#math isn't mathing

[grogumaximus]

[despairdesolationdeath]

How tf does one get a B+ at 90/100 marks on a math test?????

[critical-skeptic]

Does Majority Actually Rule?

If majority truly ruled, we wouldn’t be stuck with the ongoing nightmare that is the orange turd. Back in 2016, Hillary Clinton won the popular vote by nearly three million, and under a system where true majority rule prevailed, she would have taken office. The QMAGA lunacy—those rabid cries of 'Stop the Steal,' conspiracy theories, and violent tantrums—might have been snuffed out in its infancy. Instead, what actually governs this country is a patchwork of archaic and deeply flawed systems—mechanisms explicitly designed to cheat, disenfranchise, and favor the entrenched power of wealthy elites and their obedient base.

Take gerrymandering, for example. Districts are twisted into obscene, nonsensical shapes to ensure that certain votes carry far more weight than others, diluting the influence of dissenting voices and creating the illusion of choice. It’s a grotesque mockery of democracy, all while the charade of 'representative government' carries on for the masses. But let’s not kid ourselves—the Electoral College is the most egregious relic of this system, a rusted cog in the machinery of democratic decay. It's a mechanism so broken that its failures have become a predictable farce, celebrated only when a Republican ekes out a rare popular vote win—such moments are so unusual, they become their own news cycles.

Consider the 2020 election: Joe Biden won the popular vote by over 7 million votes—a staggering 81.3 million (51.3%) to Donald Trump’s 74.2 million (46.8%). Despite this, Biden's margin of victory in the Electoral College was only 306 to 232. Fast forward to 2024. Trump narrowly won the popular vote with 74.7 million votes (50.5%) against Kamala Harris’s 71 million (48%). And yet, suddenly, he’s awarded 312 Electoral College votes to Harris’s 226. This stark discrepancy—a narrow popular vote lead yielding an outsized electoral win—lays bare the inherent distortion within the system.

Sure, Trump won the popular vote this time around, a rare occurrence for Republicans, who have routinely lost it for decades. But when the popular vote handed Biden a decisive win in 2020, many on the right simply couldn’t handle it. Cue the insurrectionist tantrums at the Capitol, an embarrassing display of fragility masquerading as patriotism. All because they couldn’t accept that both the flawed Electoral College system and the popular vote had gone against them. Spare us the sanctimonious civics lessons and cries of "majority rules." Your hypocrisy is glaring when you invoke majority rule only when it serves your narrative. The reality? Any criticism, dissent, or inconvenient fact is dismissed with cries of fraud—introspection be damned.

The truth is, for many who scream about democracy and freedom, genuine democratic rule is their worst nightmare. It’s not about representing the majority’s will; it’s about maintaining power through any means necessary. Twisting rules, exploiting systemic rot, and gerrymandering their way to victory, all while claiming moral superiority. Gerrymandering, voter suppression, and the Electoral College are tools wielded to amplify minority rule and silence opposition. When it works in your favor, you celebrate. When it doesn’t, you rage against the system and pretend to be its victim. It’s all part of the grotesque machinery, and the numbers don’t lie: a 7 million popular vote lead netted Democrats a fragile 306-232 Electoral College win, while a 3.7 million vote lead for Republicans in 2024 inflated to 312-226. That grotesque imbalance isn’t a triumph; it’s a stark reminder of how deeply broken and manipulable the system is.

So, by all means, celebrate your so-called 'win' in 2024. The clock is ticking toward 2026, and every second that passes exposes the hollow victory for what it is—a testament to a system rigged to distort and magnify small victories while disregarding the broader will of the people. Don’t delude yourself into thinking it’s a triumph of majority rule. It’s a masterclass in gaming a decaying system, a desperate clinging to power that betrays just how terrified you are of genuine democracy.

[proshippy-fox]

can people PLEASE stop saying proshippers have to be supportive of radqueers..... yeah I like weird fiction and yeah I think people who claim to be "transmexican" or "transkorean" are being disrespectful. those aren't linked to each other.

[lastofthe20thcenturygirls]

got double helix piercings today :)) but it didn't cure my depression :((

[dragonwithanaquarium]

Lol, this is how I found out. Or, more actually only of my friends asked if I ever had a crush on someone, I said no, they said really, I said yes, they said are you aro, i said what's that, they said aromantic, I said I that doesn't tell me what that means, they explained aromantic to me, I said, I don't know, then my other friend said you might be aromantic, I said maybe but I don't really know, and now I am two years and 17 months later.

[rhbrand]

The math ain't mathing

[archived-folder]

that time of the year when i try to sell my soul to pass🙏

[mistress-of-vos]

I find it hilarious how someone can come to this platform, point out that it's definitely weird and sad to dedicate one's whole account to "redo" an indie comic and exactly that kind of people will find your post and be offended.

Idc how much the age gap between two fictional characters, aka pixels and ink on paper, bothers you. Dedicating your whole internet presence to redo an indie comic and get some sort of profit from it it's weird and it does the opposite from helping artists. If you guys really hate LO so much then stop building yourselves after the work of Rachel! Go and make a fanfic of Percy Jackson. I hate that you guys make get so loud with your "positive retellings" that are nothing but sanitized versions of the story because you get the icks at a fictional age gap. You know what that is? Pathetic.

I already made my points before, and you are only proving how annoying your group of people is. You can't demand the story to be what Rachel didn't want, because the story is HERS no matter how many redesigns you post or how many edits you make. The story will always be hers, and you guys can keep screaming and hissing for eternity.

I shall proceed to block you. I wish no more contact with accounts that profit on indie media. I will not waste more words on you antis.

I'd make a longer post (and maybe one day I will) but since Lore Olympus, the story that introduced me to webtoons is coming to an end I'd like to say something:

I can't believe it is considered problematic. It has to be one of the sweetest, fluffiest, simplest stories I have read (hence why I still like it, it's a relaxing read before bed) and somehow it got too "kinky" for mainstream. It's laughable.

Everytime I dare to click on their tag or look for the # on Twitter or FB I see people clutching their pearls as if Lore Olympus were brainwashing teenagers into marrying a non existent God of the dead and have babies with him. What the hell?

The fact that people think LO is too dark makes me laugh. A single episode of Rick&Morty, BoJack Horseman or HQS has way more explicit content and dialogue. In fact!!! If it were up to me LO would have gotten genuinely kinky!!! All it does is have some surface spicy tropes that get sugar coated to not make puritans awkward and tbh that's sad. LO and the author get terribly hated anyway for daring to portray the most common female fantasy.

And this all makes me laugh but also mad because you'd think LO at least has some genuine dark themes but no? At most we have Persephone's trauma due to Apollo's abuse and yet that topic is treated as a therapy pamphlet because people couldn't handle an imperfect victim. Hades is a wife guy who shows little to no anger. Hera was re written to be sort of a feminist so that people stopped being annoying about women having emotions.

LO is a sweet, simple story with tiny spicy things here and there that were eventually pushed aside because people couldn't handle it. I wonder how Rachel feels about this, because at the beginning the story was extremely spicy and the only crime was being published in a platform as webtoon, full of people who can't differentiate reality from fiction.

Is LO a masterpiece? Idk! I enjoy the story, it's very self indulgent for me, but I won't go and analyze every detail to see how it should be labeled as it's not meant to be a perfect media. It's meant to be an entertaining, nice story of romance and it does that job very well. This need to demand perfect writing while also crucifying authors over "dark" themes is ridiculous and contradictory.

And I keep wondering, if these people loathe LO so much, why dedicate all that time to the infinte posts they make about how they would have told the story? And all those re tellings are boring! It's always "So Persephone and Hades won't ever kiss here because she's a lesbian. Also he doesn't appear at all. And Demeter isn't an abusive mom! Oh and everyone is ugly because gods shouldn't be beautiful! And Apollo isn't evil he's uwu baby. And no toxic relationships here, Zeus is a good husband!"

Sweet Gaia, you guys wouldn't handle Saint Seiya having Athena in the body of a teenage girl with big tits and who's constantly in the edge of breaking her virginity vows. This attitude screams of jealousy and puritanism and both are disgusting.

TLDR: LO being too problematic for people is both funny and annoying. I wished it actually were as kinky and dark as people insist it is. I'd pay for a toxic romance, but that being said, I LOVE it very much as it is and it's nice to have a re telling that, while not pretending to be loyal to mythology, didn't go for a route of sanitizing all the myths. I hope that once it ends haters will move on and let real fans and the author alone. 🙏

[blooodymoon]

A "RETURN of the Number of PERSONS FLOGGED in the BRITISH NAVY, in each of the Years 1845 and 1846; specifying the Name of the Ship, the Offence, the Sentence and the Number of Lashes Infficted." Parlamentary Paper, Number 661 of Session 1847

So after reading the Post about Jopsons Flogging by @handfuloftime I went down a research rabbit hole because I wanted to see if their exist sufficient & specific enough Data to accuratly compare his punishment and put it in the historical context.

I found this Website, which features the two tables from the Parlamentary Papers and their Source.

(Note: The tables on the website are incomplete and inaccurate in certain places. I compared it from top to bottom to the original source and corrected/added the informations in my table.)

I used that as a basis for my own table, where I also added Averages and Sliders so people can search the Informations!

(Apparently Sliders don't work on Mobil, and I don't know enough about Google Sheets how to fix it :/

If anyone knows or if you find a some kind mistake please let me know.)

This week I visited the University Libary, which has access to the Parlamentary Papers Online, to see if such tables also exist for earlier years but sadly 1845 was the first year where they went into such Specifics.

(I hope to add some of the Years after 1846 to the table, when I can.)

Earlier years only had these Informations per Year:

Total Number of Punishments

Total Number of Lashes

Highest Number of Lashes Inflicted at One Time

Lowest Number of Lashes Inflicted at One Time

This Information for 1839 - 1846 under the Cut.

Also under the Read More are some Graphs, Medians & Averages about Flogging for Drunkenness Alone, seperated by Seamen, Marines, Boys and Everyone!

Hope that someone find this useful/interesting !

Corporal Punishment for Drunkenness Alone

Everyone:

Lashes Sentenced Median: 32,75 (1845: 36 | 1846: 31,4)

Lashes Sentenced Average: 31,07 (1845: 31,13 | 1846: 31,02)

Lashes Inflicted Average: 30,8 (1845: 31,02 | 1846: 30,58)

Seamen:

Lashes Sentenced Median: 36 (1845: 36 | 1846: 36)

Lashes Sentenced Average: 34,37 (1845: 34,62 | 1846: 33,86)

Lashes Inflicted Average: 34,33 (1845: 34,41 | 1846: 33,86)

Marines:

Lashes Sentenced Median: 31,4 (1845: 32 | 1846: 31,4)

Lashes Sentenced Average: 30,8 (1845: 31,09 | 1846: 31,09)

Lashes Inflicted Average: 30,8 (1845: 31,09 | 1846: 30,69)

Boys:

Lashes Sentenced Median: 24 (1845: 24 | 1846: 33)

Lashes Sentenced Average: 26,22 (1845: 24,4 | 1846: 28,5)

Lashes Inflicted Average: 25,33 (1845: 24,4 | 1846: 26,5)

Years

1839:

Number of Punishments: 2,007

Number of Lashes: 59,341

Highest: 60 | Lowest: 3

1840:

Number of Punishments: 2,026

Number of Lashes: 60,302

Highest: 48 | Lowest: 1

1841:

Number of Punishments: 2,066

Number of Lashes: 61,669

Highest: 50 | Lowest: 2

1842:

Number of Punishments: 2,472

Number of Lashes: 71,024

Highest: 100* | Lowest: 1

1843:

Number of Punishments: 2,170

Number of Lashes: 63,985

Highest: 60 | Lowest: 3

1844:

Number of Punishments: 1,411

Number of Lashes: 42,352

Highest: 72+ | Lowest: 6

1845:

Number of Punishments: 1,070

Number of Lashes: 33,511

Highest: 48 | Lowest: 3

1846:

Number of Punishments: 1,077

Number of Lashes: 32,360

Highest: 50* | Lowest: 3

*By sentence of a Court Martial.

+By order of the Commander-in-Chief for theft, in a shop at Chusan, and violence to the natives.

Sources: A "RETURN of the CORPORAL PUNISHMENTS inflicted in the ROYAL NAVY, in each of the Years 1839 to 1843, both inclusive, stating the highest and lowest Number of Lashes at each Time, and the aggregate Number of Lashes in each Year [ ]" Parlamentary Paper, Number 308, of Session 1845 + ABSTRACT of Total Numbers of CORPORAL PUNISHMENTS inflicted in the NAVY, and the Total Number of LASHES Inflicted, in each Year up to the 31st December 1846 [ ]. Parlamentary Paper, Number 661, of Session 1847

[gasps-inspanish]

The Math Isn't Mathing

TW for mentions of underage drinking

It was the annual League check in, a big meeting they had once a fiscal year at the Hall of Justice to check in on everyone’s progress and how things were going after major events that happened that year. Usually Kyle didn’t like League meetings, but having his friends who weren’t 10+ years older than him around for this meeting always meant that the meeting wouldn’t be boring.

Kyle was standing near Hal’s seat with the rest of the Lanterns (besides Guy who was sitting down on the floor next to Hal) minus Kelli who was with Titans West. The Titans and Young Justice were also in attendance and standing with one another, and the people who didn’t have a team but were still in the League database stood behind the League member they correspond to or near a friend.

“- a painting that went missing in -xx23” Bruce was saying.

“What does a painting that went missing 40 years ago have to do with our case?” Gar asked. Rachel side eyed him and Kyle almost laughed at the severity of it.

“So, that's not how math works,” Vic pointed out and Gar rolled his eyes.

“It’s close enough,” Gar argued. Rachel’s side eye continued.

“No it’s not? It’s 24 years, you're off by like 16 years” Jaime pointed out. Tim shook his head.

“No it’s not,” Tim said and Kyle was confused.

“Yes it is? I was born that year and I’m 24” Kyle said. Tim started to analyze him and Kyle was unsettled. He wasn’t that dumb, damn Tim, but Kon was also looking at him weird.

“So I know how math works,” Wally said slowly, “and yours is off. Do you not know what year you were born”?

Hal reached back and tapped Kyle on his knee, a sign to be quiet. But Kyle wasn’t going to start another brawl with Wally in the Hall of Justice or anything, and Bruce (who was one of the few people that kind of cared about keeping the meeting on track) seemed interested, so Kyle didn’t get what Hal was so worried about.

“I know what year I was born Wally, I’m not stupid” Kyle argued. Hal tapped his knee again.

“You clearly are. You weren’t born in -xx23” Wally said. Next to him, Donna, Roy and Garth looked amused but Dick looked worried. Kyle didn’t get it. He and Wally talked like this all the time. Hal being worried about his nephew getting into a fight, okay sure, but Dick knew Kyle and Wally talked to each other like this.

“Wally, I know what year I was born. It’s basic information that I use all the time” Kyle said gently in order to hopefully placate Hal and Dick. It didn’t work, as Dick still looked worried and Hal tapped his knee again.

“You were born in August” Wally said slowly, as if Kyle was stupid or a child, and that kinda pissed Kyle off.

“Yeah” Kyle said in the same condescending tone. “August 20th, -xx23. I just turned 24”.

Hal tapped him again and Guy slapped Hal’s away.

“Let him talk Hal, he’s a big boy,” Guy said quietly. His laugh and tone of excitement made Kyle realize something was wrong.

“That’s impossible. That was the year I was born and I was born in January while you were born in August so you would be a grade behind me instead of being in the same grade as me. Meaning you would have just turned-” Wally paused, eyes widening in horror when he saw Kyle had gone rigid, realizing his mistake. “Wait a fucking second, have you been lying about your age”?

Kyle opened his mouth and then closed it. He was stupid after all. He had been so used to saying he was a year older than he was, that he hadn’t realized his birth year didn’t match. He has been lying about his age for so long that he completely forgot he lied about his age in general. Everyone was looking at him now.

“I tried to warn you,” Hal said, looking up at Kyle in amusement.

“How old are you really?” Wally asked desperately.

“Older or younger?” Roy joined in as Donna looked panicked.

“Why would you lie about your age and what year you were born, but not make sure that they worked together?” Bart asked, his hand raised. Next to him, Kon was looking between Kyle and Wally like one would a fucking tennis match.

“I didn’t lie about the year. Just now” Kyle said weakly and Wally bent over, hands on his knees as if he needed to catch his breath.

“Waitwaitwaitwaitwaitwait” Wally cried out before looking back up at Kyle. “You’re a year younger than me?”?

“No! I mean, we’re the same age until your birthday in January?” Kyle offered and Donna relaxed.

“NO-”!

Guy burst out laughing at the same time as Kon. Guy had accidentally found out a bit after Hal, and Kon didn’t know until now and was just a brat. Connor was whispering something to Kelli who had this look on her face that made Kyle nervous.

Well, more nervous.

“I told you,” Hal said again. “Play stupid games”?

“Win stupid prizes” Kyle said sadly, hanging his head in defeat.

“Wait a second, you were 20 when you became a Lantern” John asked and Kyle nodded. “Kyle, we went out drinking with you plenty of times before August -xxx2”.

“Yeah” Barry said, wide eyes with realization. “We did too”!

Uh oh.

“Who cares about that!” Wally said. “He was 20 when he joined the League! You guys said you have to be 21”!

Oh yeah, Wally hadn’t been old enough to join the League when he first became Flash, having still been 20. He was never officially a part of the League team since Barry “came back to life” before he turned 21, but Kyle had said he was 21, so he was an official Justice League team member. And Wally wasn’t.

It was a beautiful day.

“Revoke his membership” Roy joked.

“I agree!” Wally said seriously, looking like he was losing his mind. “I’m older than you? You graduated university before me”?

Maybe this wasn’t such a bad thing.

“You have a fake id,” John said looking very amused. Kyle nodded. “Let me see it”.

Oh wait no, this actually sucked.

Kyle pulled it out and handed it over, and the other Lanterns and Barry and Ollie (who sat on either side of Hal) looked at it.

“I mean- it looks legit” Simon offered and Jess laughed, not even bothering to cover her mouth with her hand as she elbowed a smirking Jo lightly..

“I don’t use it anymore” Kyle said and Guy slung an arm around Kyle.

“I’m gonna make fun of you for this for the rest of your life” Guy laughed, despite the fact that he already knew. Kyle rolled his eyes.

“Yeah, I saw that one coming”.

“And you knew about this,” John said to Hal. At least if Kyle was going down, Hal was going down with him.

“Yeah, but he stopped after I found out. That’s why we didn’t go bar hopping as a group for like, four months” Hal said. John looked like he was trying not to laugh and Kyle knew he was done for. He was getting bullied for the next month at least. He hated them all. “What?“

“Hal. He still went drinking, just when you weren’t there” John said. Hal paused before doing a slow turn in his chair, looking offended.

Kyle knew Hal was more offended that Kyle was hiding it from him than him actually doing it. Oh also there was the reason Hal recommended Kyle didn’t drink for a bit, but Kyle didn’t want to think about that.

“Would you believe me if I said it was an accident?” Kyle asked sheepishly. Wally was leaning on a laughing Garth, wailing at the unfairness of the world and something about Kyle graduating University way before him, but Kyle couldn’t even enjoy it.

He would never live this down.

Notes

[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.

[shiryawashere]

television executive listen. I propose. woman. woman long dark hair, with sword... woman gets sword, yes? woman stab kill murder maim slice decapitate, with sword. mabbe feel bad about it sometime but ☝️ not all time. television executive with me yes? then, woman blonde girlfriend they kiss lots. downside is no. woman sword girlfriend

[aurora-uwu]

1 is the only number. Fight me.

[up-the-anti]

thinking about tmnt x alice in borderland ft donnies playing the king of diamonds

just cause the suits align with the brothers really well

[roseofhybrids]

keep the egg warm, teach it math