The Most Underrated Concept in Number Theory

Рет қаралды 138,117

Күн бұрын

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, I showed 4 as being built by "1+1+1+1+1" which is one too many ones.
- Lots of comments mention that exponentiation would make things have lower costs. That's true, but I wasn't including exponentiation here, because we have to define the exact mix of operations we are using at each point, not take any operations for granted (like, if we automatically allow exponentiation, why not automatically allow tetration?) so I just began with addition/multiplication, then included subtraction, then included division. You could include exponentiation if you wanted, which would change costs of course, and I'm sure there's interesting patterns to find there too (there are some OEIS entries related to that on Glenn Whitney's table I mentioned).
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.

Пікірлер: 953

@ComboClass 2 ай бұрын

@johnjeffreys6440 2 ай бұрын

sounds like Kevin from Vsauce.

@konstantinbachem9800 2 ай бұрын

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 2 ай бұрын

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

@stickmcskunky4345 2 ай бұрын

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. 2 ай бұрын

Hey you could try creating a video in numberphile!

@infinitesimalperinfinitum 2 ай бұрын

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

@TymexComputing 2 ай бұрын

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 2 ай бұрын

Lol!

@lloydgush 2 ай бұрын

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

@joefarrow1599 2 ай бұрын

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

@adamsheaffer 2 ай бұрын

@@joefarrow1599 Chastity? 😳

@soingpeirce 2 ай бұрын

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

@yurisich 2 ай бұрын

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 2 ай бұрын

@soingpeirce LOL What a wonderful way to describe it!

@lukehuntington7983 2 ай бұрын

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

@Bozeman42 2 ай бұрын

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

@ulti-mantis 2 ай бұрын

@@yurisich not astronomy, "the stars"

@EionBlue 2 ай бұрын

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 2 ай бұрын

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

@user-ec8xg3yq3f 2 ай бұрын

That seems to have already happened

@recursiveslacker7730 2 ай бұрын

“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 2 ай бұрын

Not even Ptoh would mess with the Penumbra Calculus.

@harriehausenman8623 2 ай бұрын

@@recursiveslacker7730 All hail YOG-SOTHOTH !

@undr3s1 2 ай бұрын

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

@BboyKeny 2 ай бұрын

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

@muskyoxes 2 ай бұрын

The whiteboard was retrieved from a future apocalypse

@Teapotman2 2 ай бұрын

“it’s that guy again”

@t_ylr Ай бұрын

@@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 Ай бұрын

Most unfunny comment on this platform

@anoNEMOs 2 ай бұрын

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 2 ай бұрын

@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 2 ай бұрын

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

@eric23232323 2 ай бұрын

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

@johnnye87 Ай бұрын

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

@demonking86420 Ай бұрын

I demand e be renamed to "Markiplier number"

@lincolnuland5443 2 ай бұрын

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

@gary.h.turner 2 ай бұрын

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

@Faroshkas 2 ай бұрын

What? Only 40k????

@michaelneufeld4515 2 ай бұрын

His other channel has 200k

@Faroshkas 2 ай бұрын

@@michaelneufeld4515 what other channel?

@Faroshkas 2 ай бұрын

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

@jansustar4565 2 ай бұрын

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 2 ай бұрын

Classic Brillinat

@ArtArtisian 2 ай бұрын

Dark - Internet sure is cluttered now

@RibusPQR 2 ай бұрын

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

@plasma2942 2 ай бұрын

Thank god it wasn't

@matthewlennon6289 Ай бұрын

Dimitro > Brilliant

@TheDJRiffin 2 ай бұрын

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

@DigitalJedi 2 ай бұрын

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 2 ай бұрын

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 2 ай бұрын

@@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 2 ай бұрын

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 2 ай бұрын

We found the new viewer

@curtiswfranks 2 ай бұрын

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

@alexpotts6520 2 ай бұрын

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 2 ай бұрын

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 2 ай бұрын

Sloan would approve this comment 👍

@TheBluverde Ай бұрын

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.

@AdamVollmer 2 ай бұрын

Most sane Postdoc:

@jvcmarc 2 ай бұрын

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 2 ай бұрын

Math is so rad when you're high!

@itismethatguy 2 ай бұрын

That's absolutely crazy

@harriehausenman8623 2 ай бұрын

Smart cat!

@joeblog2672 Ай бұрын

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

@aryst0krat Ай бұрын

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.

@auztenz 2 ай бұрын

Watching the first minute; pls dont burn ur self.

@servvo 2 ай бұрын

you must be new here,,, domotro never burns

@profquiz1730 2 ай бұрын

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

@AlexBaklanov 2 ай бұрын

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

@eric23232323 2 ай бұрын

@@profquiz1730 professional what? /ferris

@auztenz 2 ай бұрын

Im pretty sure he almost tripped

@m3morizes 2 ай бұрын

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 2 ай бұрын

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

@josephsummer777 2 ай бұрын

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 2 ай бұрын

@@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 2 ай бұрын

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

@MrWeebable 2 ай бұрын

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.

@SirNobleIZH Ай бұрын

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 Ай бұрын

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

@TomasIngi00 14 күн бұрын

This doesn't quite hold in general (take 10^1 > 1^10 as a simple counterexample). In fact, with b > 2e - 1, you can always find a small enough number 0 < x < e - 1 such that (1 + x)^b < b^(1 + x) (note that since b > 2e - 1, we have |b - e| = b - e > 2e - 1 - e = e - 1 > e - 1 - x = e - (1 + x) = |(1 + x) - e|, so (1 + x) is closer to e than b is). this can be seen heuristically as the left hand side can be approximated by (1 + x)^a ≈ 1 + ax for small enough x, while the right hand side tends towards a as x tends towards 0.

@SirNobleIZH 14 күн бұрын

@TomasIngi00 yeah, 1 is the exception, thanks for the counterexample. That's what math is all about

@TomasIngi00 14 күн бұрын

@@SirNobleIZH That's true, if you only count integers (and include the equal case of 2^4 and 4^2), then the result holds. Other counterexamples if we allow non-integers are e.g. 1.5^5 < 5^1.5 (approx. 7.59 and 11.18, respectively)

@nissantzvitovey 2 ай бұрын

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 2 ай бұрын

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

@Corvo-lh9iy 2 ай бұрын

bro finally got those chickens lol, lets go

@joshuazelinsky5213 2 ай бұрын

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 2 ай бұрын

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 2 ай бұрын

@@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.

@dojelnotmyrealname4018 28 күн бұрын

I want to float the concept that if we're bringing in fractions, we're no longer in the realm of integers. So at that point we'd need to start talking about Rational Complexity, no? Also, using exponetiation, division and subtraction I can bring it into Complex Complexity with 6 1's to make an i, by writing it as (1-1-1)^(1/(1+1) I don't know how to get to the transcendentals though, so it'd be Complex but with gaps in the number lines. Which is a weird concept to think about.

@MrSleazey Ай бұрын

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

@owdeezstrauz 2 ай бұрын

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 Ай бұрын

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

@joecassidy2887 2 ай бұрын

math gremlin's back

@travcollier 2 ай бұрын

More goblin than gremlin ;)

@fairygoodmuller8065 2 ай бұрын

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

@ratiogmd 2 ай бұрын

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

@ediza.8485 2 ай бұрын

Combo Class is the most underrated concept in youtube algorithm theory

@yasin_karaaslan Ай бұрын

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

@pomtubes1205 2 ай бұрын

dude these shots are underratedly amazing

@J0R1AN Ай бұрын

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!

@Maukustus 2 ай бұрын

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)

@BigBadibou 10 күн бұрын

12:20 True lool 😂

@AshburnArmorerDan 2 ай бұрын

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

@EngineerNick 2 ай бұрын

Your intros are just fantastic

@its_elkku135 2 ай бұрын

Your style is wildly creative and entertaining. I love it

@MatesMonchis 2 ай бұрын

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.

@videoDemon 2 ай бұрын

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

@alexneckoyami Ай бұрын

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

@tinkeringtim7999 2 ай бұрын

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

@defenestrated23 2 ай бұрын

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 2 ай бұрын

@@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 2 ай бұрын

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 2 ай бұрын

@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 Ай бұрын

Ffs

@fearstreak7462 2 ай бұрын

Bro wake up, new ComboClass just dropped

@ilovecats_og 2 ай бұрын

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

@TymexComputing 2 ай бұрын

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

@ilovecats_og 2 ай бұрын

@@TymexComputing ok, i get it

@BenWard29 2 ай бұрын

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

@andrewbarth8157 2 ай бұрын

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.

@X3MgamePlays 2 ай бұрын

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 2 ай бұрын

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 2 ай бұрын

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 Ай бұрын

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

@joshuasims5421 2 ай бұрын

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.

@Hotsource 27 күн бұрын

It's so great to finally see a teaser for Return to Zork revamped in VR

@leadlime29 2 ай бұрын

congrats; you invented the french numbering system

@theultimatereductionist7592 2 ай бұрын

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 Ай бұрын

I discovered something very similar, where p(n) is indicated by wrapping the number in a circle: kzfaq.info/get/bejne/edimhtCpy57YgKc.html

@keithmasumoto9698 2 ай бұрын

It must have been extremely satisfying to find that 350 million number. And those other large numbers. Subbed.

@gabrielecammarata1515 Ай бұрын

Your content is unique, clear and entertaining. Great ob!

@mcmakers8850 2 ай бұрын

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

@memyselfishness 2 ай бұрын

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.

@frankmalenfant2828 Ай бұрын

The whole thing is dope. From the form to content, without forgetting the feeling of epiphany when you finally find the keywords to quenching your curiosity. 💯

@redchief94 24 күн бұрын

This intro is how I imagine most math experiments go. My numbers always explode in my face whenever I try to do math as well.

@jordanrodrigues1279 2 ай бұрын

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

@emilyrln 2 ай бұрын

They are the perfect chaos power duo!

@EvanEscher 2 ай бұрын

I'm glad this came up in my recommended

@andrewdemos3009 2 ай бұрын

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

@DDDTTT-jm9dr 2 ай бұрын

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!

@who.u 2 ай бұрын

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.

@do811 Ай бұрын

*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.

@user-rh5lk8xh3s 2 ай бұрын

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

@louisrustenholz7642 2 ай бұрын

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!

@jackwightman3783 2 ай бұрын

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

@shefsufla Ай бұрын

This bings back some ideas i had from my childhood i used to be much more curious back then

@harriehausenman8623 2 ай бұрын

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

@frantisekjanecek1641 2 ай бұрын

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

@bennik8845 2 ай бұрын

thank god this man is just a human

@lyrimetacurl0 2 ай бұрын

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

@DLBeatty 2 ай бұрын

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

@quentind1924 2 ай бұрын

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

@frantisekjanecek1641 2 ай бұрын

@@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.

@jasimmathsandphysics 2 ай бұрын

You have some amazing sets 😂

@emilyrln 2 ай бұрын

I love the basement-looking one with all the vines dangling in a column of sunlight 💕

@alexdamman6805 2 ай бұрын

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

@ElusiveEllie 2 ай бұрын

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!

@sunnymaster6968 2 ай бұрын

great video, but you should have cut 41 seconds to make the time the first 4 digits of e.

@galactika 2 ай бұрын

i love it!!! i can't believe that you checked 50,000,000 and it was only 221,174 larger! also, i relate SO HARD to the last section about the paradox of rediscovery! i definitely think i lean toward disappointment upon discovery when i know it's something i'd be able to fully understand and handle myself, and excited/relieved when it's a complex topic i want some help with

@ComboClass 2 ай бұрын

Glad you related! In case you're not joking about the 50,000,000 thing (or in case anyone else needs clarification) I should clarify that I did not check that many numbers myself. I checked about 50 numbers manually for different combinations of operations. The "I didn't check more than 50 million numbers" line was just saying that was how many numbers I would have had to check if I were to have found the next statistic myself.

@galactika 2 ай бұрын

@@ComboClass HAHA, gotcha. i should have figured that out based on you saying you aren't familiar with coding :)

@Fire_Axus 2 ай бұрын

your feelings are irrational

@elitettelbach4247 2 ай бұрын

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

@alejrandom6592 Ай бұрын

Hey man, I only knew you for your shorts, I didn't know you actually knew math. Let me tell you, this video is amazing, I've never been interested in number theory until now. Keep it up ❤

@zeldaandTwink 2 ай бұрын

I'd like to see the same concept but allowing exponents as well 8= (1+1)^(1+1+1)

@DLBeatty 2 ай бұрын

And.... 4 as (1+1)^(1+1) still cost four! 😮

@JanJeronimus 2 ай бұрын

Adding more functions results in needing less 1s. You can even say that for every number y there even exists a functions Fn so that Fn(1) = y Allowing more functions reduces the costs at the number side. However it increases the costs at the function side. So you need to evaluate the total costs as a sum of the number side and the costs of the amount of functions you need. However you can not simply add them as it are different things and the costs of a number is not always per definition the same as the costs of a function. Regarding e.g prime numbers there is a function Fp(x) that gives the x-th prime number. So even if you don't know the next value in a sequence you already can descibe it using a function.

@APaleDot 2 ай бұрын

Yeah, reminds me of Kolmogorov Complexity.

@CruseCtrl Ай бұрын

I'm loving these new locations!

@DLBeatty 2 ай бұрын

Love your persona with the unpolished surreal props and backgrounds!

@kamikeserpentail3778 2 ай бұрын

My thoughts just go toward continuing up toward bigger operations. Exponents, tetration. How does the integer complexity change when those are considered. Like if 9 is (1+1+1)(1+1+1) with just addition and multiplication, but is (1+1+1)^(1+1) with exponents.

@ComboClass 2 ай бұрын

Some of the sequences on the OEIS table I mentioned in the episode include exponentiation in that way. Not sure if tetration has been considered.

@elf835 2 ай бұрын

@@ComboClassnow where getting into some big numbers

@kyle30312 2 ай бұрын

The cost for 11 is 7 if exponents are allowed: (1+1+1)^(1+1)+(1+1)

@elf835 2 ай бұрын

@@ComboClass 27 would become (1+1+1) tetrated by (1+1) and thus the cost would be 5

@alganpokemon905 2 ай бұрын

i totally love problems in math where it's a completely 'natural' question - something you'd actually start wondering while on the toilet another example includes: does every closed curve contain four points that can be connected to form a square?

@TymexComputing 2 ай бұрын

Or the other partitioning of THE ONE problems ;). I think the answer to this one is NO, I am thinking about a roman arc with widening side walls and with skewed flat floor and I can't find the vertically symmetrical square but maybe tgere is some skewed square ,didn't check all possible squares :)

@alganpokemon905 2 ай бұрын

@@TymexComputing So like a capital delta symbol, but the top part is arced? This definitely permits a square!

@TymexComputing 2 ай бұрын

@@alganpokemon905 Yes like that but also the floor is not horizontal but straight, skewed base at delta with top arc.

@Fire_Axus 2 ай бұрын

your feelings are irrational

@lforc25 2 ай бұрын

if you modify it slightly so that the square and the curve don't have infinitely many intersections, you can cut a circle in half and connect the two halves with parallel lines

@1234567zeek 2 ай бұрын

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

@brianpalmer967 2 ай бұрын

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

@definitelynotdolce2565 2 ай бұрын

There's something I've been thinking about ever since your video on what operations really were. Sequence/Counting > Adding (Counting in groups) > Multiplying (Adding in groups) > Exponentiation(Multiplying in groups) etc And originally I assumed that going the other way would work the same, but it breaks down by division. Division isn't subtracting in groups, hell sometimes the answer is bigger than the factors. So then, what IS division? The explanation of it just being multiplication's inverse is super unsatisfying, and I can't help but think there's more to it. It isn't subtracting in groups, but it is still grouping. Only it groups the answer rather than the factors. Would love to hear more about the subject if you're able!

@emilyrln 2 ай бұрын

I second this video request!

@stevenkinna4085 2 ай бұрын

I was looking at the same thing recently, was having ChatGPT start by explaining one and zero to me, see how far I could get. When it came to division, it actually popped out something that is entirely relevant to this question. I was specifically questioning this very thing, why division is the inverse of multiplication, and why we can't divide by zero. It said something like division is specifically the process of asking the question 'how many times can I subtract a given number from the number I started with', and that's why we can't throw zero in, because there is no number of times you can subtract zero from another number to reach anything different. I've never seen it described exactly that way before, and it actually did

@legendgames128 2 ай бұрын

I might be second. (excluding yourself)

@robbo415 2 ай бұрын

Absolutely loved this episode - the math and the locations!

@CHagen240 Ай бұрын

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

@WildEngineering 2 ай бұрын

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

@lukion27 22 күн бұрын

I have actually thought a lot about the section "What is the Largest Number We Can Build?" before seeing this video. I also came to the conclusion that you would be using 2s and 3s a lot. Wonderful video for those who like number theory and pure mathematics!

@n20games52 2 ай бұрын

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

@xgozulx Ай бұрын

the video set is so crazy, i love it

@ambu.6707 Ай бұрын

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

@toferg.8264 Ай бұрын

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!

@Jango1989 Ай бұрын

This was amazing!

@TroyWarr1980 2 ай бұрын

I lost it when you said "throdd" with such conviction 😂

@jack002tuber 2 ай бұрын

I really hope that numberphile sees this video. They sit at a desk and talk for an hour. 😄

@standard_limbo 2 ай бұрын

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

@NoPodcastsHere 2 ай бұрын

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.

@asailijhijr Ай бұрын

When you're encountering a new (to you) computer problem, you definitely want to find that you're not alone in your searching.

@Harrold251 2 ай бұрын

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?

@Rapnnex Ай бұрын

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.

@uomodibassamorale 2 ай бұрын

your best episode so far... bravo!

@vladodrobny7506 2 ай бұрын

This is so fresh and real, I love it.

@friedrichgwinner8888 2 ай бұрын

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

@Boldra Ай бұрын

I haven't seen a maths video this original since I discovered Vihart many many years ago! Well done!

@ComboClass Ай бұрын

Thanks! :)

@netcat22 2 ай бұрын

This was extremely interesting!

@KarlFredrik Ай бұрын

This video wins the prize for the most props per scene

@kyanite2507 Ай бұрын

as a computer scientist i would be absolutely delighted to find a way to algorithmically define a bunch of the functions behind integer complexity, when i find the time im definitely going to write a paper about this!