The Rare Levels Beyond Exponents

[@ComboClass]

Wow, this video is randomly blowing up again. To anyone new, welcome! I hope you stick around and check out some of my more recent episodes here: www.youtube.com/@ComboClass/videos (I also have another channel @Domotro with livestreams and bonus videos)

[@D0w0ge]

Just found this video, can you do a video on the FGH/googlology? Love the way you present information, keep it up!

[@ComboClass]

@@D0w0ge At some point, yeah I'll make another episode(s) about massive numbers and the fast-growing heirarchy will probably be included

[@OneSentenceSummary]

It was recommended to me today for the first time.

[@jesuisbon]

Yes, Hello I am one of the new people that got recommended your video and It did worked in making me watch it completely and subscribe. Damn you KZbin algorithm god!

[@jneal1347]

❤

[@raimondssmiltins8910]

Its not often you find such a small scale content creator who's as interesting and educational as you are.

[@Spax_]

seconded

[@risingSisyphus]

The way you mix your perspectives in this sentence makes it sound like you're complimenting yourself lol

[@yuelia9937]

@@risingSisyphus lmao I thought I was the only one who might've interpreted like thay

[@soyjakchud]

true

[@shreyasbhatt7112]

@@risingSisyphus wow I found this too haha

[@steelegagnon5273]

i love how low the barrier to entry is to this. your video makes it so i don’t have to have a phd and 87 years of theoretical math experience to have fun exploring this weird concept. thank you!

[@sylv512]

this isn’t really all that advanced, though

[@julesssssssss]

@@sylv512 thats basically what Steele said..

[@orang1921]

@@julesssssssss sure, but the comment is implying that, without the simplification or method of explanation in the video, it would be difficult to approach

[@HerbaMachina]

@@sylv512 I think what he means is that the video really covers everything you need to know to be able to understand the concept, and even is a really intuitive way of describing limits without any precal knowledge. It's a really well rounded video, but yeah it is mostly high school math outside the rarely talked about topics which were the focus of the video.

[@sylv512]

@@HerbaMachina I'd say this video is around a middle school level. I'm not sure what High School is still teaching exponents in 9th grade or higher.

[@RGC_animation]

Fun Fact: One of the largest number ever created, the Graham's Number, is defined using Up Arrow Notations, although many many many times bigger than a tetration, it's actually 64 layers deep.

[@reizinhodojogo3956]

graham number is g(64) in graham sequence, even g(1) is bigger than tetration, tetration is 2 arrows or ^^, g(1) already haves 4 arrows ^^^^, g(1) is 3^^^^3 g(2) is same thing but with g(1) arrows g(3) is same thing but with g(2) arrows... g(64) is graham number, it haves g(63) arrows beetwen the threes

[@pe1900]

saying 64 layers is a bit misleading i think, it’s not 64 arrows between the numbers, it’s the results of the previous layers defining how many arrows are in the last, it’s just insane

[@markzambelli]

Graham's Number... ha... that puny number pales in comparison to Tree(3)

[@pe1900]

@@markzambelli tree(g64) enters the game

[@markzambelli]

@@pe1900 (my brain hurts..........)

[@MatthewPiercey]

Domotro seems like the rare type of teacher who'll go on a wild-yet-coherent tangent when somebody asks a question and the textbook answer just isn't satisfying enough. The kind of teacher who'd be liable to talk about stuff like the _forbidden fourth state of matter_ when somebody asks what's after gas 😁.

[@eroraf8637]

Sounds like my high school physics teacher. Good old Mr. Moon, the ADD pyro.

[@RandomAmbles]

"Technically there are 85."

[@apersonthatexists6722]

Sounds like a teacher I had back in 7th grade. I’ll always remember you, Mr Roach

[@eroraf8637]

@@apersonthatexists6722 Oof, talk about an unfortunate surname. Bet that was a fun first day...

[@evrenizzet5822]

Plasma is cool, it's for when the nuclei of the atom can no longer be held together by the nuclear strong force because of the heat. So, you just have a bunch of protons and neutrons floating around. However, we can add even more heat like in conditions found in CERN, a neutron star, or the beginnings of the universe wherein the heat overcomes the nuclear weak force (made possible by gluons) and rips the quarks away from each other leaving you with "quark matter"

[@DeJay7]

My man is ON THE GRIND! Please, keep it up, you're such a rare case, this was one of the most interesting, nice videos I've ever seen. Perfect energy, just right explanations, awesome. It's crazy how much effort you put into these videos for such a small community at the moment. Hope people find you and appreciate you.

[@jackgreenearth452]

Yes! I **knew** that this existed, but every time I tried to explain it - to my mother, to my maths teacher (with a similar method to you), they wouldn't understand, or wouldn't care. Thank you for showing me that this is a real thing!

[@akunog3665]

Well, to say math is a real thing is controversial :D. I would argue math is not a real thing, but rather a construct of the mind and really a bit arbitrary at times. I am of course kidding around by being over-literal, math is a beautiful thing, and exploration of it is empowering.

[@pyropulseIXXI]

@@akunog3665 The only thing that truly exists are constructs of the mind. Math is as real as it gets

[@akunog3665]

@@pyropulseIXXI I feel like there are things that independent observers could agree on. I don't think the laptop I'm typing on right now is merely a construct of the mind. Perhaps the laptop is a poor example. Lets take the earth itself. It exists independent of a mind. Even if no humans existed the earth would be here. However, If no humans existed our math would not exist, It can only exist within the mind. And if some alien race knows math, it is a different math. There are many paths we could have taken differently to get to the same place in mathematics, a series of choices we have made, not a pure discovery. It is an interesting topic to be sure. It's not merely make-believe like faeries or leprechauns, but it's also not fully real like the earth.

[@Terrapin22]

@@pyropulseIXXI the mind is just a construct of the mind

[@invertedmind8937]

i feel like every time i tried to ask something that wasn't about the current thing they were teaching to my math teacher i would just get ignored, and i'm not even a math enthusiast

[@spongebobbatteries]

@ 5:07, The "tetration tower: a to the a to the a to the a" slaps so hard with the beat and the piano melody. That was sick!!!

[@8bitflame853]

It really was!

[@dougtheguy101]

Facts I was looking for this comment

[@ZarHakkar]

I remember figuring these out in the 10th grade. I was super into math and realized there were sequences of increasing numbers that couldn't be represented by any operations I knew. Then I figured out tetration on my own and realized there were basically an infinite amount of operations "above" addition, multiplication, and exponentation. The numbers get pretty huge and unreasonably whack pretty quick, so I figured that's why they're not used all that often.

[@jackgreenearth452]

I also figured it out on my own. It makes you wonder - if after death we have no senses and all we can do is think, forever, will I eventually discover all mathematics?

[@RodrigoRodrigues-mc4oq]

@@jackgreenearth452 it feels cool to discover that someone else in the world also had this insight about death. I also thought: if after death we would detach from from physical but keep a thoughtful, senseful soul, unconstrained on space, would I move around and observe the whole universe? Unfortunately, most probable my mind would just cease to exist with no trace of ever having existed…

[@seanlange4457]

They aren’t used all that often not cause they get big fast, but because they have uses neither in the real world nor in any other part of mathematics

[@akunog3665]

@@RodrigoRodrigues-mc4oq We don't get to know what comes next. But like you, I see ceasing to exist to be occam's razor. Doesn't mean I know any truth on the matter though, any prediction would be speculation.

[@juliavixen176]

@@RodrigoRodrigues-mc4oq @JackGreenEarth Plato (writing as Socrates) makes this argument in the Meno. That we learn mathematics in the afterlife (between physical death and physical rebirth), but we forget most of it when we are born. So learning math is really just remembering what we forget. (You know... from the realm of Platonic ideals, with perfect circles and stuff.) I looked this up, it’s called Anamnesis.

[@sykoe360]

You did a fantastic job explaining tetration by working up from the basics while also making me jam out and laugh throughout the video. Keep doing what you're doing and you'll have millions of subscribers in no time!

[@okplay9446]

This is probably my favorite part of mathematics to explore, ever since we studied roots and logarithms at school, I've been wondering what comes after, and I was surprised by how little information there was out there. So I'm glad that I found this video.

[@novamc7945]

Wikipedia has a great page on it, you should check it out

[@okplay9446]

@@novamc7945 I did, it's super interesting! Relatively new as well, as a concept in mathematics.

[@namesurname-ej1eb]

@@novamc7945 what is the name of the article?

[@novamc7945]

@@namesurname-ej1eb Hyperoperations if memory serves me right

[@novamc7945]

Yeah, just double checked.

[@GringoLingo]

5:07 “A to the a to the a to the a” was so on beat with the music I loved it

[@aogasd]

It's strangely comforting to 'sit in math class' as an adult. I missed learning about some obscure detail that gets confusing really fast and is very likely to never come up in any real practical scenario. I'm not sure why it felt different to any other math videos I've seen on KZbin. I guess most math channels don't do video, or if they do it's not as conversational as this. I should probably subscribe kek

[@Sinnester_GD_Player]

Tetration? Pentation? Hexation? I didn't even know these existed! Ty bro for the vid.

[@Ishsa]

This was a great video! I've never seen you before, but I love hyperoperations and large numbers. Thanks for exploring it in depth while keeping it engaging!

[@josiahtaylor8714]

I’m glad I found your channel again, stumbled across it a couple months ago and was really interested, glad to see you’ve kept the style and I made the right choice to subscribe this time you popped up in my feed

[@matthiasbockbreder3212]

It’s 6:18 in Germany and I love your video. Awesome! Your channel will grow fast.

[@BariScienceLab]

Great lesson! I really liked your explanation, because it was very simplistic and allowed anyone with basic algebra knowledge to understand these advanced operations! I myself didn’t know about pentation.

[@Ratimus_]

I've always maintained that if x to the second power is x squared, and x to the third power is x cubed, then x to the fourth power should be called "x tesseracted." So far, it hasn't gained the widespread usage I had hoped for.

[@AlbertTheGamer-gk7sn]

Tesserated, or zeited, as Zeit is German for time, and people think time is the 4th dimension.

[@PowerStar004]

Should x to the first power should be called "x lined"?

[@AlbertTheGamer-gk7sn]

@@PowerStar004 And x to the 0th power will be called "x pointed".

[@frocco7125]

THIS IS SO FASCINATING THIS CHANNEL IS DOPE

[@jamesieza]

Love your videos, after watching a few I came back here to my first one to say: I hope you get to 1,081,080 subscribers while retaining your unique style, something about your enthusiasm and the minimal polish is awesome. The content and writing are perfect, refreshing to have hand written notes in a world of slick KZbin animations.

[@testtesttest968]

Im glad I found this channel. Alot of intresting math stuff ive not heard about before. You explain math very well.

[@somerandomcsgoplayerlol8977]

I like this content. as a 15 year old preparing to major in math, it's nice to watch a video that isn't just about manifolds. yes, you're also right about the first part. incrementation is crucial when it comes to addition, or defining the set of naturals and such, like in Peano's axioms

[@adamdima2590]

wow, you must be really smart

[@scottym7192]

Nerd

[@somerandomcsgoplayerlol8977]

@@adamdima2590 actually no, I have a slow processing speed

[@lennytheburger]

🤓 i also do math for fun but bro please make friends for now dont worry about this

[@spyne.98]

@@lennytheburger What makes you think he doesnt have friends lol, just cause hes smarter than you?

[@nuzayerov]

As a CS student, the upwards arrow is used sometimes to use NAND logic on numbers, but I guess its fine since many symbols are repeated across Maths and Physics too

[@megubin9449]

yeah. the alternate ᵇa notation for tetration would not be any better since a polynomial (although tetration never appears in polynomial equations) like xᵇa would be confusing for whether its (xᵇ)a or x(ᵇa).

[@christopherwellman2364]

@@megubin9449 but you solved the problem yourself with the second "polynomial"! That would be an effective notation.

[@lued123]

I sometimes say that we should update PEMDAS to STEPS. That's Sets (stuff in parentheses), Tetration, Exponents, Portioning (a word I picked to represent both multiplication and division) and Sliding (the same for addition and subtraction). A lot of people end up getting misled by M and D being separate letters and will do multiplication before division even when division is further toward the beginning of the question. And of course they'll do the same thing with A and S. That's why we always see those stupid "only 69/420 people can solve this" posts on social media where they've intentionally put a division in front of a multiplication so that people who rely on the acronym will get it wrong and argue in the comments. Plus, PEMDAS doesn't mean anything while STEPS has a very relevant meaning.

[@cadenorris4009]

OR: Do what every engineer does in the real world and use fraction notation for division... Don't use the division ➗ symbol... It just leads to confusion. And when in doubt, use parentheses to make sure there is NO confusion about what should be done. In the real world we can't afford the confusion behind horizontal order of operations. I haven't used the division sign since grade school.

[@hollowshiningami3080]

@@cadenorris4009 or use x.y^-1

[@Izzythemaker127]

@@hollowshiningami3080 The "." symbol is for showing where the ones place is, if anything less than ones place is used. Do you mean "⋅" or "*"?

[@jerecakes1]

yeah most of those "oNlY 10 oUt of 10000 pEoPle" posts are just badly written simple math questions i hate it so much

[@tiedänkö]

do pople actually use pemdas

[@MikeHenry0001]

Bro thanks for this man, better than what my teacher could have explained ever💪

[@TheRenaSystem]

This was so cool and such a high quality and fun video! I was already familiar with tetration, but even so, this was so enjoyable to watch! Great stuff!! :)

[@robbiedart7422]

Man it's like you took all the topics I discovered from going on wikipedia spirals in high school/college and turned them into compelling educational videos! You're smashing it my man

[@CoelhoColho]

Thank you for the video!! You explained the topic in a very easy to understand way! Keep it up, this channel is amazing!!! Love from Brazil 🇧🇷

[@dancoroian1]

I believe your table should have growing values (in the 2s row) for tetration and beyond, analogously to 3 -- 2^^2 would be 16, and 2^^^2 would be 2^16, and so on. I can't fathom any reason why 2 would behave like 1 and stay the same, unless I'm missing something big... In fact this seems to be reflected in your next table.

[@MuffinsAPlenty]

I have been seeing this confusion in the comments a lot, so I'm copying and pasting an explanation I've given to several others. Hope it helps! I think there's a bit of confusion here, and I often see this confusion even with normal exponentiation! So let's start with exponentiation. What does a^n mean? A lot of people will tell you that this means "multiply a by itself n times". But this actually isn't an accurate description. For example, a^2 would mean "multiply a by itself twice" which would be a*a*a (multiply a by itself once to get a*a and then a second time to get a*a*a). A better description of a^n would be "a product of n factors where each factor is a". Then a^2 would have 2 factors both of which are a, so a^2 = a*a. Now, let's turn to tetration. a↑↑n means a tower of exponentiation with n levels, each level having a value of a. Then, a↑↑2 is a tower of exponentiation with 2 levels, both levels being a. So a↑↑2 = a^a. Therefore, 2↑↑2 = 2^2 = 4. I think the 16 you're getting is making the same mistake as the exponentiation thing, thinking it means "exponentiating 2 with itself two times", which would be 2^2^2. But again, this is incorrect. 2^2^2 is an exponential tower with 3 levels, each level having a value of 2, so 2^2^2 = 2↑↑3 = ³2. And, similarly, you can see that 2↑↑↑2 = 2↑↑↑↑2 = 2↑↑↑↑↑2, etc. After all, 2↑↑↑↑2, for example, means you have a tower of repeated "↑↑↑"ing with 2 levels, and both those levels are 2, so that means we have 2↑↑↑2.

[@SirSX3]

@@MuffinsAPlenty thank you so much for explaining. I was also confused, but I understand now.

[@abydosianchulac2]

@@MuffinsAPlenty Except that even at 9:10, he shows that pentation, here 2aaa2, is equivalent to 2aa(2aa2). Since you've already stated that 2aa2= 4, then the pentation 2aaa2 is equal to the tetration 2aa4, not 2aa2 again. So it should be diverging to infinity. This also fits with his assertion that the largest value a series of stacked numbers like these can converge to is e; all other series diverge towards infinity. Since 4 is greater than e, there must be an error in one of these moments of the video.

[@MuffinsAPlenty]

@@abydosianchulac2 The "exponent" in that part of the video is a 3, not a 2. Is it true that 3^3 = 3*3*3? Now, does that mean 2^2 = 2*2*2? Or does 2^2 just mean 2*2?

[@abydosianchulac2]

@@MuffinsAPlenty Nevermind, found an educational site that explains it more clearly. Thanks for your efforts.

[@avidaslan]

Thank you for making this math so accessible! I'm grateful to find you sharing the joy of new ways to understand numbers :)

[@BlackbodyEconomics]

The convergent/divergent nature of tetrations being bound by e actually makes a lot of sense - I mean, given what e actually is. Very cool video man! You just earned yourself a new subscriber :)

[@badasson8825]

This gives me an intense 2012 KZbin nostalgia. Great work dude

[@michaeldeierhoi4096]

This is a good intro to tetration of small numbers and how fast and high they increase. I needed to hear this to expand my horizons about basic math concepts I have been ignorant about.

[@ceulgai2817]

What a great way of detailing the topic while staying simultaneously informative (accurate) and inviting. I honestly feel like I could show this to my lower/middle-grade students and they not get lost. My only complaint is the use of "number" when you mean "integer" or "natural number." Using these terms interchangeably tends to cause problems later on down the road for math learners.

[@TheSwanDragons]

I was already interested, but the fractals at the end hooked me. The video was very well done, thank you!

[@guillermoratou]

Most amazing video i've seen explaining the topic!

[@josephd6115]

This is so well made and has touched on a topic I’ve been fascinated with for the longest time and explains it perfectly

[@kennethbeal]

Thank you, this is a new superpower. :) Loved the ending! While you were showing the equations bounding related to "e", I was thinking about fractals and their bounding equations -- so really cool to see you end with fractals! Life is fractal in nature, a pattern that is slowly forming for me. :)

[@somerandomashellperson7130]

Yesterday, I talked to a friend about repetitive exponentiation, and how cool it would be if it was a thing. Today, I get recommended this video. This video is amazing.

[@theforestgriefer7732]

Keep up the good work, you’re destined for greatness!

[@andaralabs]

great quality, I'm glad I found this channel, you have amazing delivery and teaching skills, decided to subscribe less than 5 minutes into the video. now I'm off to watch your other videos and look forward to whatever you do next. keep it up

[@SeanCMonahan]

Are there operations in between arrow levels? Exponentiation is a↑b, tetration is a↑↑b, etc., but is there something with fractional arrows? a(↑^(5/2))b? (It's kinda hard to describe, but can you raise the arrow operator to non-integer powers, I guess?) I'm so glad the algorithm popped this video into my feed! Immediate subscription from me!

[@ComboClass]

There's no simple standardized answer to that, but it's a deep question that some mathematicians are still pondering and testing various ways of approaching.

[@DeJay7]

That's such a fascinating idea that didn't even cross my mind. I'm gonna think about it on my own.

[@samvsmedia8680]

@@ComboClass Imagine a complex number of arrows

[@DrunkenUFOPilot]

Yes. I came up with a way to do something like that. The key idea to defining a generalized operator, call it [n], is to use iterated exponentials. x [n] y = exp^n( log^n(x) + log^n(y) ). By exp^n(x), we mean exp(exp(exp(... exp(x)...))), and likewise for logarithm. We can say log^n = exp^(-n). exp^0(x) = x. exp^a(exp^b(x)) = exp^(a+b)(x). (I hope this text-only notation is clear) We have x [0] y = x+y, and x [1] y = x*y. Nice thing about this: x [n] y is symmetric, associative, a group operation unlike ugly tetration. It turns out to be "easy" to define exp^n for any real n, for certain definitions of "easy". The solution is non-unique. Australian mathematician G. Szekeres has a paper on that, circa 1960, with a specific definition for exp^n and log^n. I invented a symmetry-based definition for a very different definition for exp^n, one of the weirdest functions I've dealt with. There's yet another definition due to Kneser, a German mathematician. Details? Slideshare, "Generalizing Addition and Multiplication to an Operator Parametrized by a Real Number".

[@DrunkenUFOPilot]

Fun tidbit of knowledge: the operation x [-1] y is similar to the "softmax" function used in machine learning.

[@waxknucklebearingjuice5592]

I failed highschool math repeatedly.... you still somehow communicated this in a way I find interesting and , if not understandable , approachable and fascinating.

[@jewfroDZak]

One of the best educational math videos I've ever seen. I love the (deliberately?) slower than conversational English tempo you used to present these math concepts. Letting the ideas breathe a little bit allows a viewer to think about it for themselves for a quick second. And I would think that that tactic would make it not only easier to digest and retain the knowledge of any math concept, I think it also highlights those moments when the transcendental beauty available to be beheld in maths really shine through....I worked with tetration back in Algebra class, but only for the one lesson in which I first I learned of it. I don't remember it ever coming up any other time in my math schooling history. That and the Numberphile video on Graham's Number (in which tetration is explained) was the entirety of my previous learning experiences on this topic. So Ive known about these operations for the majority of my life, but I never saw any of the little mind blowing ins and outs that you put up for display in the video. I really loved the charts you used that showed what the numbers 0, 1, 2, and 3 each do as they travel up through the levels of recursion. That was beautiful. And I really loved your chart and explanation of the neato fact that the limit values for a convergence producing function of a self-tetrated number can be expressed with the equation e^-e

[@DavidSartor0]

Huh. I watched the video in x2 to x3 speed, and enjoyed it. I don't think I would have learned more if I went slower.

[@jewfroDZak]

@@DavidSartor0 nuh uh

[@DavidSartor0]

​@@jewfroDZak Thanks for responding. Please elaborate, I don't understand what you mean.

[@esyriz]

I rarely comment on videos but I just found out your channel and your videos are really awesome. Keep up the good work!

[@Dark_Slayer3000]

I'm glad to have found this new channel. You're gonna have a beautiful growth with your lovely educational and entertaining content!

[@andtherefore8076]

Now I can finally understand that reighew comparison's number video. Thank you so so much:)

[@amedits7792]

1:07 from here onwards you sound like you're rapping to a sick beat, it's great lmao

[@eduardoortiz6552]

Excellent! I love how a complex topic was explained in an easy-to-understand way!

[@SeriousApache]

0:35 Rest in peace clock. This one was my favorite.

[@jacobmass5837]

Glad the algorithm blessed this video and brought your channel to me and many others

[@sidgar1]

I'm surprised you didn't touch on TREE(3) in this video. One of the ultimate examples of tetration.

[@EllieJin]

That's what I was thinking! TREE(3) or Graham's number, each have their own system of tetration, surprised it didn't get talked about.

[@sykoe360]

@@EllieJin Sounds like a great idea for another video!

[@phscience797]

The number TREE(3) is not defined via tetration. In fact, the „hierarchy of operations“ started here (generalising tetration) is very ineffective at representing it. Rather, the large TREE function is an effective bound on Kruskal‘s tree theorem.

[@ComboClass]

TREE(3) is actually constructed from a different way than tetration, based on a separate "TREE function", and we don't know it's exact size if we tried to describe it with these hyperoperations. But I'll probably mention it in a future video since it's awesome!

[@EllieJin]

@@ComboClass So it can be possible?

[@anirbanc88]

i am currently trying to translate multiplication and exponentiation in terms of addition, i searched for something and found this video, awesome man! solved a few doubts and i am happy that i am solving the problems intuitively in the same manner as you are doing it. cool! i am finding it very difficult to unpack exponents in terms of addition it would be great if you do a simple video on that also do something fun with logarithms, anti logs and stuff like that, you are awesome buddy!

[@JA7ja_]

Amazing video! Informative, easy to follow, and builds up to something spectacular! Please keep these up!!!

[@karrotka223]

YT algorith suggested me this video, glad you are starting to get recognision, your videos are extremely high quality for the views you are getting, keep it!

[@masoncamera273]

Great video. I'm wondering if anyone has figured out a hyper-exponential function, e.g., tetrating to a non-integer like e^^pi. And as silly as it sounds, I've always wondered if the concept of operations could be extended to non integers, such as half operations, and possibly even complex? Hard to say what that would physically entail but then again we've used analytic continuation on things like the factorial function before Also, interesting fact: it's worth noting that logs can turn addition into multiplication, and vice versa with exponents, I wonder if there's an analog of that with exponents, tetration, etc.

[@louisblaine4261]

If you are interested in the possibility of 'fractional' operators check out fractional differentiation and integration. This really is a thing with practical uses

[@MrCubFan415]

@@louisblaine4261 en.wikipedia.org/wiki/Fractional_calculus#Applications

[@tompo6177]

I finally understand the notation used to write graham's number, thanks!

[@tynerben]

Idea for a follow-up video: explore the corresponding inverse operations; just like log_b(x) solves for y such that x = b^y, so log is the inverse of exponentiation, there must be an operation which inverts tetration and so on.

[@Xnoob545]

Level 0: incrememting. Inverse: decrementing Level 1: addition. Inverse: subtraction Level 2: multiplication. Inverse: division Now when we get to level 3, something interesting happens. There's two inverses. This happens because it's the first operation where order matters. a+b=b+a, a*b=b*a, a^b=/=b^a (incrementing can only have one input, so it doesn't count here) Level 3. Exponentiation. Inverses: roots, logarithms Level 4: Tetration. Inverses: super roots, super logarithms

[@Xnoob545]

And I think for level 5, pentation, we have hyper roots and hyper logs

[@AlbertTheGamer-gk7sn]

@@Xnoob545 Incrementing increases the cardinality of a number, addition increases the ordinality of a number.

[@JamesDavy2009]

@@AlbertTheGamer-gk7sn I find that statement to be logically sound.

[@paolarei4418]

Level 6 : hexation inverse stack roots and stacked logs

[@destructivforce2894]

Saw this in my feed since the algorithm picked it uo, then looked at some of your other vids after this one, you're criminally underrated. Subbing to see where things go from here

[@jasonlescalleet5611]

Thank you for answering some questions I’d had for a while, and couldn’t quite put into words. One thing that I thought was interesting, and which you touched on some too, was that the different operations in the hierarch don’t work the same. Succession (which I called “incrementing” and denoted ++ because I’m a programmer) is really a unary operation: “++x”. Addition and multiplication are binary, taking two params “x+y” or “x*y” but order doesn’t matter: x+y=y+x and x+(y+z)=(x+y)+z. Exponentiation is also binary, “x^y” but order “does” matter. It seems like each tier adds some extra requirement. First you need an extra param. Then you need to make sure you’ve got the order correct. From this video it *seems* like tetration and pentation don’t add any extra caveats, but I really don’t know.

[@MuffinsAPlenty]

Interesting observations! One thing I have noticed is that it seems like nothing is lost in moving from addition to multiplication. Is there anything you have noticed being lost? And I have noticed something being lost from exponentiation to tetration. Exponentiation is right-distributive over multiplication, but tetration is not right-distributive over exponentiation. (In the hierarchy, it doesn't make sense to talk about addition distributing over succession, multiplication is both left- and right-distributive over addition, exponentiation is right- (but not left-)distributive over multiplication, and tetration is neither left- nor right-distributive over exponentiation). What do I mean by left- and right-distributive? Given a, b, and c we have a(b+c) = ab+ac. This is left-distributivity. The multiplication by a on the left of the sum distributes over the sum. Similarly, (b+c)a = ba+ca. This is right-distributivity. The multiplication by a on the right of the sum distributes over the sum. Now, a^(bc) does _not_ equal a^b * a^c in general. So a raised to a power "on the left" of the product does not distribute over the product. However, (ab)^c = a^c * b^c. The power of c on the right of the product distributes over the product. When it comes to tetration, we have neither: a^^(b^c) = (a^^b)^(a^^c), nor (a^b)^^c = (a^^c)^(b^^c). As examples: 2^^(2^2) = 2^^4 = 2^2^2^2 = 65536, but (2^^2)^(2^^2) = 4^4 = 256. (But we shouldn't expect this to work since it doesn't work for exponentiation either.) (2^3)^^2 = 8^^2 = 8^8 = 16777216, but (2^^2)^(3^^2) = 4^27 = 18014398509481984. So there is one nice algebraic property exponentiation has which tetration does not have: right-distributivity over the previous operation in the chain. Now, I don't know if there's anything lost in going from tetration to pentation because... well... I don't know of any algebraic properties that tetration has! Since I can't think of any nice property of tetration, I can't think of what could even be lost at all. But if someone finds some interesting algebraic property of tetration, it would be interesting to see if that property breaks for pentation.

[@jacobtinkle9686]

Oh. My. God. Ive been thinking what comes after exponentation for like 2 years and im super happy someone finally made a video about it!

[@Leakingdeygas]

This guy deserves way more recognition than he has.

[@Astro_weeeeee]

This whole channel feels like i stumbled across an insane mathmetician in the woods and am now just watching and listening to his insane ramblings

[@_JUNGSEIPEI_]

5:07 bro started rapping for a sec 💀

[@abhishekshekhar9116]

💀

[@nycuba7478]

(Ay, Ay, Ay, Ay) A to the A to the A to the A

[@sykoe360]

🤣

[@drewwilliamforbush2927]

Combo mixtape coming soon?

[@ComboClass]

Believe it or not, I have released many rap albums and mixtapes in the past (under other names, and I haven't shared them on these channels but will someday) and plan on releasing more before long

[@MegaMONI45]

I just discovered this channel, and didn't know how badly I needed it until now...Great job man! That was awesome :)

[@sarthakgupta1853]

Dude you need to promote your main channel more like this vid has done so much better than the other ones. All the other vids deserve to have these many views and many more.

[@ComboClass]

I didn't do any special type of promotion for this episode, it just happened to do better than the older ones. Sometimes it takes the youtube algorithm a little while to start recommending a channel to people, so hopefully it keeps getting shown to more people over time :)

[@sarthakgupta1853]

No I'm talking about the short you made. Since that channel gets more views and most people don't know about this channel.

[@ComboClass]

@@sarthakgupta1853 I usually make a short on my bonus channel letting people know when I drop a full episode here. So I didn't really treat this episode different than others. But I'm glad a bunch of people are seeing it for whatever reasons :)

[@goldencheeze]

I never thought I’d see a channel with howtobasic / buttered side down level humor combined with nice educational content Creative as heck niche you got, you’ve earned your sub

[@MattR0ss]

Some time ago I was wondering if you could have operation of level 1,5.

[@aldebaran584]

You can, but there are probably multiple "correct" solutions

[@samk4480]

Things get weird and undefined when you go outside of the common operations (addition to exponentiation) - it wasn't touched on here because it's not a big deal, but the behavior of tetration at non-integer numbers isn't formally defined. I'd imagine that a non-integer 'step' operation would behave much the same way- what is half way between multiplication and addition?

[@Xnoob545]

@@samk4480 wait, what if operation 1.5 (I'll use the symbol ~ for it) is just n ~ m = n + (average of m and 1) Because level 1, counting is n + 1, and level 2, addition is n + m

[@AlbertTheGamer-gk7sn]

As well as level e or pi, some irrational levels create weird numbers like booga-e or booga-pi.

[@0xERM]

Oh boy! I love working with numbers far beyond any practical use!

[@seize2581]

Just attended my first combo class, it was great :)

[@remcovandermeer9360]

Cool video broo. It's really mindblowing how big these numbers get. But you explaint it well. Keep up the good work!!!

[@drewwilliamforbush2927]

Oooooh so that's what happened to combo desk. Well now we have combo ramp, perfect for demonstrating the physics of rolling clocks 😂

[@emillevi7928]

Dude, I have been thinking about this pattern and what comes after it for months! Finally someone to explain it!

[@techno4826]

Thanks I was always wondering what that was, but had no idea how to google it. One day while doing assignments in math class I thought “if multiplication is just several additions, and exponentiation was just multiple multiplications, then shouldn’t there be something for multiple exponentiations?” I never got an answer to that until now…

[@Xnoob545]

you might be interested in the videos on big numbers made by the small youtuber "Orbital Nebula"

[@techno4826]

@@Xnoob545 sorry it took 2 days for me to reply, but yes the videos you've suggested are quite interesting to me. I'm on part 3 while typing this in fact.

[@emretekmen1602]

dude, you are a shining star, a one in a million. Please, please keep it up! already one of my favorite creators. I learned so much in this video!

[@thesnazzmaster]

I've always thought about if "above exponents" would be nested exponents, or just raising a number to a power n times. Really cool to see that "above exponents" does exist and has an agreed upon definition in math!

[@paolarei4418]

Like (100/hect)ation

[@neiladrian]

Never have I clicked on the subscription and notification bell this fast. Great content, Dimitri!

[@mosiekirby9513]

Don’t stop posting !!!!! Do not stop!!!!

[@FeralWorker]

Great content dude. Hope this main channel takes off.

[@therealEmpyre]

This reminds me of a video about Graham's number, which used a similar arrow notation, but with the number of arrows growing rapidly, creating a truly monstrously huge number.

[@reizinhodojogo3956]

g(1) is 3^^^^3 g(2) is 3^^^... g(1) arrows in total ^^3 g(3) is 3^^^... g(2) arrows in total ^^3... g(64) is 3^^^... g(63) arrows in total ^^3, it is graham number

[@AlbertTheGamer-gk7sn]

Graham;s number is 3 {{1}} 64, or 3 expanded to 64.

[@paolarei4418]

Imagine g(g(g(g...... For g(g(g..... Times Or exponentaded or tetrated or pentaded or..... g(g(g.....taded (or irrationalGrahamAted) It would be extremely extremely extremely extremely extremely.... x10↑↑↑↑↑10 big

[@17.11Acts]

Your visual work is AWESOME. Thank you

[@NoNameM9]

this is literally my first video of you and its really good

[@Ahmgcats]

Pretty neat how much math is out there that you may never run into just depending on what you're studying. An intuitive concept and really well explained, yet something I never ran into in my math degree given I mainly worked in discrete math. Would be great to have links in the bio to proofs of some of the facts about limits towards the end (for example tetration of infinite height of sqrt(2) converging to 2).

[@elitettelbach4247]

This is incredibly fascinating! And very well explained. Awesome job!

[@TwentySeventhLetter]

I remember first learning about hyper operations and wondering why 2 stayed the same while 3 rocketed towards infinity, and the introduction of e into my mathematical vocabulary both illuminated and further confused the subject 😄

[@jimmyh2137]

2 stays the same because you always take 2 copies, 2+2=2x2=2^2=... same thing, you're taking "2" two times. 2^2 means multiply 2 by 2, two times, which means adding 2 to itself two times. Just make it 2x3, 2^3 and so on instead :D

[@tobiasursmartimuller1657]

What a great video. Your explanations are so interesting to listen to and I hope I can learn a little bit from your videos. Thank you!

[@wadejonaitis426]

You're so close to 1k subscribers.

[@ComboClass]

Yup this video helped me pass it! I actually have a lot more subscribers on my bonus channel and on another website, since short videos are so popular these days, but this channel is where my main projects go and where my coolest subscribers are haha :)

[@wadejonaitis426]

@@ComboClass I'm so glad your main channel hit 1k. Your such an underrated creator and make high quality videos.

[@devbites77]

Fascinating stuff! You have a very engaging and interesting communicative style.

[@GynxShinx]

I find these unendingly interesting how fundamental they are and how much people take the first few for granted.

[@josheco4]

super good and underrated channel w

[@GaryCrasher]

Now i wanna know: What are the inverse operations to these? Is there any applications? Time to go down a research rabbit hole i guess

[@asheep7797]

Superlogs, slog(x) Notes: slog(x) = slog(log(x)) + 1 slog(x) = slog(10^x) - 1 correction: slog(x) = slog(log_b(x)) + 1 slog(x) = slog(b^x) - 1 (b = base)

[@michaeledwardharris]

Sorry, dumb question, but how is 2 to the nth-tration still 2? Related timestamp is 7:05, that seems to contradict the following screens which show the correct values for the 2 sequence. Feel like I'm missing something here.. altogether awesome video though, your presentation style is awesome af!

[@wolfboy414_lac]

the following boards show higher tetrations, not nth-trations. any n higher than 2 on 2^^n would cascate into infinity. the reason 2 to any tration is 4 is because the actual tration stack is always only 2 numbers high, i.e. always equal to its lower tration, so it just ends up as 4, always.

[@michaeledwardharris]

@@wolfboy414_lac very nice! Thanks for the clarification.

[@yglyglya]

2^^^2 = 2^^2 = 2^2 = 2*2 = 2+2 Second 2 means write a two "2" times

[@FirstLast-oe2jm]

educational AND genuinely hilarious, good work with this video :)

[@torak456]

At 7:00 I'm a bit confused. You're saying 2↑↑2 = 2↑2 = 2² = 2 × 2 = 4 But at 9:10 you show that 3↑↑↑3 = 3↑↑(3↑↑3), so surely it follows that 2↑↑2 = 2↑(2↑2) = 2↑(4) = 2⁴ = 16 Hopefully those characters render correctly on your device.

[@MuffinsAPlenty]

While I can understand the temptation to think that 3↑↑↑3 = 3↑↑(3↑↑3) implies that 2↑↑2 = 2↑(2↑2), this is not true. That would be like saying 3^3 = 3*(3*3), so surely it follows that 2*2 = 2+(2+2). Of course, you can see this is false since while 3^3 = 3*3*3 is true, 2*2 is not equal to 6. And I hope this example helps illuminate the error in your thinking. The reason 3↑↑↑3 = 3↑↑(3↑↑3) is because it's a special case of a↑↑↑3 = a↑↑(a↑↑a), and this is true because of the definition of pentation as repeated tetration. By definition, a↑↑↑3 is a tower of tetration with 3 levels where each level has value a. (Much like how a^3 is a tower of multiplication with 3 levels where each level has value a.) On the other hand, a↑↑2 is a tower of exponentiation with 2 levels where each level has value a, so a↑↑2 = a^a. Substituting a = 2, we get 2↑↑2 = 2^2, by definition of tetration. The fact that 2↑...(n arrows)...↑2 = 2↑...(n-1 arrows)...↑2 isn't a really deep result. It just follows from the definition of the operation as being a repeated version of one fewer arrows but with only 2 levels. It's just that this is a new operation with weird notation, so it takes a while to think about.

[@benjaminlum5894]

I'm new here, and judging by your lab coat, i was expecting a new entertaining chemist who popped up in my recommendations, but I am not at all disappointed! Keep it up!

[@martinshoosterman]

While the real valued function X^X is commonly not defined for x=0, in this video you are talking about operations on natural numbers. For operations on natural numbers there is a definitive choice for 0⁰ that is typically used. Usually 0⁰=1. Especially since these all have recursive definitions, it's much easier for them all to have the same base case. (so tetration, and higher is also defined for 0). Also if 0⁰ is taken as undefined, then tetration and further would be undefined whenever 0 is in the first argument. Which is really undesirable, so it's much better to just settle on the most natural definition

[@MagicGonads]

it's not just a choice for natural numbers, it's the only choice consistent with the definition of the set A^B being the set of functions from B to A, meaning |A^B| (the size of this set) is the meaning of exponentiation on natural numbers. There is only one map from a set of 0 elements to a set of 0 elements, the empty map, so 0^0 = 1.

[@martinshoosterman]

@@MagicGonads Having the definition be consistent is technically a choice, but yes I agree, it does seem more appropriate to not call it a choice given that it follows directly from the set theoretic definition.

[@budderman3rd]

The sum series of e^x proves 0^0 = 1. It shows: e^0 = 0^0/0!

[@AlbertTheGamer-gk7sn]

Same with negative numbers as well.

[@lebronejhames7265]

Wow this was super educational and entertaining especially for a maths video. Great work I’m astonished you only have 5 thousand subscribers keep on the grind and you’ll make it to 1 million in no time. I’ll be sure to follow the journey

[@poklet]

Thank you. This is something I always wondered about. And succession was something that I'd never considered. But there it is, it's the basis of everything… or is it? Is there something lower than succession that succession is based upon????

[@ComboClass]

Succession is usually the first "operation" used to define systems of arithmetic, along with considering "identity elements" (like in the typical system of arithmetic, 0 is the additive identity element and 1 is the multiplicative identity element). There can be other considerations like having "inverse elements" and things, but that's getting into a deeper topic for another video.

[@jchin1117]

I'm a recent subscriber. You give me serious old school Bill Nye vibes. Thanks for your content