to scratching and grinding against the pool floor.

_______________________________________________________________

PHYSICAL EXAM / TREATMENT RECORD: Cherry

Species: Oceanides. Sex: Hermaphrodite

Age: 30+yrs

Recent illness/injury: scale Rot on running up tail and fins

Behaviour: (normally) hyper, Social. (Recently): withdrawn, Depressed.

PHYSICAL EXAM Examined by: Dr Bayley Quin

time:1:08m

Date: 26/05/XXXX

Wt 221.32 Kg. Temp 24.2°c

GA:

Red, black and Grey Oceanide. Eyes: blue

INTEG: scales have been showing severe signs of Scale rot,

RAVS Animal Condition Score: bad

Reason for RAC Score assigned: to be kept in Pool 2 until medicine Takes effect.

Addtl Comments from (-----l : Cherry has fallen really badly into Depression, request for Shimmer to be allowed to visit and stay with until recovered.

____________________________________________________________________

Addtl comment from (Dr Bayley Quin): Permission granted, I'd ask that Strom, Screech And Whistler be let out of Pool 7.

______________________________________________________________________

The centre is rather quiet in the afternoon with only a few people being there this late. "Afternoon Nola, how are you handling the flyers?" They call out to the older woman.

Nola sighs wearily as her younger co-worker approaches. "Those rusalka give me more trouble than all the others combined, I swear," she grumbles accent coming out thick.

Gesturing out toward the large pool the three currently rest in, she continues, "They hardly stay put long enough for a proper check! Always zooming about." She huffs.

"Whistler is a right negodyay," Nola remarks with a shake of her head. "I'll keep an eye on them tonight, but going to go check Big man and make sure he eats today, then I'll feed the demons before letting them out, Quins gave me the go ahead to let them back out to the pods" they state with a chuckle.

Nola nods sagely at the caretaker's plan. "You do that, dear - wouldn't do to have the big lug go hungry. Just take it slow and watch Him. one wrong move and..." She mimes a massive set of dentae snapping shut for emphasis.

“I will Nola, he's not that bad, he's just a grumpy big lug, ain't the first time I've fed him, think it's the reason Quin has me feeding most of them.�� They remark while scooping fish into the buckets and barriers on the cart out of the fish pools.

"Hey there Big Man" they say softly walking to the edge of the water standing there to see what mood the large grey Oceanide is in. "Are you going to try and eat me again?" They call out. Within the pool's cool depths, a massive form begins slowly stirring their approach. After all, the Oceanides could feel the vibrations of their steps. The Meg glances toward them with a rumble, crimson eyes glinting with annoyance.

His fins and gills flare briefly as if to try and scare them off but after it doesn't work a grumbling noise comes from him. They slowly sat at the edge waiting for him to come closer, even if he didn't let them touch it at least gave them the opportunity to check his wounds to make sure they were healing.

They bring the bucket of fish to sit beside them almost like bribery. "Don't you growl at me" they call out with a glare while grabbing a fish and holding it up to see what they had for him. With immense leisure he slowly prowls closer through the water.

He circles the tiny creature slowly, letting the churning wake of his bulk send ripples lapping at the pool's edge in a rumbling wave of threat. But as they make no move borne of fear, merely proffering bribes of fuel and medical care, a grudging respect glints within those warped features.

They put on a brave face while lowering the squirming fish into the water, holding it by the mouth. "Come on Big man, just want to see how you're healing" they say softly while waiting for him to approach.

His maw parts, releasing a snarl that vibrates the waters ominously - yet he doesn't charge or attack. He swims close, browling again before taking the Fish from their hand.

The fish vanishes into his mouth followed by crunches, a deep rumble leaves his chest and he glares at them waiting for another.

His form seems to ripple and swirl the water as his tail flicks slowly. They are gentle while reaching out very slowly to touch his face checking the fresh scars. Their other hand grabs another fish, holding it out for him to eat. "Gentle big guy, I'll get you more, no ones stealing your fish" they hum.

A disgruntled rumble resonates from his massive frame as their hands trace the recent wounds. The fish vanishes as quickly as the first before a low whistle leaves him as he lets his head rest on their lap as they feed him more fish.

"You need to stop getting into fights with Big blue, gonna get yourself hurt even more. You two bulls are a nightmare sometimes he's not trying to take your pod or fish" they state softly bring the bucket closer so Megatron could pick out the fish he wanted to eat.

His dentae snap down on another before he grabs for the squid and clams, the crunching of the shells echoes in the quiet night. A low, primal snarl rumbles up from the depths of his chest which has them tensing up. His gills flexes and flare, as he sniffs the air.

"Easy big man, easy" they coo, tracing the flared fins of his face. "That angry at him huh?" They nearly whisper. "The more you two fight the more you both end up back in here till your wounds heal, no ocean swimming till you heal"

he hisses, tail lashing hard enough to send waves crashing against the pool's rim. growls leave him and he quickly flicks around away from them. They sigh as he darts back into the water, sitting there for a while before tipping the rest of the bucket into the water so he still had fish, kelp and clams to eat. He submerges himself in the pool's cool concealment once more. They give the large mer one last look before grabbing their cart and leaving him.

Making their way to the next lagoon. The three sea skipping Oceanides head immediately snap towards the newly-arrived scent of food. The trine observers with fierce scarlet eyes, powerful frames arching and twisting through the water with the grace.

Screech breaks the surface first, fins flared with a Shrill. circling nearer the waters edge.

Whistler calls out in a little song of delight to see them wildly, breaking the surface with a splash. "Hello you three, giving Nola trouble again?" They ask rather disapprovingly.

The trine draws near as one, awaiting food eagerly. They sigh as the three mers hiss, thrill and rumble at them. "Yea, yea, I know dinner time." They hum while moving to the cart. "Don't you drag me in!" They warn before tipping the barrel of fish into the water. No sooner do the fish spill into the water than a trill rings out. powerful arms snake about their waist - and with a mighty splash, Whistler drags them into the water!

The seeker coos and thrills madly, spiralling through the depths with his prize clasped close against slick violet and black scales. Bubbles stream from his grinning maw. A deep snarl leaves the Blue Oceanides, only for Whistler to let out a soft whine and he tries nuzzling against them. They cough and spit water, trying to swim to the side not wanting to get caught in the feeding frenzy.

Storm appears quickly snagging them from the purple Mer pulling them safe against his body. It's a deep rumble from Storm, screech subsides with a sulky snarl but quickly swims off to catch fish. They cling onto the blue Oceanides as he slowly helps them back onto the land. "Thank you Storm" water drips from their now drenched uniform. They glare at the other two mers.

The other two continue to cavort and squabble over fish. Storm calls out softly to them which earns him a quick glance. They move back over to another bucket grabbing it grumpily, moving to sit in front of the Blue mer. "Well since those two want to be demons I guess you get the treats" they mutter while pulling out a handful of clams and sea weed for him.

Screech slinks nearer across the waters with a loud purr. Whistler chitters excitedly, zipping through the shallows to plaster himself against his pool mates flank, whining pleadingly.

What really makes the other two mers whine is when they pull out the Squid. Letting Storm eat it smugly. Screech shrill warbling of outrage pierces the quiet air the moment the squid is given to the blue mer. shrieks, and gills flared along with his wings which twitched in barely restrained fury. Whistler writhes and chitters jealousy. He eyes the succulent mollusk. "No you two want to be demons, no treats." They huff out not falling for the thrills and chritters.

A little after they finish feeding they move to open the canal that leads back into the cove. All three shoot up in excitement, jetting out the moment the gate is open. Excited thrills and calls leave them which others echo back at them.

_________

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[mindblowingscience]

An international team of physicists has successfully measured the size of a certain type of neutrino to a certain degree. In their paper published in the journal Nature, the group describes experiments they conducted that involved measuring the radioactive decay of the element beryllium. Neutrinos are subatomic particles with a mass very close to zero. They also have a half-integer spin and rarely react with normal matter. To date, three kinds of neutrinos have been identified, each by association with an electron, muon or tau particle. Physicists have become more interested in neutrinos over the past several years because it is thought better understanding them may lead to a better understanding of why there is more matter than antimatter in the known universe.

Continue Reading.

[chococrepes-art]

THREE IS A GOOD GUY THEORY (and Two will snap eventually theory)

So you remember BFDIA 11 episode "Lots of Mud", right? Where Fries goes and opens the door with the number 1 on it and then imidietly gets crushed, then GolfBall goes to the door with the 2 on it and struggles to open the door but she gets burnt after managing to open it (even faster than Fries got crushed might I add) and then TennisBall, the only one that was afraid to go and the last one to go, goes to the door with the number 3 on it and finds out that it's perfectly safe?

Well I was thinking, what if that scene of that episode is foreshadowing for what is going to happen in BFDI:TPOT?

First up: Fries going to the door with One on it, he goes there without any fear and then gets crushed imidietly

This is pretty obvious already, but if my theory is correct, then this symbolizes and is foreshadowing the fact that One is sketchy and dangerous from the start of their appearances, and will stay that way for all of Tpot or at least a long time if not forever, as well as the fact that once One arrived here they took their time with stalking and doing anything bad or dangerous to the contestants even if One was bad from the start.

Also at the beginning nobody cared about people getting kidnapped and didn't listen to Basketball or Fanny until it was already too late, similar to how Fries didn't fear opening the door until it was already too late.

Next up: GolfBall goes to the door with the number Two on it, once again without any fear, and struggles to open it, but once it opens it imidietly incinerates her

This is probably a foreshadowing about how Two is slowly loosing it and is eventually going to snap, and once they snap Two will became instantly dangerous in the moment even if it's hard to get them to snap. Even before Tpot 14 I've seen people say that Two is starting to subtlety get more mad and annoyed every episode, and now that Gaty is out and in danger of being kidnapped things will probably get worse.

(sidenote: notice how Fries has no issue with opening the door but it takes a bit to get crushed, compared with GolfBall who has trouble opening the door but then imidietly gets incinerated, like how nobody's feared One's kidnappings until it was too late, and how (almost) nobody's going to fear Two untill they snap most likely)

And Lastly: TennisBall, who after seeing what happened to Fries and GolfBall is trembling with fear and hesitant to open the door with the number Three on it, TennisBall even admitting that he's scared, but then the door opens without any trouble and there's nothing bad or dangerous there, TennisBall being surprised but glad that it's safe

If my theory is right, then this is foreshadowing how Three will seem scary and intimidating to the viewers and maybe even contestants and objects at first, but once we actually meet Three they will not be dangerous or evil but instead quite normal or even good one way or another. The fact that TennisBall opened the door easily probably means that Three is good from the start without need of a lot of stimulation to be a good person and the fact that TennisBall is so scared probably means that Three seems and will be intimidating to both the viewers and the contestants but is actually fine for the most part, like a big plot twist the writers are trying to pull.

Also another point I have is that Three appeared at the end of Bfdi:Tpot 11 "Out of the blue" where Three closed their own jail door with themself inside on purpose, instead of imidietly escapeing the chance they got, witch might mean/goes to show that they truly feel sorry and/or guilty and/or remorse for what they've done to get there in the first place. Most people would have run away imidietly weather they felt bad or not or wouldn't have felt bad about what they've done at all.

(and if anyone was wondering, and I think that's very interesting considering both Bfdia "Lots of Mud" and Bfdi:Tpot "Out of the blue" are both the 11th episodes of their seasons respectively, and bfdia is season 2 while bfdi:tpot is season 5)

Hey guys!!! Thanks for making it to the end of my theory :) Today I decided to remake a theory I made a bit back about BFDI:TPOT and it's narrative, specifically about what's going to happen with the algebraliens One, Two and Three, because I don't like the way I wrote it the first time (but I won't delete that post, don't worry)

[physicallyimprobable]

what's the 3-dimensional number thing?

Well I'm glad you asked! For those confused, this is referring to my claim that "my favorite multiplication equation is 3 × 5 = 15 because it's the reason you can't make a three-dimensional number system" from back in this post. Now, this is gonna be a bit of a journey, so buckle up.

Part One: Numbers in Space

First of all, what do I mean by a three-dimensional number system? We say that the complex numbers are two-dimensional, and that the quaternions are four-dimensional, but what do we mean by these things? There's a few potential answers to this question, but for our purposes we'll take the following narrative:

Complex numbers can be written in the form (a+bi), where a and b are real numbers. For the variable-averse, this just means we have things like (3+6i) and (5-2i) and (-8+3i). Some amount of "units" (that is, ones), and some amount of i's.

Most people are happy to stop here and say "well, there's two numbers that you're using, so that's two dimensions, ho hum". I think that's underselling it, though, since there's something nontrivial and super cool happening here. See, each complex number has an "absolute value", which is its distance from zero. If you imagine "3+6i" to mean "three meters East and six meters North", then the distance to that point will be 6.708 meters. We say the absolute value of (3+6i), which is written like |3+6i|, is equal to 6.708. Similarly, interpreting "5-2i" to mean "five meters East and two meters South" we get that |5-2i| = 5.385.

The neat thing about this is that absolute values multiply really nicely. For example, the two numbers above multiply to give (3+6i) × (5-2i) = (27+24i) which has a length of 36.124. What's impressive is that this length is the product of our original lengths: 36.124 = 6.708 × 5.385. (Okay technically this is not true due to rounding but for the full values it is true.)

This is what we're going to say is necessary to for a number system to accurately represent a space. You need the numbers to have lengths corresponding to actual lengths in space, and you need those lengths to be "multiplicative", which just means it does the thing we just saw. (That is, when you multiply two numbers, their lengths are multiplied as well.)

There's still of course the question of what "actual lengths in space" means, but we can just use the usual Euclidean method of measurement. So, |3+6i| = √(3²+6²) and |5-2i| = √(5²+2²). This extends directly to the quaternions, which are written as (a+bi+cj+dk) for real numbers a, b, c, d. (Don't worry about what j and k mean if you don't know; it turns out not to really matter here.) The length of the quaternion 4+3i-7j+4k can be calculated like |4+3i-7j+4k| = √(4²+3²+7²+4²) = 9.486 and similarly for other points in "four-dimensional space". These are the kinds of number systems we're looking for.

[To be explicit, for those who know the words: What we are looking for is a vector algebra over the real numbers with a prescribed basis under which the Euclidean norm is multiplicative and the integer lattice forms a subring.]

Part Two: Sums of Squares

Now for something completely different. Have you ever thought about which numbers are the sum of two perfect squares? Thirteen works, for example, since 13 = 3² + 2². So does thirty-two, since 32 = 4² + 4². The squares themselves also work, since zero exists: 49 = 7² + 0². But there are some numbers, like three and six, which can't be written as a sum of two squares no matter how hard you try. (It's pretty easy to check this yourself; there aren't too many possibilities.)

Are there any patterns to which numbers are a sum of two squares and which are not? Yeah, loads. We're going to look at a particularly interesting one: Let's say a number is "S2" if it's a sum of two squares. (This thing where you just kinda invent new terminology for your situation is common in math. "S2" should be thought of as an adjective, like "orange" or "alphabetical".) Then here's the neat thing: If two numbers are S2 then their product is S2 as well.

Let's see a few small examples. We have 2 = 1² + 1², so we say that 2 is S2. Similarly 4 = 2² + 0² is S2. Then 2 × 4, that is to say, 8, should be S2 as well. Indeed, 8 = 2² + 2².

Another, slightly less trivial example. We've seen that 13 and 32 are both S2. Then their product, 416, should also be S2. Lo and behold, 416 = 20² + 4², so indeed it is S2.

How do we know this will always work? The simplest way, as long as you've already internalized the bit from Part 1 about absolute values, is to think about the norms of complex numbers. A norm is, quite simply, the square of the corresponding distance. (Okay yes it can also mean different things in other contexts, but for our purposes that's what a norm is.) The norm is written with double bars, so ‖3+6i‖ = 45 and ‖5-2i‖ = 29 and ‖4+3i-7j+4k‖ = 90.

One thing to notice is that if your starting numbers are whole numbers then the norm will also be a whole number. In fact, because of how we've defined lengths, the norm is just the sum of the squares of the real-number bits. So, any S2 number can be turned into a norm of a complex number: 13 can be written as ‖3+2i‖, 32 can be written as ‖4+4i‖, and 49 can be written as ‖7+0i‖.

The other thing to notice is that, since the absolute value is multiplicative, the norm is also multiplicative. That is to say, for example, ‖(3+6i) × (5-2i)‖ = ‖3+6i‖ × ‖5-2i‖. It's pretty simple to prove that this will work with any numbers you choose.

But lo, gaze upon what happens when we combine these two facts together! Consider the two S2 values 13 and 32 from before. Because of the first fact, we can write the product 13 × 32 in terms of norms: 13 × 32 = ‖3+2i‖ × ‖4+4i‖. So far so good. Then, using the second fact, we can pull the product into the norms: ‖3+2i‖ × ‖4+4i‖ = ‖(3+2i) × (4+4i)‖. Huzzah! Now, if we write out the multiplication as (3+2i) × (4+4i) = (4+20i), we can get a more natural looking norm equation: ‖3+2i‖ × ‖4+4i‖ = ‖4+20i‖ and finally, all we need to do is evaluate the norms to get our product! (3² + 2²) × (4² + 4²) = (4² + 20²)

The cool thing is that this works no matter what your starting numbers are. 218 = 13² + 7² and 292 = 16² + 6², so we can follow the chain to get 218 × 292 = ‖13+7i‖ × ‖16+6i‖ = ‖(13+7i) × (16+6i)‖ = ‖166+190i‖ = 166² + 190² and indeed you can check that both extremes are equal to 63,656. No matter which two S2 numbers you start with, if you know the squares that make them up, you can use this process to find squares that add to their product. That is to say, the product of two S2 numbers is S2.

Part Four: Why do we skip three?

Now we have all the ingredients we need for our cute little proof soup! First, let's hop to the quaternions and their norm. As you should hopefully remember, quaternions have four terms (some number of units, some number of i's, some number of j's, and some number of k's), so a quaternion norm will be a sum of four squares. For example, ‖4+3i-7j+4k‖ = 90 means 90 = 4² + 3² + 7² + 4².

Since we referred to sums of two squares as S2, let's say the sums of four squares are S4. 90 is S4 because it can be written as we did above. Similarly, 7 is S4 because 7 = 2² + 1² + 1² + 1², and 22 is S4 because 22 = 4² + 2² + 1² + 1². We are of course still allowed to use zeros; 6 = 2² + 1² + 1² + 0² is S4, as is our friend 13 = 3² + 2² + 0² + 0².

The same fact from the S2 numbers still applies here: since 7 is S4 and 6 is S4, we know that 42 (the product of 7 and 6) is S4. Indeed, after a bit of fiddling I've found that 42 = 6² + 4² + 1² + 1². I don't need to do that fiddling, however, if I happen to be able to calculate quaternions! All I need to do is follow the chain, just like before: 7 × 6 = ‖2+i+j+k‖ × ‖2+i+j‖ = ‖(2+i+j+k) × (2+i+j)‖ = ‖2+3i+5j+2k‖ = 2² + 3² + 5² + 2². This is a different solution than the one I found earlier, but that's fine! As long as there's even one solution, 42 will be S4. Using the same logic, it should be clear that the product of any two S4 numbers is an S4 number.

Now, what goes wrong with three dimensions? Well, as you might have guessed, it has to do with S3 numbers, that is, numbers which can be written as a sum of three squares. If we had any three-dimensional number system, we'd be able to use the strategy we're now familiar with to prove that any product of S3 numbers is an S3 number. This would be fine, except, well…

3 × 5 = 15.

Why is this bad? See, 3 = 1² + 1² + 1² and 5 = 2² + 1² + 0², so both 3 and 5 are S3. However, you can check without too much trouble that 15 is not S3; no matter how hard you try, you can't write 15 as a sum of three squares.

And, well, that's it. The bucket has been kicked, the nails are in the coffin. You cannot make a three-dimensional number system with the kind of nice norm that the complex numbers and quaternions have. Even if someone comes to you excitedly, claiming to have figured it out, you can just toss them through these steps: • First, ask what the basis is. Complex numbers use 1 and i; quaternions use 1, i, j, and k. Let's say they answer with p, q, and r. • Second, ask them to multiply (p+q+r) by (2p+q). • Finally, well. If their system works, the resulting number should give you three numbers whose squares add to 15. Since that can't happen, you've shown that the norm is not actually multiplicative; their system doesn't capture the geometry of three dimensions.

[kbandtrash]

Less Than Three (Hongjoong x Reader)

~Rachel~

@sorryimananti-romantic hahaha remember how I said I was the slowest writer ever? Apparently that's not true because it only took me a month and a half to write this. I need help

Content: fluff, a whole lot of delusional one-sided pining (the cute kind though) (from him), silly middle school antics from both teachers and students, betting, friends to lovers

Summary: Kim Hongjoong teaches middle school math and finds himself absolutely smitten with you, the math department's newest hire. You're the last person to find out.

Note: This is set in an American-style middle school because that's where my teaching experiences have been and I figured my best work would come out of the system I'm most familiar with

Word Count: 11.5k

Why did Hongjoong choose middle school?

He asks himself this question every day as well

If he wanted to be a math teacher, high school was clearly a better option

But no he had started out in a middle school, promising to take a high school position as soon as one opened up

And now he was actively avoiding high school principals that wanted to recruit him

Is it because he found out the math is easier to explain and the kids really aren’t that bad?

Yeah sure

But this year he had another reason to stay

See his math department had a position open up this summer and as he got to take part in interviews to select a new hire

He fell in love

Now that was not the whole reason he advocated to hire you

In fact your credentials as a math teacher

The places you had taught before

Your praxis exam scores

Your recommendation letters

The professional development programs you were a part of

Test scores from your previous classes

The way you understood math

And especially the way you believed that every student had the potential to pass your class

All of these made you the obvious choice and everyone else agreed

But he was struck several times with Cupid’s arrows throughout the interview and that absolutely did not hurt his opinion of you as a potential colleague

On this particular morning his excuse to talk to you before classes started for the day was a test for your 7th graders

Which they would be taking next Friday

Your feedback always ended up being valuable anyway so there was no need for this to be an “excuse” but it got to be one anyway

You were typing something up when he knocked on the open door to your classroom to announce his presence

You finished typing your sentence before you looked up, and when you saw Hongjoong leaning against the door frame with some papers in his hand, you greeted him with a smile and a wave

He wondered if it would look unprofessional if he grew out his hair to cover his ears

You looked gorgeous today (every day) and he knew there was no stopping his ears from turning bright red

“I have the unit tests for 7th,” he said, stealing a chair from a desk to sit backwards on while he talked to you

“Oh good!” you replied. “I’ll take a look at those and get them back to you by lunch”

Buy lunch he thought

That sounded like an excellent idea

He should buy you lunch

He didn’t say that though

“Works for me. How was your weekend?”

And he stalled and stalled and stalled until the first bell rang and he had to get out to hall duty

At lunch he walked in again while you were putting in assignment scores and munching on some pretzel sticks

“I made a key for that new assignment we decided to add for 8th grade. Can you check answers if you get the time?”

His heart did a little flip as you covered your mouth with your hand while you finished chewing

How could you be so graceful and polite?

And your hands were so pretty wow

“Yes just as soon as I finish putting these in. And—” you turned your chair around and grabbed something from behind you— “here are the 7th tests from this morning. I added a couple notes on the integer operations review questions, but other than that, I think they’re great”

He unfortunately did need to go and eat his own lunch and fulfill other teacherly lunchtime duties so he couldn’t stall as much as this morning

But he read your notes on the way back to his classroom

And boy he could have lived in your handwriting

The pen that you used was the perfect instrument to capture every letter, every dot, every loop in a way that encapsulated your personality

And as soon as he caught himself thinking that he knew he needed to ask the home ec teacher to smack him over the head with a frying pan because wowie

He would need professional help in order to date you like a normal person at this point

Anyway he was practically skipping back after lunch because he came up with another question to ask you

There were students there so instead of using your first name he caught your attention by calling you Miss (L/N)

Which for some reason gives him more butterflies than calling you normally

This time luckily instead of giving you a new task it was a task he could do for you

“How many copies of the activity page should I put you down for?”

“Ooh good question”

You put your finger over your lips as you counted on your other hand and he had to actively look away and think about other things

There were students present after all

“Which classes are you doing it with?” you ended up asking

He used the opportunity to set a hand down on your desk and lean forward in a cool pose

“I’m making the eighth graders do it but it’s extra credit for the seventh”

“That sounds like a good plan to me. So sixty for the eighth graders and then how many do you think I would need if it’s optional?”

Oh boy he loved it when you asked for advice

It gave him the chance to look cool and smart and he always got to play it off as the humble senior teacher

“I do half just to be safe. Plus then you have extra if you want to do it again later in the year or even next year”

And then your face did that thing it did when you liked a suggestion he gave

Your face lit up with your eyebrows raised and your mouth in a little “ooh”

There were students around there were students around

“Okay then sixty for the eighth graders and forty-five for seventh. A hundred and five?”

He scribbled the number down on the copy request form

“Excellent. I’ll run this down then”

He gave you a smile and a wave as he went to attend to his own class

Which you returned

And he was sure his heart would never physically recover from the stress it was under this year

You didn’t need to know that you were the only other teacher he was sharing this particular activity page with

He would give it to the other teachers if they asked him for it but he wasn’t going to go out of his way for this one because it was kind of silly and short

But he would do anything to make your first year at a new school just a little bit easier

Back in his classroom he tried to get class started when one of his eighth graders had the audacity to raise his hand and say the following:

“You just walked back from Miss (L/N)’s class huh?”

Mr. Kim squinted suspiciously at the student

“Yeah…why”

“You’re way nicer when you go to her class before our class starts”

Before he had any time to figure out what that meant, the class exploded in giggles and shouts of agreement

“Yeah you smile a lot more this year than you did last year” a girl who had had his class the previous year as well agreed

And then from the back corner

One of the notorious trouble makers stood up

Pointed his finger up at poor Mr. Kim, who was now considering taking a high school position again

And shouted “Mr. Kim has a crush on Miss (L/N)!!!”

There was no longer any hope for this class

Exponent rules? Down the drain

Hongjoong was now a laughingstock

Irredeemable in front of a bunch of 13-14 year olds

So he stood with his hands clasped behind his back, lips pursed, and his eyes dangerously blank

It wasn’t long before kids started shushing each other

Even once they were quiet, he held their attention for a second longer

“If you have two to the fourth times two to the third, what’s the product?” he asked clearly, writing the problem on the whiteboard in new black marker

Not addressing the issue would not stem the tide of curious teenagers for long but it would have to be a band-aid solution until he figured out how to do this properly

Because he couldn’t allow rumors to spread that he liked you because firstly unprofessional and secondly what if you heard them and thought he was weird

But he couldn’t lie and allow rumors to spread that he definitely didn’t have a crush on you because firstly. unprofessional kids don’t need to know about that stuff

Like they think they’re all mature and old enough to date when they can’t even drive yet like can you imagine as a teenager asking your mom to drive you to your date with your girlfriend. Embarrassing

BUT SECONDLY

What if you heard the rumors and maybe you did like him back and it crushed your heart to hear that he didn’t like you

While the kids were working on their assignment and he was working on his computer he decided the best way to explain this to them

If they ever brought it up again

Was that they’re always in a better mood when they get to talk to their friends, right?

You’re a good friend and it’s fun working with you

The other math teachers are fun too but you’re close to his age so it makes sense that you would be closer

Yep

Good explanation Mr. Kim

Literally just one day later the kids bring it up again and they do not buy his explanation

They don’t even pretend to because their brains are not developed enough to have that kind of social tact

And the rumor doesn’t exactly spread like wildfire around the school but everyone knows about it you know what I mean

Except you actually

You’re usually very aware of everything going on in your classroom but somehow this particular subject escaped your attention

You learned early on that for some reason kids don’t have shame anymore in spilling their guts to the classroom about what they’re gossiping about

So the usual “if you can’t wait to tell your friend later then tell the whole class now” classroom management tactic is now useless

So you didn’t do that instead you just asked them which problem they’re helping their friend with and if they’re not helping their friend with math then they should get back on task

You didn’t think that any middle school level gossip could actually be that interesting anyway so yeah you had no idea that kids were shipping the teachers

And you didn’t want to know

So you stayed blissfully ignorant

While Hongjoong was lowkey agonizing over it

Now you and Hongjoong tend to eat lunch in your own classrooms rather than the staff lounge

Because why use up any more social energy than necessary right

But there is a vending machine in the staff room and Hongjoong was craving something sweet one day

And when he walked in the other teachers in there suddenly went awkwardly silent

And he knew

They were talking about him

Yeah he should take a high school position next year

Two of the younger teachers, Jung Wooyoung from the history department and Choi San from the phys ed department, broke the silence giggling to each other

“Sorry Hongjoong” San apologized “the kids are just hilarious these days. They’re so excited about you and (Y/N) it’s adorable”

Hongjoong gave them a look

“Adorable?”

“Well firstly you” Wooyoung pointed out “I never thought I’d see you fall head over heels but I totally get it. She’s super cool”

“And secondly the kids” San continued “any time you guys are talking in the hall I have all of the girls talking about it during warm ups. The boys all think of you as their role model when it comes to liking a girl”

“That reminds me!” Wooyoung interrupted “I actually did have a boy confess to a girl in my class the other day and you wanna know what he told her??”

Hongjoong’s eyebrows furrowed in a glare, absolutely sure that his reputation was never going to recover

“He said ‘I like you more than Mr. Kim likes Miss (L/N)’”

The teachers tried to tone down their laughter out of respect

Hongjoong wanted to leave immediately but this stupid vending machine was having issues

“And then you know what she said back?”

Wooyoung was having trouble holding himself together

His eyes were shining a little with tears of laughter

“She said ‘That’s impossible. No one can like anyone more than Mr. Kim likes Miss (L/N)’”

The other teachers could not hold it back anymore

The vending machine finally gave Hongjoong his candy bar

(Plus the extra one he bought for you)

So he made his way swiftly back towards the door

“Oh no no wait” Wooyoung called after him

“Tests to grade” Hongjoong lied simply

“No come on you have to hear this”

Hongjoong stopped at the door, fighting the urge to classroom manage his coworkers

He looked back both unimpressed and expectant at the same time

“Okay everyone raise your hands if you betted on before Christmas break” Wooyoung called out

Some of the teachers looked at him like he had just asked out a widow at her husband’s funeral

But they raised their hands sheepishly anyway

“Great yes and after Christmas break?”

The other half raised their hands, most of them looking like they wished that had eaten lunch in their own classrooms

“These are bets on when you’re going to ask her out” Wooyoung explained cheerfully

“Wasn’t someone also trying to get bets going on whether you would confess first or she would notice first?” San asked

“That was also me” Wooyoung said with a grin

The lounge door opened again and in walked the principal and one of the vice principals

Hongjoong thought this was finally the end to this stupid conversation

But no

“Oh Hongjoong!” the principal greeted him with a pat on the arm “I’ve got $50 on you dating before Christmas. Don’t let me down”

Waiting until the end of the school year to take a new position would simply not be soon enough

Hongjoong needed to find a new job or retire within the month

But then on the way back to his classroom he remembered he had an extra candy bar for you

So he made a detour

But what greeted him was a closed door with the window on the door covered

He hadn’t ever seen your door shut like this before so he wasn’t sure what it meant exactly

Maybe you went out to lunch today?

Then he could just nab one of your sticky notes and leave the candy bar on your desk with a positive message

He would rather give it to you himself, but the idea of you finding the candy bar on your desk later made his chest feel all warm and cozy so it was fine

Your door was unlocked, so he pulled it open just enough so he could see inside

And you were in there actually

Slumped in your desk chair with your head in your hands

Oh

Oh no

Hongjoong had been there before

Every teacher had been there before

The work of a teacher isn’t as easy as most people would like to believe

Especially for middle school, and even more so for math, it takes someone with unending patience and courage to help kids learn every day

And some days that patience and courage runs a little thinner than other days

Even experienced teachers wonder from time to time if it’s worth the soul that they give to their classes

The classes that seem sometimes not to notice one way or the other if you’re teaching or not

You hadn’t noticed that you weren’t quite alone anymore, so that left Hongjoong with a choice to make

Did he let you have this moment to yourself?

Or did he try to help you through it?

He closed the door as softly as he could

And then he gave a little knock to give you some privacy and some time to gather yourself

Then he opened the door and stepped halfway through

“The vending machine spat out two candy bars instead of one” he lied with a grin, holding up the extra candy bar and wiggling it between his fingers

You had sat up and were resting your chin on your hand curiously

But there was definitely a downtrodden aura about you that he couldn’t miss

“That’s pretty lucky,” you replied

He took that as permission to enter and on his way to your desk he snagged his usual chair to sit backwards on

“How’s today going so far?” he asked as if he knew nothing, setting the candy bar down on top of your closed computer

“Oh, you know,” you sighed

He chuckled, opening his candy bar, not making eye contact with you to relieve some pressure for you to keep up an act

“That good, huh?”

You laughed in return

“Just got my butt kicked by a bunch of seventh graders, so yeah, it’s kind of whatever right now”

Ah the teacher equivalent of “I want to quit my job and hide in my bed for the rest of my life”

“Oh, yep, I’ve been there more than once” he said with a nod “and sometimes there’s really nothing you could have done better, you know? Kids are just like that sometimes. It makes me glad I’m not an elementary school teacher”

“Oh my word yes” you agreed, finally picking up the candy bar “at least I can kick these gremlins out after 45 minutes and I don’t have to see them again until the next day. I can’t imagine being with the same class all day”

Did Hongjoong end up squandering his entire eating time just to talk with you and make sure you felt better?

Yes

He would have to sneak bites of his sandwich in between activities during his afternoon classes

But like it wasn’t the first time he’d had to do that and heaven knew it wouldn’t be the last

At least he had a good reason today instead of something stupid like he was lost in the test grading sauce and forgot to eat

He proceeded to get his butt kicked by his afternoon classes because he couldn’t wipe his lovefool smile off his face

But it was okay because once again it wasn’t the first time and heaven knew it wouldn’t be the last

Not long after came the hallowed and hated teacher inservice day

Professional development day

Both a huge waste of time and a relaxing little work day

The administrators usually planned a series of workshops for the morning that all looked. well. kind of stupid but it was all planned with good intention

Then it was lunch time

And then after that you had until the end of contract hours to do whatever pretty much

The math department liked to go to lunch together and then have a really productive planning meeting until people got bored and then it was individual prep time pretty much

Mostly it was just kinda nice to be at school without kids there

This year was no exception to all of this

But Mr. Principal had $50 on the line and decided to play matchmaker about it

For each of the workshops the teachers were split into groups

And you and Hongjoong had been placed together every single time

Somehow you genuinely thought it was just good luck

Hongjoong was hyperaware of every other teacher looking and pointing and giggling and you were so peacefully oblivious to it

The first workshop was about medical emergency training, specifically training teachers in case of allergy or diabetic emergencies

As soon as the nurse finished with the epipen instructions you leaned over to Hongjoong and whispered “like this?” 

And stabbed him with the fake epipen right in the thigh

Hongjoong swore he was going to die right then and there because why was that so attractive???

“Four…five…six…seven…eight…” you counted out, holding the epipen in place for the full ten seconds

And like a good patient he sat like a statue because his circuits were absolutely fried

You glanced up at his face and your expression immediately dropped

You checked the epipen all over to make sure it was just a trainer

(It was)

“Hongjoong are you okay?? Your face is all red did I do something wrong?” you worried

He unfroze and tried to undo the damage of his Little Moment but the nurse was already over at the table taking a closer look at him after dismissing the rest of the teachers to practice

She looked at you and then at Hongjoong and then at you again and back at Hongjoong and a funny look of understanding came over her face

“Are you Mr. Kim the math teacher” she asked

He nodded awkwardly

“And are you Miss (L/N) the math teacher” she asked

“Yes?” you answered

She patted Hongjoong on the shoulder

“He’ll be fine in a minute or so” she reassured you, glancing up and away somewhere else

You both followed her gaze over to the principal, who gave her a bright smile and a thumbs up

Which looked innocuous enough to the untrained eye

But to Hongjoong this was just another in a long list of embarrassments

The principal had told the guest workshop speakers about him

And he had been spotted that easily

After that was administration-provided snack time

Hongjoong offered to grab sodas if you would grab chips and stuff

And at the soda table he was greeted by Wooyoung and San, who called him affectionately Mr. Traffic Light

Hongjoong resisted the urge to react publicly

Even though there were no children around he refused to allow himself to be caught cursing them out

But he brainstormed violently about the ways he could get back at them without getting caught

Truly his middle school teacher powers of ignoring were activated because those two were practically dancing around him trading one-liner after one-liner and Hongjoong paid them no mind

Until he realized they were following him back to the table where he was sitting with you

He did not need them teasing him around you so he tried to shoo them off

But Wooyoung gleefully turned his attention up to the projector screen where the groupings for the next workshop were displayed

Truly your presence was the only saving grace this day had to offer

They were in this group with you two

The computer teacher/school tech support guy was leading this workshop and he was showcasing how one might use ai in their classroom

And all four of you were totally zoned out because firstly ai in a math classroom?? For what

Chat bots are notoriously bad at math

Ai in San’s gymnasium? Once again no practical application

And Wooyoung honestly just wasn’t that interested because none of you were listening so he didn’t feel like he had to either

So he decided flirting with you was more fun

Just to make Hongjoong mad

But for all of his whispered pick up lines and compliments you had approximately the same response as to this ai workshop:

Playing gamepigeon with Hongjoong under the table

(Which Hongjoong had initiated by the way)

(And you had perpetuated after destroying him at the mini golf game)

Aka you ignored Wooyoung pretty well

So Hongjoong got to glare over at him with the peace of mind that your attention was fully his right then

San tapped your shoulder and asked for your number

Which you gave him a little too easily for Hongjoong’s continued peace of mind

But then San just made a group chat for the four of you to play uno together for the remainder of the workshop

So Hongjoong decided that actually this setup wasn’t so bad

The last workshop was unfortunately much more important so the four of you couldn’t continue slacking off

The principal was running this one and it was genuinely for the betterment of the school environment

Plus San and Wooyoung were sent off to other groups

So it was the two of you and a few other senior teachers that were going to have a discussion about one of the behavior initiatives that the school was trying out

You were the only two math teachers in the group, so when the principal passed out data that had been collected about this behavior initiative, the other teachers sort of automatically passed the papers to you

(And the science teachers in the group but that’s irrelevant)

There was only one copy of each dataset, so that meant you got to share

And that meant that you got to scoot your chairs close together to look at the paper at the same time

And even better the text was kind of small so you had to bring the papers close to your faces to read them

So the two of you were shoulder to shoulder

Practically cheek to cheek, your heads almost touching

To read this data and explain it to the less number-savvy teachers

Today was actually pretty fantastic so far, Hongjoong thought

He also wondered if you could physically feel the heat coming off of his face but that was not something he was going to let bother him right then

He was going to live in the moment

And perhaps thank the principal later

After a surprisingly productive and insightful discussion

It was finally ~lunch time~

The math department gathered up in the department head’s classroom to decide which of the nearby restaurants to choose to go to

And no one could agree

Not a single person was feeling like eating the same thing

Except Hongjoong he was agreeing with whatever you said

Not just because he wanted you to have your way but because whatever you said sounded good to him too

Maybe it sounded good because you said it but nonetheless

And then the department head said the following fateful words:

“How about you two just go ahead and we’ll try to decide on something for the rest of us

Hongjoong was suspicious right then and there that this was a setup

The department head probably had money on Before Christmas

But Hongjoong was absolutely not going to let this opportunity slip through his fingers

“I’m cool with that” you answered before Hongjoong could gather his thoughts enough to say anything

Then you turned to him

“I need to go grab my purse from my classroom. Do you need anything from yours?”

“Oh, yeah, just my wallet” he answered totally on autopilot

You stopped by your classroom first and then his since his was closer to the front of the school

He was so excited to finally realize that dream of his from a few weeks back

He was gonna buy you lunch

And not only that it was just the two of you going out to lunch

It didn’t matter that this wasn’t a date

Or that technically you were just gonna buy food and eat it back at school

It was special to him to go somewhere with just you 

That wasn’t school

Seriously he was so happy he couldn’t stop smiling when he told the cashier that your orders were together and he was going to pay

You kinda elbowed him and told him he didn’t have to do that

But he just shrugged and turned his smile on you

“But I wanted to” he replied

You accepted that

“It’s on me next time” you promised

And oh man he could have died right there he felt so complete

Except if he died then there would be no next time

But you were saying next time and it made him think that wow yeah this would happen again

You meant if it ever happened again but he was determined that it was a certain for the future

Because even if it didn’t happen by chance then he would make it happen himself

When you got back to the school the rest of the math department was gone so you started eating without them

And that was a magical moment of peace too

Just the two of you

No one around to tease Hongjoong

He could almost pretend you were just normal friends and he wasn’t head over heels for you

Then the rest of the department came back and they all had food from the same restaurant

So Hongjoong had been right to be suspicious earlier

It was most definitely a setup to get the two of you by yourselves

And he liked it so he wasn’t going to complain about it

Even though it was embarrassing

Well as fun as professional development day ended up being

Parent teacher conferences were not

Hongjoong was looking forward to more Mr. Principal shenanigans

Like maybe when all the teachers were in the gym yours and his table could be next to each other

And he could talk to you when neither of you had any parents

And when the night was wrapping up

And he could walk you out to your car because it was dark

And maybe treat you to dinner because the school-provided food wasn’t usually that great

But no

None of that could happen

Because someone had decided to have teachers stay in their classrooms for conferences this year

The parents would have the wonderful opportunity to get lost in their child’s school trying to find all of their classes

What a joke

They were probably going to get so many complaints they would switch it back to normal next semester

But the one time Hongjoong was looking forward to parent teacher conferences

Of course

Was the one time they had to switch it up

The one silver lining in this was the teachers complaining about it together

Hongjoong’s favorite email he had ever received was now from Wooyoung

Who was replying to the email notifying teachers of the different setup this year but just to Hongjoong

In all lowercase:

“hey loverboy u see this crap

math teacher romancephobic fr smh”

And then with his full professional email signature at the bottom

This precious email was moved to Hongjoong’s funny emails folder, which was usually reserved for unhinged student emails

He did not reply to it

San and the other phys ed teachers were joking about how nice it would be to have the gym to themselves for once but they were in agreement that this probably wouldn’t last

Of course the math department had tests scheduled across all the grades right before conferences

Which of course left everyone grading like crazy

And of course the kids trashed the classrooms the day before

And of course the head janitor ended up getting sick

So it was up to the teachers to make sure their classrooms didn’t look like trash

Even though they had 150+ tests to grade and a pile of late work to grade

And they had 24 hours (7 of which were going to be spent teaching, and hopefully 7-8 would be spent sleeping) to make this all work before parents started showing up at 4 o’clock tomorrow afternoon

What a time this was going to be

So Hongjoong picked up his pile of tests and went to your classroom

Predictably you were grading tests

He stole a student desk and moved it close to your desk

“Which tests are you working on?” he asked

“Eighth” you replied, not looking up from the test you were currently grading

“How does this sound” he began “I’ll take your seventh and you take my eighth so we don’t have to switch answer keys?”

Still barely looking up, you handed him three binder-clipped stacks of paper

“As long as you don’t mind working to music” you replied, your eyebrows raising as a little smile played on your lips

He took your tests and handed you his eighth grade tests

(This worked out nicely as you both had three classes of seventh and two classes of eighth. He was tricking you into letting him take the heavier load ohoho so sneaky)

(He was just lucky you were grading eighth instead of seventh first)

“Don’t mind?” he snickered, uncapping his favorite felt-tip grading pen “I’d prefer it”

And that’s how speed grading turned into karaoke

Grading went almost certainly slower than it would have if you had worked alone but it was way more fun this way

Of course he ended up with sixty some more tests to grade than you

So when you finished you left for a bit and came back with snacks

As well as his pile of late work

He tried to protest but no no

“You’re not nearly as sneaky as you think you are Mr. Kim” you teased him “You thought you could fool a math teacher into thinking we had an equal workload here?”

“That wasn’t the point” he whined, trying to put the cap back on his pen and inking his finger instead

“Then what was the point?”

I love you that’s the point

But what was the point actually

What was a point he could believably tell you without giving himself away

“Just…because” he said convincingly, shrugging his shoulders and settling back into his tests “You look stressed these days. Wanted to do something nice I guess”

You brandished his late work stack again

“So I’m going to do something nice too”

Yeah he probably wouldn’t ever feel this way about anyone ever again

For him at this point it was you or nothing

But the problem was he was willing to let it be nothing for far too long

He would never make a single move unless he knew you were okay with it

Because if he and his stupid heart ruined whatever you had going now then it would really be nothing

He wouldn’t give up the something he still had

He was like a curve approaching an asymptote

He could get infinitely close, but he would never actually touch you

After settling your gradebooks for tomorrow you started by picking up your classroom together

This did go faster with music by the way

And then you picked up his classroom together

And then he realized Hongjoong realized he could have his wish

The one about walking you out to your car and taking you out to dinner

Walking you out was easy but taking you out was another story

He would have to suggest it himself

A little known fact about teachers is that they became teachers because they don’t know how/don’t want to talk with other adults

Kids don’t judge you if something comes out of your mouth a little different than you meant it

Other adults are mean and judge you over silly things

This unfortunately meant that he didn’t know how to ask you to go to dinner with him without making it sound like a date

He was kinda just hoping a lil Kdrama moment would happen and one of your stomachs would rumble really loud so he could laugh it off and say you should go eat together

But you got closer and closer to your car and no tummy rumbling

You got to your car and no tummy rumbling

You opened your door and said goodbye and still no tummy rumbling

“Wait!!!”

He could have slapped his own mouth

You were a little startled but it stopped you from getting in your car

“Hmm?”

“Just uh…it’s kinda late and snacks are great but they’re not that filling…so do you wanna um…” he trailed off and did not finish his thought

“Wanna what?” you asked

Big boy pants Hongjoong come on

“Go grab dinner? Or something?” he finally spat out

He must not have seen your face light up in the dark

“No pressure” he added when you didn’t answer within 0.05 seconds

“No that sounds great! I was thinking of grabbing something on the way home anyway and it would be way more fun to do it with someone else” you accepted

He let you pick the place again and you drove separately because after you would be going in different directions

But it ended up being a nice little fast casual restaurant

And it didn’t feel like a date really but it kinda felt like a date but no it didn’t

It just ended up being a comfortable little outing between friends

Hongjoong knew he couldn’t stop smiling

And he wondered if you already knew how he felt about you

He was sure he didn’t smile like this at anyone else

And you were many things but dense couldn’t possibly be one of them

He knew he was the opposite of subtle

But if you were willing to spend time with him like this then that meant he didn’t make you uncomfortable

Honestly he was so used to the way you made him feel at this point that loving you from afar felt like second nature

Being friends with you was enough if he could keep loving you like this

The waiter came by and asked if it was one check or two

And Hongjoong was fully prepared to pay once again

But you beat him to it

You were already prepared with your card and everything

“I owe you for last time, remember?” you told him with a cheeky grin

He shook his head

“You don’t owe me anything”

“Well then next time it’s on you”

Hongjoong started to smile again

“Next time?”

“Well, yeah” you said with a shrug, now a little bashful “This was fun. We should keep doing it”

“It is fun” he agreed with a laugh “It’s nice to hang out with people outside of school for once”

You laughed at that

“Oh boy tell me about it. Nobody told me that teaching would ruin my social life”

And things just kind of continued like that for a while

Before you had come to the school, Hongjoong had lowkey felt like he was kind of going nowhere with his life

Any time he spent at school outside of his contract hours felt like an infringement on his personal time

Or not even at school, just away from home

He had been hoping that taking a high school position would give him back a little bit of the passion for teaching that he was losing

He hadn’t even been teaching that long it’s just that the profession really is like that

Indescribably rewarding and incredibly draining all at the same time

(Especially these days. Sometimes you really wonder if it’s worth it)

But having something to look forward to every day besides a favorite class ended up being what he needed to love his job wholeheartedly again

It wasn’t just you he had fallen in love with

He had also found new friends in an unlikely place

The math department was like family and he enjoyed their company dearly

But that weirdo phys ed teacher and obnoxious history teacher had turned into excellent friends frighteningly fast

Sometimes they joined you and Hongjoong on your dinner outings

And honestly it was such a blast

An amendment to my previous statement about teachers being teachers because they don’t like talking to other adults:

Other teachers often do not count as other adults

Sometimes they do when you have to have grown up conversations

But gossiping about students does not count as grown up conversation

There were some eighth graders that all four of you had

Hongjoong not currently but he’d had them for previous classes and they were now in your class

Was it a little embarrassing as 20 somethings to have your main source of gossip be preteens?

Yeah but oh man there is nothing like finding out which of your students are lying to their other teachers about the work they need to do

Or what they’re like in other classes

(You and Hongjoong, and sometimes Wooyoung, tended to have very different opinions about some students than San did so it was double fascinating)

Who they’re friends with outside of your class

Or crushes they have on each other

Basically if you spend every day around preteens you gossip like them too

It was now November and starting to get uncomfortably cold outside

So San had offered his home as a little gathering space for you all to order food and hang out

It was almost like a little Friendsgiving

“Any of you have (male student name, obnoxious connotation)?” Wooyoung asked, taking a sip of his soda

You immediately scoffed

“I’m about to write an email home about that kid”

Hongjoong was surprised

Annoying kids existed in every class but you usually had something nice to say about them at first at least before you got into the bad behavior

He hadn’t had this particular student but he was intrigued as to why he annoyed you so much

Wooyoung and San also both looked surprised

“He’s great in gym—like cooperative, doesn’t do stupid stuff too often—” San said “but that doesn’t usually mean anything about how they are in math class”

“No he’s great in history too” Wooyoung added “finishes all his work on time, helps his friends with theirs if he finishes”

Your eye might have twitched a little bit

You let out an unamused laugh and crossed your arms over your chest

Hongjoong was fascinated and quite enamored with this new side of you

“If he has time to help his friends then he should be working on the seven late assignments I’ve been reminding him about. He hasn’t turned in anything for two weeks”

Wooyoung gasped and covered his mouth

“You’re joking”

“I’ve talked to him about it twice and I warned him if I had to remind him a third time then I would email his parents” you said, shaking your head “Like I asked him if he’s understanding the material, if he’s got something going on at home, if he needs some help, and he’s giving me nothing to go off of”

Wooyoung smiled devilishly

“He’s been lying to me, then. I ask him every day if he has other classes to do stuff for and he says no. I will absolutely be getting on him about that”

San shook his head

“Teenagers” he sighed

Everyone nodded and repeated what he’d said

“Teenagers.”

After a moment of silence, Hongjoong spoke up

“How about (female student name, pleasant connotation)?”

The mood lifted immediately and everyone gave their own version of the word “aww”

“She is the highlight of my whole day” you said

“Seriously she’s so polite and she tries so hard even when she’s having a hard time” Wooyoung agreed

“Super athletic too” San added

“Ooh and (male student name, pleasant connotation)?” you said to another chorus of agreement “He’s kind of a punk sometimes but he’s another one that always does his best”

Hongjoong smiled

Complaining was fun, but he loved the light in your eyes when you talked about the parts of the job that you loved

Anyway as I said it was November and the Before Christmas faction of teachers was starting to get nervous because there was no sign of anything happening

They saw you walk out together more than usual but they didn’t know that you were meeting up outside of school and stuff

The kids were also more riled up about it than usual

To the point where Hongjoong wondered if the other teachers were inciting chaos on purpose

It was getting bad enough that one of his classes almost failed a test across the board—on a unit about rounding and converting fractions to decimals of all things

As in the easiest math ever

So Mr. Kim had to resort to drastic measures

At the beginning of all his classes, he drew a box on the edge of the whiteboard

“This is the nonsense box” he explained with a teacherly smile

That is to say frustrated but still filled with love for his students

“Every time one of you is talking about anything that is not related to class, a tally mark goes in the box. Each tally mark represents an extra fifteen seconds you get to sit in your seat after the bell rings”

A chorus of protest arose

“I don’t want to hear it” Mr. Kim shook his head “How many of you are planning on retaking last Friday’s test?”

About half of the hands in the room came up sheepishly

“Exactly. It’s because we’re constantly off topic that no one is able to listen and learn in here. We can do better, okay?”

And then immediately from the back of the room

“Ooh, Miss (L/N) just passed in the hall!!!”

And chorus of “Ooooooh”

Whether it was true or not, Hongjoong was happy to draw his first tally mark without a word

Just that same teacherly smile

Another round of protest came and he drew another one

After the third tally mark, they shut up

“Good. Let’s talk about coefficients. Has anyone heard that word before?”

Once again I will say it was November

And the month after November is December

And December is the month of Christmas

Not just Christmas break

But Christmas itself

And that meant that Hongjoong now

After coming back from a brief Thanksgiving break

Had only a few weeks to find you a Christmas present

Now he wasn’t thoughtless like this wasn’t the first time it had crossed his mind

It had just stressed him out wondering if he would have the guts to confess his feelings for you or if he would be outed somehow first

Plus in case y’all didn’t know teaching (especially teaching around the holidays) is stressful

He just imagined Christmas as this far off date that was too good to ever come

And so he hadn’t even had time to think about what to get you

Something for your classroom?

Something for your home?

A cute accessory?

A fun math shirt?

Not a gift card though that was far too impersonal

Nothing seemed good enough for you

If he were to deliver his feelings with a gift like any of these, it felt insufficient

Most lunches the last bit of November and the first week of December

(If not spent pestering you)

He spent fretting over his Amazon cart with his head in his hands

And then the most unfortunate miracle occurred

The heaters toward the math hall decided working at full capacity was a waste of tax dollars

And the weather was shaping up to be quite uncomfortable

Everyone started to bring blankets and stuff but it was never quite enough

Your classroom was especially cold, since it was the farthest down the hall

So Hongjoong was gifted an opportunity

He went and found one of those soft and cozy electric blankets

In a color he assumed to be your favorite considering how much you wore it and how many of the little trinkets around your classroom were that color

And he packaged it like he had meant to give it to you for Christmas anyway

Then he brought it for you the next day

You were sitting at your desk in your full winter outdoor gear with a blanket that did look cozy but thankfully was not electric

And your teeth were practically chattering as you waved hello

He still hadn’t taken off his coat or his gloves either actually

He set the present on your desk

“I was saving this for Christmas but I think you might need it more now” he told you with a grin

Your curiosity was suddenly piqued

You opened the gift cautiously, glancing up at him every few seconds

But as soon as you felt the material of the blanket, you perked up

And upon discovering that it was electric you could have cried

“I haven’t been able to feel my fingers since last week” you told him gratefully “Seriously I was trying to figure out how many space heaters I would need to buy to survive the winter”

“The department head has one in her classroom” he said “If you grab your old blanket and your laptop then you can let your new blanket heat up here while we hang out over there”

What a beautiful suggestion

The department head raised her eyebrow at the two of you coming in with blankets

But she smiled too when she saw you settling down in front of her heater

“Don’t tell anyone” she said “but sometimes I take a nap over there during lunch”

“Oh I see exactly why” you agreed, sitting cross-legged and setting your computer in your lap “I can literally feel my bones thawing out”

Hongjoong settled down a respectable distance from you

But he thought someone else was pranking him when he felt a blanket drape over his shoulders

He looked around, startled, and then he realized it was the other side of your blanket

You had thrown your blanket over him to share, even though he had one of his own

And now you were pulling his arm to get him to scoot closer to you

Was this a dream??

He would not be happy if his alarm rang

But no it was real and he happily obliged

You were sitting shoulder to shoulder again

Just like at the professional development workshop

Except this time it was so warm and cozy and there was definitely a much less professional vibe

The department head raised her eyebrow at you once again

But Hongjoong was way too enamored with your shy smile to notice

After just a few minutes in this cozy little haven

Your time was unfortunately cut short

Not by students arriving to school

But by Wooyoung poking his head into the classroom

“Oh I’ve been looking everywhere for you guys”

He stepped into the classroom and greeted its owner accordingly

She nodded as if to allow him permission to enter

“They have hot chocolate down in the teachers lounge” Wooyoung informed them “I figured you icicles back here in the freezer rooms would want some but it looks like you were hiding a—” he looked down at the two of you skeptically “campfire back here”

Wooyoung clearly thought he meant something by that but no one else knew what he was trying to say

So his last comment went ignored

“Do you want hot chocolate?” Hongjoong asked you “I can go get some for us”

“No it’s okay I’ll go with you” 

Oh Wooyoung saw exactly what was going on here

Hongjoong did not

Hongjoong was a little confused—was his help becoming an awkward burden to you?

While Wooyoung saw the truth

You just wanted the excuse to walk with him

Now Wooyoung had two choices

He could step back and let you two walk down together, maybe say something to speed this math teacher romance along

Or he could third wheel

And who was Wooyoung if he didn’t pass up the chance to annoy?

Plus he had taken an oath once he started collecting bets that he wouldn’t try to swing the competition one way or another

And since telling you about the hot chocolate while you were together felt like pushing the competition faster, it felt right to pull it back a bit by getting in between you for a few minutes

So he offered hands to both of you to help you off the floor

“Let’s all go together then” he said “You should also find out if your campsite director wants any”

“Campsite director?” Hongjoong asked, shutting his computer

You also closed your computer and set it aside, looking around to figure out who Wooyoung was talking about

“Your gracious host this morning” he clarified, giving the department head a charming little wave

You both took Wooyoung’s hands at the same time and almost pulled him down with you in the process of standing up

As you straightened your pants, you asked the department head if she wanted you to bring her back any hot chocolate, to which she responded yes please

Wooyoung insisted on walking in the middle

And Hongjoong wondered if all of the patience he had acquired as a middle school teacher was going to be spent on not wringing this guy’s neck this morning

Fortunately Mr. Jung acted enough like one of Mr. Kim’s students that he was able to pretend he was one and just let it go

Unfortunately by the time they got back to the department head’s classroom kids were starting to show up so that was the end of artificial campfire cuddle time

But there was a silent agreement as you picked up your blankets and computers that you would be doing this again sometime

The first few weeks of December didn’t necessarily fly by but they did go by quickly in hindsight if that makes sense

And before everyone knew it

Christmas break was upon you

And there was no further progression of the math department couple

The Before Christmas teachers had gotten antsier and antsier right up until the day before break started

But thankfully no one had been worried enough to interfere

Hongjoong decided to stay in his classroom that day

(As if he didn’t stay in his classroom every day)

Because people had been giving him disappointed looks all over the school and he was tired of it

Even the principal had made a trip to Hongjoong’s classroom the Friday afternoon before everyone left

Just to tell him how disappointed he was that he hadn’t had the guts to make a move even when he knew the principal’s precious money was on the line

And ask him if you were secretly dating already and just didn’t want to make a big deal about it

But ultimately to wish him luck and a nice break

A few minutes after he left, you popped into his classroom

You had your bag and your coat and it looked like you were ready to head home

“Are you so ready to get out of here?” you asked

He stopped immediately in the middle of his task and slammed his laptop shut

Contract hours had ended a whole minute ago and he had wasted a whole minute still doing work? On break?

“Say no more” he said, standing up and putting his coat on “We should have left five minutes ago”

“Wholeheartedly agree” you replied “I would have but the principal visited me and I felt like I had to look like I was doing something”

Hongjoong froze

The principal had visited you too?

Why?

For what purpose?

Betting purposes?

This technically didn’t still count as Before Christmas Break right?

“That’s weird haha he came and visited me too, like ten minutes ago” he said with the normalest most unbothered tone he could manage

(He tried)

(His voice was not quite an octave higher than usual)

(Which is considerable improvement since it’s usually an octave and a half)

“Maybe he’s just making rounds” you said with a shrug “It’s nice to see how much he cares about this school. Definitely one of the better principals I’ve worked with”

Hongjoong relaxed perhaps too visibly

“Yeah if you need a guy to have your back he’s got it no question”

“Anyway if you don’t have any plans for the day after Christmas you should spend it with me”

If Hongjoong had been drinking something he would have choked

Even if he’d had plans in the first place he would have canceled them for you

“My day is wide open” he said “As is almost all of my break”

You smiled and his stomach did a flip

He wondered if it was ever going to stop doing that

He hoped not

“Mine too. If you’re bored, you can probably text me and I won’t be doing anything”

Was it cringe that he was now looking for technically his second Christmas present for you?

Yeah maybe but that was his lifestyle now

To quote N.Flying’s Lovefool “if they call me a lovefool it’s okay as long as it’s for your sake”

Even if you didn’t see him that way that was A-Okay

Anyway he found some cute little math-themed trinkets like pi earrings and a right triangle pin that said “I’m always right”

Because actually who are math teachers if they don’t like puns

And the morning after Christmas Day he wrapped them up all nice for you

And he got ready to meet you at a little restaurant you now frequented together

But this was actually like the first time he was meeting you on a whole day off so he had to figure out how to dress not like a teacher but still kinda nice

He had no idea

All of his pants were teacher pants

His shirts? All teacher shirts

He was way too good at dressing for his job

So he wore jeans instead of his usual khakis and hoped that was different enough?

He also layered one of his graph paper teacher shirts with a plain t-shirt underneath and didn’t button it up

He looked himself up and down in his mirror, trying different poses to make sure he looked like a Normal Guy

In the end he decided his hair was the only issue

He only knew how to style it in a teacherly fashion

And leaving it unstyled wasn’t an option

So he looked up some tutorials on some easy styles but he just could not see himself as anything other than a math teacher

So he gave up and just prayed that he looked okay

You looked perfect of course

The difference was subtle in theory but the way you dressed, the way you did your hair and makeup, it made for a world of difference

Oh you looked so gorgeous he was never going to get over it

Your eyes practically sparkled when you met him out front and man he thought before that he couldn’t be more in love with you but he was wrong

You did seem a little more awkward today than usual and he couldn’t figure out why

Like you almost seemed nervous

What for? It was just him

You got your food and he suddenly remembered the gifts in his pocket

“Oh!” he exclaimed, pulling them out “I know I gave you the blanket a few weeks ago but I found these and I knew you had to have them. Merry double Christmas?”

You covered your mouth to hide your laughter and your bite of food

“That’s so funny because I found something for you too”

And you pulled a little package out of your bag

Down bad wasn’t even a good descriptor anymore and neither was head over heels like there had to be a stronger silly description of being in love and if there wasn’t it would have to be invented for him

You had found him a shirt that said “Math is hard. So is life. Get over it.”

(Tbh an actual shirt that I own)

And he promised you he would be wearing it the day you all got back from break

As for his silly little gifts you adored them

You put the pi earrings in immediately and started brainstorming what do do with the other things out loud

He could not have been happier with himself

And then you caught him staring at you

You paused in your excited little ramblings

He sat up a little straighter, wondering what to say next

But you smiled and looked down at your plate

“You’re pretty cute, you know that?” you told him, unable to meet his eyes for more than a second

In front of you, his first instinct was to deny the possibility and he followed it without thinking

“Eh, no I’m not—not in comparison to you, anyway”

Hmm if he wanted to keep his feelings secret then that was not the thing to say

But you deserved to know it

Not just cute but beautiful

Heartstoppingly so

You took the compliment well anyway

“Have you ever thought…” you began, trailing off

You watched him as he waited for you to continue, his eyes wide and curious

“Never mind” you dismissed with a casual wave of your hand

“No hey what were you going to say?” he asked

“Nothing I just had a weird thought for a second”

“No no come on! You know I wouldn’t judge you for anything”

You hesitated again and he could see that same nervousness he’d noticed before

“Have you ever thought…I don’t know” you put your elbows up on the table “of us as more than friends? Like dating maybe?”

Once again he was going to be very upset if his alarm clock went off now

But even if this was a dream, there was no harm in saying it out loud

“Every day since the day I met you” he answered honestly

You blinked like you didn’t believe him

“You’d better not be joking because I mean this like I’m risking our whole professional relationship here—”

And then he realized

You literally had no idea about him

Genuinely no clue that the whole school knew exactly how you had him wrapped around your finger

Except you

“—on the slightest chance that you might feel the same way—”

“Hey” he stopped you softly “I would never joke about how I feel about you”

Embarrassment began to set in for both of you

“Seriously,” he said, holding back a laugh, “you can ask any of my students, any of the teachers. I think you were the last one to find out that I have a massive crush on you”

“Oh no you’re kidding!” you exclaimed, your hands coming up to hide your face

“I really wish I were—Wooyoung has two rounds of bets going with the teachers about when and how we would end up dating. It’s just about me though—I don’t think anyone else knew about you”

You buried your face further in your hands

“That is so embarrassing” you whined “Seriously I might kill Wooyoung when we get back from break”

“I’ll help you” he promised

You ended up deciding to tell the other teachers on account of the bets they had placed on you

But you asked them not to tell the kids

Hongjoong was comfortable dealing with them at this point and he didn’t see the point in subjecting you to the attention he got about it

Speaking of the bets

No one won any of them technically

The two categories were before/after Christmas break and you notice/he confesses

And since you confessed during Christmas break well

As tempting as it would have been to say that you two deserved the money

(Especially on those teacher salaries)

You just made Wooyoung give it back to everyone who had put money down

And honestly? Very little about your at school dynamic changed

The kids still teased Hongjoong every day about his obvious soft spot for you

But he didn’t care because he knew better than they did anyway

You did actually start to notice now that kids were gossiping about you and Hongjoong

And it was really funny actually

Especially when he stopped by your classroom and the kids went dead silent watching you

Or when you left his classroom and you heard the kids explode with their weirdo little preteen comments from just outside in the hallway

Wooyoung insisted on telling you about every time you were mentioned in his classroom

This included the story about the girl rejecting a boy because quote no one can like anyone as much as Mr. Kim likes Miss (L/N) enquote

And you about died from laughter and embarrassment

Because how had you missed every single sign thrown your way??

It was so obvious now that you were dating him and you knew why he spent as much time as physically and contractually possible in your classroom

Because like I said very little about your at school dynamic changed

He still treated you almost exactly the same way

Except now if you were having a bad day he could hug you and kiss you on the head and tell you that everything would be okay and you were a good teacher

And if he got cold he could come to your classroom and have you sit on his lap while you shared your blanket

You were very very careful with any displays of affection by the way like it was only behind a locked door that you would even dare

Because firstly unprofessional

A literal breach of the code of ethics more likely than not

And secondly what if the kids saw you???

The other teachers were whatever like you didn’t really want them to walk in on you either but at least they were other adults

But the kids??? 

There were already too many rumors flying around the school about you and they did not need a even whispering of confirmation

The end of that came of course when you got engaged like a year or so later idk and you showed up to school with a ring on your finger and the kids went wild

“Mr. Kim she has a ring!!!”

“Mr. Kim what are you going to do?? She’s gonna get married!!”

“Mr. Kim you must be heartbroken”

You had prepared for this together

You had known it was coming so you knew you had to be ready for the chaos it was going to cause

So you had decided that he would also start wearing a ring to match even before the wedding

So to all his very concerned students he got to hold up his hand and say in the coolest most chic manner possible

“And who do you think gave her the ring huh?”

It was like setting off a nuclear bomb of middle school gossip but it was so worth it

Anyway breaking the chronological flow going back in time because this needed to be the last scene

The cutest change with your at school dynamic now that you were dating was now you could exchange secret messages on sticky notes with the papers you traded

He started it by handing you a test key to check with a sticky note on the top that you assumed was a label for what the test was

But on closer inspection it was a pickup line

“The limit of my love for you is like the limit of 1/x as it approaches 0; it doesn’t exist”

So you wrote back on the same sticky note “well mine is like 1/x^2 and it approaches infinity so there” and handed it back to him once you checked his key

Not to be outdone he wrote you a new note

“Girl are you a 30° angle inscribed in a circle because you’re acute-y pi”

Oh that one was bad

You had to give him something worse

“If we’re both math teachers, how come we have so much chemistry?”

You handed him that one in between classes

And as he read it he had to disguise his sudden laughter as a cough because there were kids around and they didn’t need to be curious about what he was laughing at

His next sticky note had a crease down the middle horizontally

128√e980

You recognized it immediately but you folded it in half to reveal the secret message anyway

“I love you”

So you gave him back “I hope you like fractions because you’re my other half”

You stored all these away in a little file on your computer titled “Valentine’s Day Math Jokes”

Maybe for some future Valentine’s Day activity

But mostly just to keep them all somewhere safe where you could look at them any time you wanted

Without some kid being like “oooh Miss (L/N) whatcha lookin at”

Your favorite note from him was about as simple and dorky as they came

Much like Hongjoong himself actually

Simplify 2x+6i<2(x+9u)

First you distributed the 2 on the right

2x+6i<2x+18u

Then since there was a 2x on both sides, you could subtract them and cancel them

6i<18u

Then divide by 6

And the answer made you smile every time

i<3u

(so I know how I wrote this so fast actually. I just have a goal to write 250+ words every day and uh pretty much every day of the month of January ended up dedicated to this one. Someone said Math Teacher Hongjoong and I (graduated in December with a math teaching degree, student taught in a middle school for 4 months) went feral over it whoops)

Masterlist

[lilith-hazel-mathematics]

Hi there, I’m not a mathematician, so idk if my question is silly, but why is it that integers in set theory are defined as a composition of all the previous integers instead of something simpler like for example just the set containing the previous integer? Is there something that makes the way we official define integers useful, or was it something like “this works, good enough”?

Hey, thanks for your question. What you're describing is actually Zermelo's construction of the integers, which uses the successor function S(n)={n}. You may recognize the name "Zermelo" from his greatest hit "Zermelo-Frankel set theory", the all-time most popular theory of sets, and arguably the de facto foundation of all modern mathematics. So, you are in good company in thinking that's the obvious implementation. However, the Von Neumann implementation (used in Hazel's "count to 100" challenge) is considered to be superior, which I think is for three main reasons.

The first and probably weakest reason is its convenience in defining the "<" relation. Under the Zermelo convention, you need to define "<" using the Recursion theorem, but it's much easier to prove Recursion when you already have "<" at your disposal. Having to prove the Recursion theorem without ever mentioning "<" is a pain in the ass. It's not terribly difficult to prove Recursion if you have access to the axiom of infinity, but it's way harder from a finitistic set theory, and that difficulty is compounded if you can't mention the "<" relation (I've done it and it's fucking annoying). I'm not a dirty finitist*, but I do have a deep interest in reverse mathematics, so these kinds of pragmatic concerns are important to me. Under the Von Neumann implementation, we simply have n<k if and only if n∈k, and the successor operation S(n)=n∪{n} isn't much more complicated. (*finitism is an unpopular but nonetheless respected mathematical philosophy, me calling it "dirty" is a joke.)

The second reason is its relationship to cardinality: as a set, a finite ordinal n contains exactly n elements, provided we use Von Neumann's implementation. For example, 0={} has 0 members, similarly 2={0,1} has 2 members, and so on. Besides being elegant for obvious subjective reasons, it's also pretty convenient in certain technical aspects. For example, there's an important set theoretic operation called set exponentiation, where Y^X denotes the set of all functions f mapping X→Y. In the finite case, it holds that |Y^X|=|Y|^|X|, hence the name and notation. This overlaps with the ordinary notation for the set of ordered pairs, X^2 = {(a,b) : a,b∈X}. This is compatible with the set exponentiation operation, but only if we use the convention 2={0,1}. That is, the ordered pair can be thought of as a function with domain {0,1}, in the sense that (a,b)[0]=a and similarly (a,b)[1]=b. This doesn't work under Zermelo's convention. A similar issue arises for triplets at 3, quadruplets at 4, and so on, essentially forcing the Von Neumann implementation all the way up, if we want this elegance. There are other technical roadblocks I'm sweeping under the rug (roadblock shaped like "a function is a set of ordered pairs"), but this answer is long enough already.

The third reason I'll say, and certainly the strongest, is its relationship to infinite ordinal numbers. Infinite ordinals are unbelievably important in set theory, for like a trillion reasons I could talk about for an entire decade, so it's pretty important to have some kind of implementation. The least infinite ordinal is named ω, which is the least nonzero ordinal obeying the property ∀(n<ω), n+1<ω. In other words, ω has no immediate predecessor. Under the Von Neumann implementation, we can simply say ω={n : n is a finite ordinal}, albeit the formal way to say "n is a finite ordinal" is a mouthful of logic. This extends to larger ordinals, in the sense that Von Neumann's convention allows us to easily construct ordinals of any infinite size. There's no good way to extend Zermelo's implementation to the infinite ordinals, so it's just not as good.

TL;DR Zermelo's convention works, but it's very slightly more inconvenient in almost every conceivable way. Nobody actually writes out Von Neumann ordinals by hand (except as a fun joke), so there are no downsides.... unless you don't have Axiom of Replacement, since then you have to use a completely different third convention which almost nobody knows about.

-Lilith

[weepingicegiant]

HUGE TPOT 12 SPOILERS!

BUCKLE UP CHAT. CAUSE THIS IS GONNA BE A LONG THEORY!!!!!

Where do i begin?

So heres the thing, I watched TPOT 12 then took a shower right after. While taking said shower i took a moment to think about everything that went down. And thought of the wildest theory ever..

• XFHOV

LETS START FROM THE BEGINNING,

One and Three, right?

These two fellas have FINALLY appeared after 15 years! Why is that?

Here is the scene when X found out their value: 7!

Four, Seven, and X are celebrating and generally very happy. Who isnt happy?

Three and one.

Theyre clearly angered by this whole situation, but why? Who are they angry at?

Theres three possible options:

1. Seven. But that wouldnt make sense, huh? Basically every algebralien has SOME sort of grudge held against him, so its nothing new. Scratch that!

2. Four. Its possible, four has done alot of things. But thats not who im personally going to focus on. Well, partially at least.

That leaves one more person!

3. X. But why them?

"Theyre just a little silly dude, they did nothing wrong!" And youre right. Thats exactly why theyre a target.

• THREE AND ONE'S CAPYIVITY

Youre probably wondering, why am i focusing on X, and not four? Four is more of a suspect, after all.

Think about it this way,

Three is trapped in a prison inside of Four's school, so four put him there! (They are the only one with control over that place anyway.)

One is trapped inside of the moon. There isnt really any solid proof to prove my theory with this one, but she came out during TPOT. She clearly had the ability to escape and didnt seem tired out and/or surprised that she finally left her prison. One actually seems happy and collected, even going as far as "greeting everyone" once she's out. She chose to come out at this time, just like how she chose to mess with Two's show.

One also confirmed that shes an "old friend of Two's." If One is suddenly against Two after all the years of them knowing each other, Two clearly did something to her.

• MUTUAL CONNECTIONS

And youre STILL probably wondering, "Chim! You still didnt explain what X has to do with all of this!"

Four and Two hate eachother, this hatred being caused by Two when they stole more than HALF of the bfb cast.

(They seem to not be familiar with eachother when they "first" meet. Im not exactly sure why here HELP.)

But anyway!

Who does Four have a good friendship with? X!

Who does Two have a good friendship with? Also X!

X is what keeps them sane around eachother, the only main reason why they stick around.

If anything bad were to happen to X, Four and Two clearly have the power and ability to get back at whatever or WHOever caused this harm.

One and Three did something to X, and it resulted in Two and Four snapping.

• THREE'S PERSONALITY

Three has basically ZERO information on himself, we know nothing about the guy. But what we do know, is that hes agressive.

In the Number Playground Chronicles, we get an article that explains an event that takes place, Three being apart of it. The article reads:

(To reduce confusion, ill be adding the names for you guys to differenciate whos who.)

"(Five)Integer did not pick up the ball when dropped, and Three Integer, the person playing with him, became impatient. (Five)Integer was angry at (Three)Integer because, Three Integer could simply pick up the ball and throw it to (Five)Integer, and (Three)Integer and (Five)Integer could keep playing.

"FOAMING THREE INTEGER"

The horrible, tragic incident happened at 10:13 AM. Three Integer became Upset.

Three Integer at 10:16, Angry.

Three Integer at 10:31, Furious.

Three Integer at 10:24, and "Foaming."

Three Integer at 10:39, when he started to produce smoke.

(Five)Integer picked up the ball at 6:17 PM."

This event did truly happen, we can see the beginning of it play out in the first few seconds of XFOHV:

Three was so angry and REFUSED to even touch the ball. Five had to go pick it up themself HOURS after the incident.

Three also WILLINGLY closed the cell door after it was opened. He couldve escaped, yet he didnt. I have two possible reasons for this:

1. He's afraid of Four catching him, so he followed orders and stayed put.

2. Three's gone insane after a decade and a half of being all alone, to the point that he WANTS to stay inside.

• ONE'S MOTIVES

One seems like a friendly character, shes smiling in basically the ENTIRETY of her screentime, (minus the part when she conversed with Fanny.)

But something about her smile isnt right, its almost disturbing. The way she grins in the oddest situations,

She is seen with a list during the post-credits scene, with four names on it that are all crossed out, meaning they are "completed."

□ Bell.

□ Bomby.

□ Fanny.

□ Ice Cube.

Notice how all four of these contestants were in some sort of distress during that moment, and One helped them out! In exchange for a "favor."

1. She helped Bell escape elimination by removing her string (something that annoyed Bell constantly due to contestants activley climbing it.) And hiding her.

2. She helped Bomby escape elimination by hiding him.

3. She gave Fanny a new mouth, discarding the need to spend hours at a time searching for it in the ocean.

4. She gave Ice Cube a new pair of legs, allowing her to walk again.

What exactly does One need these favors for? Revenge against Two, of course!

● ONE AND TWO'S FRIENDSHIP

Theres a popular theory stating that Two was kicked out of the equation playground, this would clear up the confusion as to why they basically NEVER appear in the subscriber specials.

Maybe this is because Two hates math! They said it to Gaty in one of the episodes.

I also believe that One was also kicked out. Why, you may ask?

Take a look at this scene in the beginning of TPOT 11:

One's picture was hidden underneath Seven's. As if nobody(COUGH COUGH. Espcially four) wanted her to be mentioned so they simply hid her.

Maybe this is how One and Two became friends, two rejects.

● RANDOM THINGS THAT GET THEIR OWN SECTION CAUSE IDK WHERE ELSE TO PUT THEM

1. How the HELL did One get a mouth and... legs?

2. Judging by One's little room, she probably really likes space and astronomy, maybe thats way the moon was where she was sent.

3. Kinda freaky to think that One was there in the moon the ENTIRE time. Throughout every single episode of the series from BFDI 1A to TPOT9, and we never knew.

Yeah tbh idk what else to say this was just a little info dump cause my mind was PACKED. anyways yeah tell me if i missed anything anf let me know about your little theories and opinions on mine! :3