The Rare Levels Beyond Exponents 

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

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

It was recommended to me today for the first time.

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 RU-vid algorithm god!

❤

seconded

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

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

true

@@risingSisyphus wow I found this too haha

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

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

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

@@markzambelli tree(g64) enters the game

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

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

"Technically there are 85."

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

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

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"

this isn’t really all that advanced, though

@@sylv512 thats basically what Steele said..

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

@@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.

@@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.

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?

@@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…

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

@@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.

@@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.

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.

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

@@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.

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

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

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

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

@@novamc7945 what is the name of the article?

@@namesurname-ej1eb Hyperoperations if memory serves me right

Yeah, just double checked.

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.

@@cadenorris4009 or use x.y^-1

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

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

do pople actually use pemdas

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

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?

@@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

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

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.

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

@@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.

@@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?

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

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.

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

@@ComboClass Imagine a complex number of arrows

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".

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

It really was!

Facts I was looking for this comment

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).

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

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

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

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

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

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

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

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.

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

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.

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!

@@ComboClass So it can be possible?

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

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

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

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

Level 6 : hexation inverse stack roots and stacked logs

wow, you must be really smart

Nerd

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

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

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

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.

@@wolfboy414_lac very nice! Thanks for the clarification.

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

We have not defined tetration or higher by a similar set of laws as he have with exponentiation (see Laws of Indices). Sure we could define super roots as tetration by a fraction but we're getting into nested radicals. A super logarithm would be easier to define as how many levels are in the power tower to get a base number to the super power in question?

💀

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

🤣

Combo mixtape coming soon?

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

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.

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)

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

@@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.

In the x^^4 part, it only shows the number of digits it would have, not the number itself

@@SariRomero-wo6sz Also known as the "Extensible-E notation".

Well. basically like square root. Inverse operations are equations to solve. When you do "5-3", you are looking for what you've add to 3, to get 5. It's not so obvious,. because you are comfortable with subtraction. Division is tougher. Doing 51 divided by 17 is trying to find out how many times you can add 17 to itself to get 51. All algorithms boils down to just that. For roots, same thing. You are trying to solve "which number can be multiplied n times by itself to yield x ?". And so, for tetration, you are looking for a number which when raised n times to itself yield x. All hyper operation will look like those.

@@programaths Perhaps you could do video demonstrating this? 😉

@@SOBIESKI_freedom That takes time ^^

2 tetrated to a number larger than 2 would be larger (like one of the whiteboards shows) but 2 tetrated to 2 is an exponent stack 2 high so just 2^2. And 2 pentated 2 is a tetration stack 2 high so just 2^^2 which is the same example as above. And etc., so 2 (any hyperoperation) to 2 = 4

2 x 2 is not 2 + 2 + 2

@@ComboClass Thank you so much for this reply! Reading others people’s explanations, I still could not grasp it, but now I can.

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 :)

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

@@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 :)

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.

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 :)

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

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.

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

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

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