The Most Underrated Concept in Number Theory

This is probably my favorite video I've made yet. It's about an underrated mathematical concept known as "integer complexity" and my personal journey to discover it.
0:00 - Introduction
1:20 - A Mathematical Question I Stumbled Into
3:23 - Discoveries Among the First Dozen Numbers
6:49 - What is the Largest Number We Can Build?
11:19 - Number Webs With Mysterious Gaps
13:54 - Incorporating Subtraction and Division
17:23 - How I Found the Name of This Concept
21:00 - Further Directions We Could Take This
24:40 - A Philosophical Question I Stumbled Into
27:27 - Outroduction
Clarifications/corrections:
- When I draw the web of which numbers you can build with eight ones I forgot to draw a dot for 14, which should also be on that web.
- During one of the whiteboard b-roll shots, it showed 4 as being built by "1+1+1+1+1" which is one too many ones.
Special thanks to all of my supporters on Patreon! (Supporting the show not only helps me keep improving my content, but also lets me avoid needing to incorporate any product placements from brands in episodes)
Evan Clark, Max, George Carozzi, Peter Offut, Tybie Fitzhugh, Henry Spencer, Mitch Harding, YbabFlow, Joseph Rissler, Plenty W, Quinn Moyer, Julius 420, Philip Rogers, Ilmori Fajt, Brandon, August Taub, Ira Sanborn, Matthew Chudleigh, Cornelis Van Der Bent, Craig Butz, Mark S, Thorbjorn M H, Mathias Ermatinger, Edward Clarke, and Christopher Masto, Joshua S, Joost Doesberg, Adam, Chris Reisenbichler, Stan Seibert, Izeck, Beugul, OmegaRogue, Florian, William Hawkes, Michael Friemann, Claudio Fanelli, The Green Way, Julian Zassenhaus, Bailey Douglass, Jan Bosenberg, Brooks Boutwell, David Irvine, qe, George Sharabidze, Jack Dwyer, Fredrik, Dave Brondsema, and Chandler Smith!
If you want to help support this channel (and get your name on that list) and get some bonus content, check out the Patreon here: / comboclass
Combo Class Discord server: / discord
Subreddit: / comboclass
If you want to try to help with Combo Class in some way, or collaborate in some form, reach out at combouniversity(at)gmail(dot)com
In case anybody searches any of these terms to learn about them, some topics in this video include: number sequences, the OEIS (or "Online Encyclopedia of Integer Sequences"), Mahler-Popken "integer complexity" and extensions to it, my fun personal mathematical journeys, and more.
This episode was directed/edited/soundtracked by me (Domotro) and was filmed by Rishi Amutas and Carlo Trappenberg.
Disclaimer: Do NOT copy any dangerous-seeming actions you may see in this video, such as any actions related to fire.

[@ComboClass]

This is probably my favorite video I've made yet. It's about an underrated mathematical concept known as "integer complexity" and my personal journey to discover it. 0:00 - Introduction 1:20 - A Mathematical Question I Stumbled Into 3:23 - Discoveries Among the First Dozen Numbers 6:49 - What is the Largest Number We Can Build? 11:19 - Number Webs With Mysterious Gaps 13:54 - Incorporating Subtraction and Division 17:23 - How I Found the Name of this Concept 21:00 - Further Directions We Could Take This 24:40 - A Philosophical Question I Stumbled Into 27:27 - Outroduction (see video description for more links and info!)

[@johnjeffreys6440]

sounds like Kevin from Vsauce.

[@konstantinbachem9800]

here is a programm that generates all numbers with complexety 25 or lower by using multiplication or addition. maxlen=25 numbers=[(1,)] representation=dict() representation[1]=(1,"1") for w in range(2,maxlen+1): newnum=[] for i in range((len(numbers)+1)//2): a=numbers[i] b=numbers[-1-i] for x in a: for y in b: if x+y not in representation: n=x+y newnum.append(n) representation[n]=(w,f"({representation[x][1]}+{representation[y][1]})") if x*y not in representation: n=x*y newnum.append(n) representation[n]=(w,f"({representation[x][1]}*{representation[y][1]})") numbers.append(newnum) print(f"found {len(newnum)} new numbers of length {w}") for i in range(1000): if i in representation: length,r=representation[i] print(i,length,r) else: print(f"could not find {i}")

[@KenFullman]

I can't help wondering if this might form part of the proof of Collatz conjecture.

[@stickmcskunky4345]

Absolutely riveting! I have been digging into the sequence (of integer complexity) for a few days and found some cool stuff, but you brought up the e thing and I hadn't even realized that was why the (2^a)(3^b).. that's so cool.

[@daniel_77.]

Hey you could try creating a video in numberphile!

[@infinitesimalperinfinitum]

I'm glad you're free-range, I'd be terrified to see what would happen if you were contained

[@TymexComputing]

The fires are locally unstable, if it starts it finishes long before it could start again :) lastly I've seen some math freethinker video about the planet diameter that would need to be to be Able to keep an alcohol vapour flame to keep up with sunset cycle or so on, Mr Mould

[@publiconions6313]

Lol!

[@lloydgush]

Industrial mathematicians aren't good for your arteries...

[@joefarrow1599]

Looks to me like he's been kept caged for a long time

[@adamsheaffer]

@@joefarrow1599 Chastity? 😳

[@undr3s1]

"Dad, why is there smoke in our neighbor's house?" "Damn it, it's that guy again"

[@BboyKeny]

"Why don't you stop him dad?" "Son, he might be reckless but his math checks out"

[@muskyoxes]

The whiteboard was retrieved from a future apocalypse

[@Teapotman2]

“it’s that guy again”

[@t_ylr]

​@@Teapotman2 like guess I can assume nobody died cause the videos out lol, but I was genuinely concerned about how big that fire was and how close it was to a structure 😅

[@xX_dash_Xx]

Most unfunny comment on this platform

[@EionBlue]

I fully believe that one day Domotro will discover some sort of eldritch mathematical concept that opens his mind to some horrific elder god which will drive him absolutely mad. And nobody will notice the difference.

[@defenestrated23]

I was about to say, how do you know that hasn't already happened?

[@user-ec8xg3yq3f]

That seems to have already happened

[@recursiveslacker7730]

“There's evidence in the arithmetic record that the study of formal systems reached a pernicious apex in the Long Before. Advancements made by mathematicians such as Russell, Gödel, Eisencruft, Atufu, Wheatgrass, and System Star contributed to the understanding of notions like undecidability, pointed regularism, and abyssalism. Upon reaching this minimal degree of mathematical maturity, equipped with sophisticated grammars, researchers set out to experiment with the limits of expressibility. They contrived bold research programs and galloped into the mathematical wood, unwitting of the dangers that brood there. The record is even scarcer than usual, due to the efforts of successive generations to obfuscate the venture. As best as I can gather, at some point in the course of inquiry, a theorist from a mathematical seminary called the Cupola formulated a conjecture on the fragility of formal semantics. The conjecture ripened to a broader theory, out of which spawned a formal system called the penumbra calculus. In the few fragments of texts that predate the obfuscation, it's stated that, in the penumbra calculus, certain theorems are provable, but are falsified upon the completion of their proofs. As much as this result is at odds with the systems of thought I've encountered in my own inquiries, I find little reason to doubt the veracity of the authors. Nevertheless, it's certainly a peculiar property. The Cupola theorist's results erupted into a grand investigation into the expressibility of the penumbra calculus. The conclusions were troubling. Pushing further, researchers constructed sister systems with alternate axioms. These systems were still more fragile, with the systems' inference rules themselves unraveling upon the completion of certain proofs. Convinced that their discoveries were made possible by some idiosyncrasy of self-awareness, but synchronously fearful of the implications of their results, some schools of theorists engineered complex automated deduction systems to probe boundary theorems and launched them into neutron stars. The outcome is undocumented, but the result convinced theorists across the Coven to abandon research and blacklist anyone who studied the penumbra calculus and its derivative systems. Peculiarly, support for the injunction was unanimous. Of note, even the spacefolder Ptoh agreed to abandon its investigation into the forbidden calculi from the reaches of its bleak star. Though the manner of its consent was not without controversy; to announce its accord, it inverted the color charge of quarks in a small region of space, causing a research station to collapse in on itself. Nonetheless, Ptoh's consent is testament to the degree of existential anxiety that could cause investigation into the penumbra calculus to go dark.” - On the Origins and Nature of the Dark Calculus

[@recursiveslacker7730]

Not even Ptoh would mess with the Penumbra Calculus.

[@harriehausenman8623]

@@recursiveslacker7730 All hail YOG-SOTHOTH !

[@soingpeirce]

This is the most flat earth theory vibe I've ever seen on a video that has actual substance

[@yurisich]

I imagine he also has an interest in the cosmos and geology. So technically he can introduce himself as someone who studies astronomy, crystals, and number theory. Bonus points if he switches out the lab coat for a woven hemp poncho.

[@DLBeatty]

@soingpeirce LOL What a wonderful way to describe it!

[@lukehuntington7983]

I read this reply and chuckled and then he said "threeven" and I understood it

[@Bozeman42]

The difference being that a flat-earther will never actually do the math.

[@ulti-mantis]

​@@yurisich not astronomy, "the stars"

[@anonemos]

I like to imagine that he's just going through his day and suddenly he thinks up a next sentence to say in his video, so he records it in any place where he is at that moment.

[@DLBeatty]

@anonemos, I kinda thot of it as steam punk b4 steam power was invented -- but looking at it thru a Picasso lens.

[@M42-Orion-Nebula]

1. You don't have enough clocks. 2. The "e" reveal was beautiful.

[@eric23232323]

I loved seeing the mini connect four in the middle of the clocks lounging on the chaise lounge.

[@johnnye87]

It had never occurred to me before that 3 is equal to both e and pi (to 0 decimal places).

[@jansustar4565]

I thought that the whole "i couldnt find the numbers in the millions since i wasnt a programmer" segment would lead to a Brillinat sponsor.

[@redpepper74]

Classic Brillinat

[@ArtArtisian]

Dark - Internet sure is cluttered now

[@RibusPQR]

"Now before I go into how I found out about numbers that large, let me tell you about" today's sponsor, Brillinat.

[@plasma2942]

Thank god it wasn't

[@matthewlennon6289]

Dimitro > Brilliant

[@lincolnuland5443]

How does this guy only have 40k subs!? He lit stuff on fire and then started talking about math.

[@gary.h.turner]

Well, there are people who think that's weird and therefore refuse to subscribe! 😱

[@Faroshkas]

What? Only 40k????

[@michaelneufeld4515]

His other channel has 200k

[@Faroshkas]

@@michaelneufeld4515 what other channel?

[@Faroshkas]

@@michaelneufeld4515 what other channel?? I didnt know he had another one

[@curtiswfranks]

The OEIS is one of the most-important websites ever.

[@alexpotts6520]

Worth remembering as well, while it's a website these days, it's actually far older than the web. For decades it was stored on paper in filing cabinets.

[@ConManAU]

Also don’t sleep on their superseeker function - send an appropriately formatted email with a sequence and it will not only check if it matches any existing entry but it will perform a variety of transformations to try to find a match.

[@harriehausenman8623]

Sloan would approve this comment 👍

[@TheBluverde]

Thanks to this website I finally found out that someone had already discovered the sequence "0, 1, 10, 2, 100, 11, 1000, 3, 20, 101, 10000, 12, 100000, 1001, 110, 4, 1000000, 21..." before me.

[@auztenz]

Watching the first minute; pls dont burn ur self.

[@servvo]

you must be new here,,, domotro never burns

[@profquiz1730]

don't worry, he's a highly trained professional

[@AlexBaklanov]

There's also a highly trained professional cameraman behind the camera that always have his hose ready to make the scene wet =)

[@eric23232323]

@@profquiz1730 professional what? /ferris

[@auztenz]

Im pretty sure he almost tripped

[@jvcmarc]

I was bizarrely discussing about this same thing with my cat the other day while high. However, I wasn't able to find anything about it on the internet and ended up forgetting it. It's such a mystique coincidence for you to have posted this video so close to those thoughts, thank you

[@Ring13Dad]

Math is so rad when you're high!

[@itismethatguy]

That's absolutely crazy

[@harriehausenman8623]

Smart cat!

[@joeblog2672]

Did you ask your cat at the time? Perhaps you missed something from him / her.

[@TheDJRiffin]

Never heard of "throdd" and "threeven" before, love it.

[@DigitalJedi]

I think it's fun to extend this to other numbers. For example with 4, you get the funny possibility that fodd sounds like quad, but every thing made of quads must be feven.

[@tinyturtle1898]

Its a funny way to describe dividing and taking the remainder. It has its own operator "modulo" and is defined as % in programming languages. So if your Integer has remainder 0 when divided by 3 its "threven", and remainder 1 is "throdd" but what is remainder 2 called?

[@linktristen5]

​@@tinyturtle1898pretty sure that is still throdd. The only threeven numbers are those evenly divisible by three. Just like the only even numbers are those evenly divisible by 2.

[@BridgeBum]

As far as I know they are terms he made up. He specifically defined "threeven" in a previous video as divisible by 3 but it is such a great portmanteau that no definition is required. Truly inspired.

[@kenthartig7065]

We found the new viewer

[@owdeezstrauz]

Friend: 📞 "Hey D, wanna come out tonight? Have some fun with people?" D: "No way, I'm figuring out how many 1's go into numbers tonight and for 2 weeks straight."

[@cdorman11]

That's the thing about being the most interesting person at a party. It takes weeks of experiences to be interesting for 15 minutes.

[@m3morizes]

On that last philosophical point: there's also this feeling that if the new thing you discovered hasn't been researched before, that it's perhaps too contrived or useless or uninteresting. It's not too difficult to discover a sequence that isn't on the OEIS, but you have to ask yourself if you were motivated purely by the math, or if you were motivated by finding the lowest hanging fruit of undiscovered (uninteresting) math. If something you discovered has been researched before, at least you know it's important enough to have been worthy of the time and effort of serious mathematicians. If not, you're left wondering how important your discovery really is, even if it is new. To analogize this to exploring islands, if you discover a new island, and it's extremely rich in natural resources, nature, hospitable for humans, spacious, etc., chances are, someone has already discovered the island, and people already live on it. If, however, you discover an island that truly no one has discovered before, chances are, it's probably just a small rock full of bird poop. Should you be excited in the first case or in the second case? The second case is your original discovery, but it's also less meaningful and useful than the first case, but the first case isn't your original discovery. Rather than be disappointed in either case, may as well be excited in both cases. Like you said, if it hasn't been discovered before, that's an awesome feeling. Even if the reason for it not having been discovered before is that it is somewhat contrived, it was clearly interesting enough for you to stumble across naturally. You should feel proud in that case. If the thing you discovered has been discovered before, and researched before, you should feel proud that your mathematical intuition is well-honed enough to tread the same paths that the giants before you have. You should also feel excited that you can skip to the front of the line. The foundations and path has been built for you, so you have no excuse to not rush to the frontier as fast as you can (by learning and studying and catching up on research) so you can start making new discoveries from a different (more developed, new) starting point.

[@josephsummer777]

Minor point: bird poop islands have been very important to humans, lucrative even.

[@josephsummer777]

Quoting: Seabird poop-sometimes called guano-was the “white gold” of fertilizers for humans for millennia. Rich in nitrogen and phosphorus from birds’ fish-based diets, the substance shaped trade routes and powered economies

[@m3morizes]

@@josephsummer777 I was half joking when I said bird poop because of that, but maybe that's a point in and of itself. Even what seems like the most useless of discoveries can turn out to be a life-changing resource.

[@rlstine4982]

I went from number theories to bird poop islands. Thank you, KZbin comments 🤗

[@MrWeebable]

Flawed logic. Implies both the valuable and worthless islands have all been discovered before you, but the worthless islands were just never claimed or inhabited. So the discovery wasn'tnovel in either case. However, the assumption is that somebody has been there before, which isn't necessary. Over time it becomes more unlikely but still possible to discover truly novel things, both valuable and worthless novel things. Some unclaimed sequence in OEIS may have never been found before you found it.

[@joshuazelinsky5213]

Hi! I'm one of the mathematicians who has done some work on Integer Complexity, especially work with Harry Altman. A few quick notes: A related fun open problem: do powers of 2 have the obvious complexity? That is for any n>1, is the complexity of 2^n just 2n? Also, I gave a version of the Gaussian problem as a research problem to a student group a while ago. There work is I believe still under review. Edit: Your thoughts about rationals are interesting. I think you are correct that that problem has not had much work (or at least if there is work on it, I don't know of it). To some extent, your 5/6 example seems to be taken advantage of "Egyptian fractions" which are ways of writing a number as the sum of fractions of the form 1/a for various distinct a. In particular, for 5/6, you are using that 1/2 + 1/3=5/6. Frequently it seems that using an efficient Egyptian fraction representation for a number will give rise to a low cost way of writing that fraction.

[@aadfg0]

For the open problem: yes assuming a strong version of the generalized Catalan conjecture. Conversely, if we can prove this then we have some control on the subcase |2^m - 3^n| = k in the conjecture.

[@joshuazelinsky5213]

@@aadfg0 Yes, they are closely related. It turns out also to be related to what the base 3 expansion of power of 2 can look like. The claim implies roughly speaking that powers of 2 cannot have disproportionately many zeros in their base 3 expansion.

[@Corvo-lh9iy]

bro finally got those chickens lol, lets go

[@nissantzvitovey]

This just showed up in my feed, first time watching one of your videos. As a math major myself, I truly admire your mad mathematician vibes, with your calculations done in the wold with no clear uses, the true calling of all mathematicians.

[@dynamotexan]

Just hang around with this guy. He started way back with much more simple principles and systems and it has been fun seeing the progression; from the maths and the mad mathematician

[@SirNobleIZH]

That revelation with e reminds me of when i realized a^b will always be greater than b^a so long as a is closer to e than b

[@Yora21]

Yeah, but you probably didn't pull the proof out of your coat.

[@yasin_karaaslan]

Thank you for filming this outside. It is strangely comforting to see surfaces illuminated by the sun or places covered in mud due to rain, maybe it's because I don't go out much

[@Maukustus]

really interesting how all of the low numbers feel like they have a pattern until you reach a big number and that pattern just shatters, incredible how common it is (talking about the primes being one bigger than the ones below)

[@ediza.8485]

Combo Class is the most underrated concept in youtube algorithm theory

[@aryst0krat]

This feels tailor-made for people with ADHD. There's never a moment to get bored because there's always something new on screen to pay attention to while listening.

[@ratiogmd]

Very entertaining. You're the Explosions&Fire of math

[@joecassidy2887]

math gremlin's back

[@travcollier]

More goblin than gremlin ;)

[@fairygoodmuller8065]

Ma, there's a weird cat outside. It's spouting maths at me, the weird fuckin thing

[@MrSleazey]

This guy also gets a good workout while making these videos; lots of hiking, climbing, carrying stuff, stomping out fires off camera.

[@MatesMonchis]

I was inspired to come up with the following generalization: Let the cost of one 1 be zero, the cost of addition be A and the cost of multiplication be B. What's the cheapest that we can buy an integer for? The integer complexity defined in the video is equivalent to letting A = B = 1 (and adding 1, as you set the cost of 1 to be one, while it would be zero for me). I think a very interesting case comes up when we set B = 0. The first few values of the cost of n are equal to the optimal values for something called "addition chains". The first value where they differ is n = 23. I think I might study this more in detail! Thank you for letting me know about this topic.

[@andrewbarth8157]

Small correction, but for the mysterious gaps at around 12:04, 14 has 8 one's constructed by 7*2, or ((1+1+1)(1+1)+1)(1+1). Great video.

[@theultimatereductionist7592]

I have studied the unique way to write a positive integer using only the prime function: p(n)= the n-th prime, and the integer 1. 2=p(1) = () 3=p(2)=p(p(1)) =(()) 4=2*2=p(1)*p(1) =()() 5=p(3)=p(p(p(1))) = ((())) 6=p(1)*p(p(1)) = ()(()) This can be mapped into the number of ways of Dyck numbers: counting the number of ways of a string of Xs and Os (X = left parentheses O = right parentheses) so that in any initial segment there are more Xs than Os.

[@quinn7894]

I discovered something very similar, where p(n) is indicated by wrapping the number in a circle: kzbin.info/www/bejne/eaPZhKCum52nhtU

[@pomtubes1205]

dude these shots are underratedly amazing

[@AshburnArmorerDan]

You've got Weird AL energy. I hope you recognize how big of a compliment that is.

[@videoDemon]

_for a messy guy - in a messy neighbourhood - you've sure got a clean lab coat_

[@alexneckoyami]

Disappointed you didn't go through with the haiku you started

[@do811]

*This* is what the internet is for in the area of math, just putting out there ideas & concepts you have encountered/thought of and hopefully eventually someone will find it and can think on it, give a new set of eyes. We can advance with mostly unheard of or new ideas that no one would have really taken the time to dive into, that would have been forgotten otherwise. Occasionally, inevitably someone will think of something new. Instead of forgetting about these things and just going “huh. Whatever.”, *document it* in some way. Even just write it down in a notes app or something, but preferably in a way that will make other people think about it.

[@X3MgamePlays]

16:30 Well yeah, you build up with multiplications of 2 and 3. Of course a division would cost even more one's. Moments later, WTF?! HOW??? Also, I expected you to mention something like "to the power of". (1+1)^(1+1+1) = 8 Which reduces the number of ones from 6 to 5. Great video. Happy THE squirrel is still around. Have you named it yet?

[@quentind1924]

Why would we stop to powers ? What about tetration or even stronger ones ? The question i had myself is about if substraction was allowed or not, since i’m pretty sure numbers 31 or 63 would be more optimized by taking (the formula for 32/64)-1 than the formula for 31/63 without substraction. Same for numbers like 62, that would be the most efficient by doing 31×2, with 31 having a substraction in it’s formula Edit: i literally wrote that when i was 1 minute before he was talking to it, nvm

[@ethanbottomley-mason8447]

Division is helpful since if you have your number, call it n, and you are trying to make it, then there are four basic ways, you write n = a * b, a + b, a - b, a/b. If division is ever useful, then some number needs to have a minimal cost of the form n = a/b. Now how can this happen? You want a number n such that bn has a small cost. If bn is very close to a number of the form 3^k, then its cost will be small. In fact, I would not be surprised if the cost of numbers of the form (3^k+1)/2 have this as their form with the least cost for k sufficiently large.

[@gaborszarka7596]

@@quentind1924 when you introduce new operations, information is added to the recipe, therefore the complexity must go down

[@joshuasims5421]

This video was enthralling, I'm so glad you survived filming it. But most of all, your philosophical point at the end was important for anyone working in research.

[@its_elkku135]

Your style is wildly creative and entertaining. I love it

[@fearstreak7462]

Bro wake up, new ComboClass just dropped

[@ilovecats_og]

WAKE UP! -ITS THE FIRST OF THE MONTH- COMBO CLASS UPLOADED!

[@TymexComputing]

I also love it, i just dont accept that arsonistic approach

[@ilovecats_og]

@@TymexComputing ok, i get it

[@BenWard29]

Nope. I’m getting a restraining order. This comment causes cancer.

[@ElusiveEllie]

The fact that you just found some interesting little problem, worked out some solutions, and struggled to find out other ways people had experimented with this idea until you found the OEIS, and THEN you learned that plugging in your solutions brought you to this whole concept... It is so freaking cool and I'm so happy for you that you found it like this. This is insanely fun!

[@J0R1AN]

Incredible coincidence that this video found its way into my recommendations. A few weeks ago I encountered exactly this problem while working on an idea about bypassing some filter in a program that blocked digits, but allowed string like ‘true’. By using things like ‘true+true+true’ which equals 3, I could create numbers that I needed. But it would take a lot of text to create, say, the number 100. I realized that through multiplication and parentheses I could create larger numbers more quickly but had to figure out an algorithm to generate these. It ended up being a pretty messy brute-force algorithm but I could generate the equations for the numbers I needed with exactly this idea!

[@EngineerNick]

Your intros are just fantastic

[@memyselfishness]

This was a rather interesting concept. It reminds me a bit of floating point arithmetic. In fact, in many ways, your question about non-integer numbers is a question of floating point representations. Floating point might not be the right word to use, but I think you might understand what I'm saying.

[@tinkeringtim7999]

This guy has totally embraced his neurodiversity in engaging people and I'm so here for it! Brilliantly crafted content and delivery. Subscribed.

[@defenestrated23]

I have a very similar flavor of spicy brain (light stuff on fire and geek out about math) so this channel is pure gold

[@tinkeringtim7999]

@@defenestrated23 Me too, although I think I have a twist that prevents me from being able to actually string those clips together into a whole thing other people can see.

[@thecoolv130]

Lol I could be wrong but as far as I know this guy never said anything about being neurodivergent. Maybe he makes a video like this because he has a unique personality and perspective on the world, just like anyone else. Funny how we can essentially call people mentally ill with no blowback using this new PC word “neurodiversity”.

[@tinkeringtim7999]

@thecoolv130 He doesn't have to say it to be true. It's more common than you think for people to not realise or find out until they're 40. He's either a truly incredible actor or he is on the spectrum, heck he's the kind of person one would be better off calibrating the tests to!

[@MarceccMC]

Ffs

[@harriehausenman8623]

The way you inspire curiosity - brilliant as ever! 🤗 Especially how you show genuine research and exploration is fun ! 🥳

[@andrewdemos3009]

you're my hero part of the reason I'm going back for my bachelors in mathematics!!

[@user-rh5lk8xh3s]

I really liked it. Super interesting. This channel is underrated. Good job man

[@elitettelbach4247]

Absolutely loving the aesthetics and contents of this video! And now I'm considering the implications of the terms threven and throdd.

[@who.u]

This reminds me of some recent homework I was doing for computer science in university! First we had to write a dialect of the esoteric language P′′ (P double prime) in C. Then we had to write code in this language to do something basic like add two numbers together or print a message to terminal. Each character of the code is a single instruction/operation such as incrementing or decrementing by 1, or performing something in a loop (functionally the same as parentheses for multiplication). Long story short, I went down a bit of a rabbit hole trying to optimise my program to make it as short as possible and I found similar patterns to yours, especially around Euler's number.

[@DLBeatty]

Love your persona with the unpolished surreal props and backgrounds!

[@brianpalmer967]

This video was waaaay better than I thought it would be. Never seen this guy before, but I'm officially subscribed~

[@DDDTTT-jm9dr]

It's crazy to see a video on my home page of the almost exact topic that I was investigating yesterday. I was trying to find an answer to the problem: What is the minimum of ones that I need to use to form all the numbers? (using addition, multiplication, exponentiation, tetration...) Great video!

[@CHagen240]

so many vibey creative choices in the presentation. love it.

[@alexdamman6805]

Dom keep going! I love your work. This complexity subject is very promising.

[@louisrustenholz7642]

This is super interesting! This sounds a bit related to height functions in diophantine geometry, where they are used to quantify the "complexities" of solutions (complexities of rationals, Weil complexity of algebraic numbers, etc.), although integer complexity seems to have a more "combinatorics" feel to it. I wonder if the connection goes beyond face value, that would be super cool!

[@Rapnnex]

Love your enthusiasm for presenting obscure mathematics with the most chaotic energy possible. Changing the set of available operations from {+, *} to instead include all hyperoperations {+, *, ^, tetration, pentation, ...} might be an interesting way to extend the topic, to me it "feels" less arbitrary than cutting off the set of operations off after the first two.

[@WildEngineering]

i dont think ive ever witnessed a video like this before, you got me hooked

[@NoPodcastsHere]

Really powerful point about being grateful for those who have already solved the problems we're working on. I think the fire and destruction is probably unnecessary, what you do is already cool enough.

[@robbo415]

Absolutely loved this episode - the math and the locations!

[@toferg.8264]

Wow! I feel like each number has had a tarp over it my whole life, and you lifted the tarp, letting me see the framework inside. Thank you!

[@1234567zeek]

Thank you very much! This was top-notch unique content. Nice work!

[@jordanrodrigues1279]

Have you considered doing a collab with Explosions and Fire? Something like "doubling cubanes"?

[@emilyrln]

They are the perfect chaos power duo!

[@nivolord]

Nice! Integer complexity is something I have been thinking about for a long time, and is fascinating. I often ponder what should be the "cost" of multiplication, in bigger families of operations including exponentiation and up. The helpfulness of subtraction in reducing complexity is both a marvel and a menace when trying to fix a set of operations. Thank you for this great video! Edit: Let me just mention that my introduction into this topic was via Kolmogrov Complexity, which studies more generally how much resources it takes to describe an object. Integer complexity seems to be a precursor to this wider topic of complexity of mathematical objects.

[@mcmakers8850]

Giving me Explosions and Fire energy. Glad to be a new subscriber.

[@n20games52]

Loved the video and look forward to investigating the channel more.

[@mhm6421]

10:35 For anyone seeking explanation, imagine you split 10 into x pieces. In this case he set x=4 which results in (2.5)(2.5)(2.5)(2.5)=(2.5)^4 Using this pattern we can define this as: f(x) = (a/x)^x =e^(x ln(a/x)) (The variable a is the amount of ones we can use, in this case it would be 10) Which if you derive you get: a^x (1/x)^x (ln(1/x) - 1 + ln(a)) = 0 First two cannot be 0 since x is a real number -ln(x) - 1 + ln(a) = 0 x=e^(ln(a)-1) x=a/e But remember that we split "a" ones into "x" pieces So every piece would have the value a/x, plugging x=a/e You get that it should be exactly e!

[@syfp4769]

I ran into this concept in a coding project I did a few years ago. Cool to see someone else has researched it

[@Harrold251]

Y'know I've been watching your channel for a few years, it always very enjoyable having maths explained by a dishevelled madman. . . . I've only just realised you only have 40K subs, WTH?

[@eric23232323]

DUDE. I just found this video, first of yours, I'm subscribed now to both of your channels. LOVE! I'm only half through (we're on a dinner break) and I'm on the edge of my seat for the ending. Your delivery and presentation is TOPS. Love it so much. FWIW you remind me of my (decidedly off the beaten path) fraternity at RHIT. I'm sure you and they would get along like gangbusters. I'm only sad to see you don't have a hundred or more videos for me to watch. I'll have to patiently drag it out and watch one a week. Do you have a patreon? WHY NOT? :-) (PS. Auto-correct changed that to "Do you have a pattern?" Gold.)

[@pennywerner9192]

He has a Patreon

[@ambu.6707]

rly interesting how this method is used to construct integers irrespective of bases. rly looking forward to the video on eulers constant. its role in number bases is rly intriguing and im excited to learn more

[@xgozulx]

the video set is so crazy, i love it

[@haukzi]

Complexity in comp sci is a measure for the amount needed of some abstract resource. That can be time, space in bits, area for a circuit, circuit depth, gate count, edges in a graph, or how many "ones" you need for integers.

[@standard_limbo]

Just want to say, you are the most chaotic math KZbinr and I love your stuff man

[@CruseCtrl]

I'm loving these new locations!

[@uomodibassamorale]

your best episode so far... bravo!

[@porglezomp7235]

The most satisfying situation I got to was when I was doing some personal programming languages research during university and got to the point where some of the questions I was coming up with were things that people were only investigating in the past few years. I could still go look at other people's thoughts about it, but it was confirmation that I really was approaching new stuff.

[@Juttutin]

I'm super pleased with the algorithm for putting this video of yours into my feed.

[@pancholopez6892]

You can extend this concept by adding ones and twos to get a number, for example 2 needs 1 number, thats 2, 3 needs 2 numbers 1+2, and so on. Also you can keep adding numbers, like 3,4,5, etc. Great video!

[@vladodrobny7506]

This is so fresh and real, I love it.

[@onlyguitar1001]

This is cool. I always had trouble making sense of shuffling patterns. In high school I would shuffle poker chips and looked at how many times it takes for a stack to return to it's original permutation when doing perfect shuffles. I made a small program on my calculator to do this but I never understood the math behind it (I think it's some group theory stuff). I didn't pursue mathematics but I think I might learn more about this from the OEIS.

[@friedrichgwinner8888]

I'm so happy I clicked on this video. You are such a cool guy, I am gonna watch all your videos now

[@CliffSedge-nu5fv]

I remember doing something very similar to this when designing a game that uses dice and cards to make various numbers. It had a simple "make a big number to beat the target" premise, but there were a few ways to get a bonus to add on or multiply your number, so I tried to estimate the probabilities of beating different targets considering the different combinations of addition and multiplication possible depending on what bonuses the player had. I ended up making a similar looking list of different expressions that totaled the same amount.

[@ariaden]

One possible variation: Given an ordered sequence of N rational numbers, find a "shortest" set of polynomial equations (only allowed coefficients are 1 and 0, so instead of "1*x*y" you write "x*y" and you skip "0*x*y", but you can implement subtraction by moving terms to the other side of an equation) such that the set of equations has a unique solution with the first N unknowns having the values from the given sequence. Example for sequence 6, 11 can be this set of equations (unknowns x, y, z, w, u, v): v = 1 z = v + 1 w = z + 1 x = z * w u = x * w y + 1 = u

[@frantisekjanecek1641]

4:14 The "cost" of number 4 is 4, but the example for number 4 in the column has 5 ones.

[@bennik8845]

thank god this man is just a human

[@lyrimetacurl0]

in the thumbnail it's (1+1)(1+1)

[@DLBeatty]

The extra '1' was just a tip that I left the good doc.

[@quentind1924]

It’s obviously an editing mistake, it doesn’t even add up to 4

[@frantisekjanecek1641]

@@quentind1924 Of course, this is a very silly little mistake. I am learning english and my math background is not the best, so I am happy for any mistakes I find. The videos from this channel are great.

[@jackwightman3783]

Seen 20 seconds of the video, subscribed as I knew this was a channel for me

[@HaileISela]

that was a fascinating topic and i love the overtones of resonance between this and synergetics and the dozenal numbering i have tuned into i suppose i could see how in spherical terms one sphere as a unit, two as the unit pair and three as the unit eye, triangle, would be the respective relationships. four would of course be unique in being the foureyes with a unit volume, and six, dozen, and so on and so forth. thank you for that beautiful insight

[@maynardtrendle820]

I love to see the OEIS coming up in casual instruction!🤴🏼

[@dev_X_]

Very insightful. thank you

[@JerusalemStrayCat]

Your use of the word "efficient" at 8:28 when you were talking about the relationship between integer complexity and the number _e_ made me think of radix economy (or whatever it's called - the ratio between the length of a number and its base). The most "economical" base is _e_, so maybe that is related to integer complexity somehow? Granted, _e_ shows up everywhere, so it may be a coincidence. Also, I really feel the bit at the end about "discovering" something that has already been researched. I'm a research intern working on a project in a field that i don't know much about, and I recently found a published paper on pretty much the same experiment I was doing. It was basically the exact paper I am trying to write except, y'know, better. However, the silver lining is that I have to think more deeply about my research to come up with something it has or does that isn't in the other already-published work, and I'll come away with a better understanding of what we're trying to study.

[@CaptainWizard3000]

Looking forward to your upcoming video about games!

[@Catt0]

This was extremely interesting!

[@kittybeans8192]

We can reduce the tools in our toolset too, just as you added the ability to subtract and divide, we can take away multiplication, parentheses, and even addition. Even with such limitations, we can still build the number 1. I dunno why, it seems kinda trivial, and yet, feels kinda beautiful to me.

[@joyce8120]

reminds me of the Fundamental Theorem of Arithmetic. fascinating video

[@user-fo7fd6yj9p]

Your mind seems wildly tempered in wonderful ways, I would love to sit down and talk shop about large number theory and complex relative maths. The video was a wild ride and is worth recommending to a friend.

[@user-fo7fd6yj9p]

Additively I'd like to give you another indexing idea that I think will lead to the future of number theory, if Pi is infinite we could theoretically have all numbers derived from Pi and using either their integer value to return different values (0 gives 3, 1 gives 1, 2 gives 4 but say 0~2 would be 14) to eventually have computers look at all numbers as their related index of pi, depth of relation of numbers and since it goes on indefinitely you could get all values eventually when they are all tallied.

[@gabrielperry1716]

In computer science, there is a similar idea that relates to entropy or the amount of information stored in a sequence of 0/1 bits. It's closely related to information compression like zip files. Basically, it's a measure of how quickly you can describe an integer, specifically how many symbols you need to communicate or encode that integer. It's also related to first-order logic, there's a few neat videos on Rayo's number, particularly by Numberphile. Rayo's number is the smallest number that is bigger than any number you can describe using a google symbols (in a specific logic notation, first-order set theory).

[@joeblog2672]

Cool topic. I can't recall if I have come across this before. Certainly not in school. Such a tantalizing thing mathematics is. A brilliant universal language that taunts you into believing that all of its universe of subtopics possess some degree of real world relevance even when most show no sign whatsoever!

[@mjohnson2807]

11:11 e goes into 10 about 3.678 times, so if you multiple e to that power, you get a maximum of 39.5986256445

[@WriteWordsMakeMagic]

Fascinating video. I wonder what your experience was finding the sequence online after thinking at first that it was something that was just in your head. I had a similar experience when investigating grafting numbers

[@WriteWordsMakeMagic]

Just realized the last section of the video covered this question, haha

[@fireisplays6074]

i love ur channel

[@electra_]

I'm really curious to hear how you rediscovered this while desining a game - it reminds me of something I was working on.

[@coltenh581]

Such a great one. Thanks

[@ConlangKrishna]

A similar question is: Which sizes of coins do we need to minimize the amount of coins that a person needs to carry to pay any price without geeting change back? Or put differently, what are the most effective coin sizes to minimize the number of coins in circulation? I have been playing around with a duodecimal system, where you have coins with values of 1, 3, 6, and 10 (meaning a dozen) of a currency. And then get to the next order of magnitude by multiplying by dozens, so you get 30 (three dozen), 60 dozen, and 100 (twelve dozen = one gross) of a currency. This would minimize the needed amount of coins to pay any price. Interestingly, as you mentioned e. There are currencies that use(d) 2.5 of a kind, e.g. the Dutch Guilder used 1, 2.5, and 5, before going to the next order of magnitude, 10, 25, and 50. Fascinating video!