Numbers too big to imagine

In mathematics, tetration is an operation based on iterated, or repeated, exponentiation. By using operations such as tetration, pentation or hexation we can create enormous numbers. Graham’s number is one of the most famous big numbers, but there are many even bigger numbers.
Chapters:
00:00 First Hyperoperations
00:35 Tetration
01:26 Infinite Towers
02:12 Higher-level operations
03:23 Graham's Number
04:45 TREE(3)
07:00 Giant Numbers
Music:
@AlekseyChistilin
LEMMiNO - Cipher - • LEMMiNO - Cipher (BGM)

[@digitalgenius111]

From 7:26 all the greater-than signs (">") should be pointing in the other direction ("

[@Gregory_12]

🐊

[@Therealpro2]

[@newlineschannel]

Nice ur a googology fan, been working on many projects for 7 yrs now!

[@andidyouknow8208]

Your good im happy when i get a b on my algebra honors two tests

[@mhmmyes9620]

💀

[@ChessGrandPasta]

it still amazes me to think that if you were to pick a random positive integer the chance that it's bigger than Graham's number tends to 100%.

[@no_name4796]

Yeah infinity is just that big lol

[@alansmithee419]

The idea of generating a completely random positive integer seems bizarre to me, because no matter what result you get there should be a 100% chance that the number generated should've been bigger, since there are always infinitely many more integers larger than it but there must be finitely many smaller than it (otherwise you don't have an integer; all integers are finite). If you generate 3 such random numbers, does each have to be bigger than the last? It should be a 100% chance right? What if you look at the third number first, and then look at the second number you generated? There should now be a 100% chance that it is bigger than the third... I don't think the concept itself makes sense.

[@cc1drt]

to “pick” a random positive integer it needs to first exist. The irony of this is functionally speaking the chance that its bigger than graham’s number is ZERO

[@alansmithee419]

@@cc1drt The probability of an action resulting in a certain outcome being 0 also requires the action to be completable. So really the probability is not 0, but "NA" (Not Applicable - as in the question can't be answered)

[@alansmithee419]

@@randomaccount2448 If it were at all possible, you would be guaranteed to pick an integer, because you only have integers to pick from. You can't pick something that isn't in the set.

[@soup9242]

I find it funny how TREE(1) is 1, TREE(2) is 3, and TREE(3) is some ungodly huge number.

[@abendbg]

YEAH IKR

[@DeetotheDubs]

A perfect example of, "Boy, that escalated quickly."

[@Jipsy7969]

And tree 4 is your weight in tonnes

[@dustypaladin9216]

Imagine tree 4

[@Noneyettocome]

I didn't understand that tree number 🧐

[@shawnheim5043]

Tree 1: 1 Tree 2: 3 Tree 3: Unimaginably huge number beyond the realm of human comprehension

[@poucher]

lmao

[@dough9512]

What about Tree 2.5: ?

[@ckv1985]

​@@dough9512undefined

[@zaviyargul]

41 seconds in got huge numbers

[@moonbeamskies3346]

I can personally attest to the fact that tree(3) is a large number because I stayed up all night with pens and paper testing it by drawing trees and I never came to the end. I had well over 400 trees drawn in that time and didn't seem to be near the end of possible trees. I fell asleep and dreamed of trees combinations.😮

[@pedrocoelho5562]

a true scientific mind! Don't take things for granted, proof is required :)

[@melonneleh777]

Wow! I really respect the dedication 🫡

[@edgepixel8467]

Nice. Keep going.

[@thesenate1844]

I still don't quite understand the rules on how TREE works. What does "not embedded in previous tree" mean exactly?

[@ckv1985]

​​@@thesenate1844you cannot steal the tree

[@user-zb1wc3rz9f]

Even though TREE(3) completely dwarfs g63, I love Graham's Number because you can somewhat appreciate just how insanely massive it is when you express it in terms of how many 3s and exponentiation towers exist even just in g(0). In comparison, the tree function is like... "Yeah here's some confusing rules, we go from 1, to 3, to practically-but-not-quite forever"

[@poruzu]

Fax

[@Vhite]

Personally I prefer TREE(3), since it's based on relatively simple rules that are able to bloom into such a big number without touching the infinity.

[@alexandertaylor7316]

​​@@Vhite my issue with TREE(3) is that you can say it's larger than Graham's number, but there isn't really an easy way to show it, so the default answer is "believe me bro".

[@user-je3sk8cj6g]

@@alexandertaylor7316 Well, THREE(3) is demonstrable, but you basically need a math PhD... So it is indeed "believe me bro" for at least 99.999% of everybody. On another topic, I dare say, mathematicians overthink waaaaay too much...

[@microscopicallysmall]

@@user-je3sk8cj6g 3(3)

[@Kormit537]

Then there's penatration

[@Raj10896]

I wonder how big it could be or such just end at 6 inchs

[@shankar383]

Hexation Hexation Octation​@@Raj10896

[@unnamedscribble]

underrated comment 🤣

[@eclipseshetheyneos588]

Mathmatitions don't know about that one

[@dheerajmalik6989]

after that, there is migration

[@niviera7807]

I opened KZbin to listen to some music and here i am watching a man teaching me math

[@RoyaltyInTraining.]

I never thought a number could scare me, but G1 is already so stupidly and mindbogglingly big that it does the trick.

[@blackjacktrial]

What about g3^^^^^^3?

[@kunalkashelani585]

And when you figure that any upcoming number is practically so much bigger than the previous one, that it's just ridiculous!

[@reshmidas8152]

Beware this vid can giv u a numberphobia

[@liam.28]

look up "busy beaver function"

[@Sahl0]

g0 is insane

[@marasmusine]

I do like how we can't write down Graham's number even if we inscribed a digit on every particle in the visible universe, but we do know what digits it ends in (last digit is 7).

[@sylencemouse1860]

How do we know that

[@johnhawkins5314]

​@@sylencemouse1860 well every power of 3 ends in 1, 3, 9 or 7 starting at the zeroth power. So as long as you can show that Graham's number is 3^(4n+3) or 3^(4n-1) then you know it ends in 7 Now I don't understand Graham's Number well enough to show that, but presumably, that is how it would work

[@Redditard]

much appreciated!! @@johnhawkins5314

[@ChemEDan]

@@johnhawkins5314 TREE^g63(g63) where the exponent acts like it does in sin^

[@WaltonGFilm]

​@johnhawkins5314 I have a similar theory. Well stated. Basically, math, patterns, observe and compare said pattern to which "power of 3 digit" each of the earlier phases of G would land on. Then yeh......??

[@Norv09]

Finally, a good way to measure the ratio of chips to air in a lay’s packet of chips.

[@Aerma]

I love this video - explains complicated topics extraordinarily simply. Would love a part 2 covering even bigger numbers :)

[@Soothsayer_98]

lol says you, at 1:50 my brain turned off and i didn't catch anything past that

[@InsaneI]

Geometry Dash reference?!

[@LexxGee1234]

*what do I expect*

[@nidadursunoglu6663]

The fact that its easier to imagine infinity than a really big number is insane. Its easy to know infinity goes forever but its almost impossible how big would a pile of 7 tetrated by 7 number of apples

[@ryomaanime4563]

because you don't imagine infinity, you imagine something that doesn't end It's close but not the same, it help to understand what it is but you don't imagine it Anyway human brain is bad with big number. And it doesn't have to be this big before the brain goes "yeaaaah something like that maybe, doesn't matter when it's this big" Just imagining a 20km thing is hard as heck. You can try to picture it next to thing that size but it's already to a point where the only thing we could compare to are pictures made from the sky And it's downright impossible to understand how big are the earth, the sun or the solar system. Very small number aren't easier tbh

[@thezone5840]

A google to the googleth power. Infinity as it will take beyond the heat death of the universe to calculate those numbers.

[@pyropulseIXXI]

This makes no sense; infinity is not 'imagining something going on forever.' First off, you cannot imagine that, because all you are doing is imagining something going, then ceasing to imagine that, so you haven't gotten anywhere close to imagining forever, and lastly, infinity is an infinitely large entity, not a 'process that keeps going.' So you are so terrible at imagining infinity that you have fooled yourself into thinking you could more easily imagine infinity than a really large number, which only speaks to the fact that imagining infinity is far harder than imagining any finite number, no matter how large.

[@tubegerm6732]

​@@pyropulseIXXIfound a pseudointellectual! infinity is definitely easier to imagine than tree(3). infinity is easy, it's infinity, and basic logic that we take for granted stops working there. everyone knows that, simple. but with numbers like tree(3) there isn't anything fundamentally different bewtween them and say, 31. they're both just positive integers. but the scale pf tree(3) is so unimagineably massive, that it becomes easier to think about it as just being "basically infinity" dispite having much more in common with integers that we use every day than with infinity. and that there's the rub. we think of tree(3) as being equivilent to infinity, because that concept is easier to comprehend than tree(3)'s true size.

[@minecraftveteran7410]

Hello there! I think the reason Infinity is easy to understand, is down to the basic understanding we have on the concept of Infinity. We may know it as "never ending", but once you start building up your foundation from there, contradictions start appearing everywhere. But then you realise the exact same thing can be said for TREE(3) or g(64). In conclusion; we might have a better understanding of these large numbers than Infinity. I hope you can see my view, and thanks for reading!

[@EnerJetix]

This video felt like a combination of Numberphile’s videos on the topics, but with neat animation as visuals instead. Very well done

[@megubin9449]

didnt think id see you here

[@EnerJetix]

@@megubin9449 we seem to all be getting recommended the same underrated math channel

[@qwertek8413]

It would be easier to just say it felt like a 3blue1brown video.

[@EnerJetix]

@@qwertek8413 yeah, but that wasn’t the first thing I thought of

[@idogaming3532]

Why do KZbin views freeze at 301?

[@livingthemcdream]

Just so you know, you just explained exponentiation better than literally every teacher I have had up until now in less than 30 seconds

[@KiatHuang]

The best exposition I've seen so far on large numbers with precise descriptions and excellent graphics. The narrator's voice is perfect for describing mathematics in English. At university I always preferred maths lecturers who did not have English as their mother tongue - less fluff, focus on diction.

[@ycajal]

This is mind-boggling in so many tree levels

[@madamada219]

I see what you did there😎

[@Mountain_2]

Cant even comprehend level 0

[@New-Iron-Edits]

​@@Mountain_2Gotta be in 4th grade.

[@RedGallardo]

That's why I hate it when people so recklessly use infinity as a number to count with. Infinity is way bigger than any of these numbers. Infinitely bigger. In fact, tree(3) n-ated by tree(3) where n=tree(3) would still be infinitely smaller than infinity. Which is why it's pointless. They say "infinity+1 is bigger". I say it's not, infinity already contains infinity+1 and infinity+infinity and infinity power infinity, and tree(infinity). It's not limited with any finite answer so assuming anything may be bigger is just illogical. But it's easy to imagine. A perfect mathematical circle has infinite sides. All possible trees in the palm of your hand.

[@-Oddity]

​@@RedGallardoThe more you think about, the less infinity seems like a number and more like some incomprehensible eldritch horror from another dimension.

[@galacticdiamondz6425]

7:41 You need to swap the > signs for < signs.

[@carlosmirandarocha8905]

Yes lol, I was wondering about this too

[@Szy96335]

The pinned comment already says about that

[@aiyazashraf]

Exactly

[@carealoo744]

Thank you for finally explaining simply what an up arrow notation actually is, I've been trying to figure that out for a while:)

[@CoolGuyVan]

People like you are able to make math more interesting 👍

[@NevertahnProduction]

if you understand math in the first place, that is

[@vincentjiang6358]

This video is severly underated

[@greenlll121]

​@@vincentjiang6358nuh uh not the video only the guy who made it is also underrated

[@steffenbendel6031]

but only finitely more interesting. Maybe around the factor TREE(TREE(3))

[@pyropulseIXXI]

math is more interesting on its own; what you just admitted is that you are not interesting and need someone else to program your mind with ideas that are interesting on their own. This guy is not making math more interesting; he is literally just talking about the math, and the math is interesting on its own. I am amazed at people such as yourself

[@mike1024.]

I get that the TREE concept is a bit hard to describe, but I feel like some semblance of how we know it is finite yet much larger than Graham's number would have been appreciated.

[@Z3nt4]

The problem with trying to explain it is that the explanation itself requires a much deeper understanding of mathematics than it seems. I'll go on a -slightly- pedantic rant and then try a metaphor to explain it anyway, and apologies if at some point this comes across as condescending. It's not, I'm just trying to _really_ make it as simple as possible. Apologies also to whoever this oversimplification might offend. To most people, mathematics is just another science subject out there, but the reality is that it goes so deep and is so vast as to, in my opinion, be larger than all the other subjects (physics, chemistry, engineering...) combined. The mathematics taught at highschool level feels comparable to learning to say Ni hao, which is "hello" in Mandarin and Cantonese, and calling that being fluent in all the Chinese dialects. A lot of the proofs out there, even for things that seem like they should be "easy" to talk about, require a completely different dialect of mathematics to talk about. You need to peel it back to the abstract logic and go from there. An example of one such dialect (first order logic) would be the following sentence: ∀x ∀y ∀z x ≤ y∧y ≤ z → x ≤ z (for every X, every Y, every Z such that X is smaller than Y and Y is smaller than Z, it follows that X must be smaller than Z). It expands the concrete analysis of, say, an equation, to an abstract observation about variables without worrying about what those variables actually are. For this specific problem of the TREE function, we need to take another step back into second order arithmetic, which is used to further expand and talk about some properties and relations between mathematical objects. For instance, the sentence ∀P ∀x ( Px ∨ ¬Px ) would fall under this category (for every formula P and every variable X, either that formula with that variable is true, or _not_ that formula with that variable is true). It is within this dialect of mathematics speaking about properties of objects that we can construct a proof both that the TREE function is finite for any finite values passed to it and that TREE(3) is much, much larger than Graham's number. Rant and semi-formal explanation over, I'll put it in software terms, which bears striking resemblance to mathematics on many levels but is much easier to grasp: Picture a random mechanic in a random videogame that you can toy around with, familiarize yourself with and learn to use (can be something as simple as jumping). But to know _how_ and _why_ it works the way it does beyond "press this button and it jumps", you need to learn the programming language it's coded in, and go dive into the code. And then you might realize that just from the code you don't fully grasp how it does what it does, and you need to actually _learn how the programming language itself is built_ and go almost all the way down to how the machine functions at a physical level in order to know how the actual code works, and only then fully understand the mechanic. TREE(3) is one of these mechanics, it's concept is very simple, but to actually know how and why it works the way it does you need not only to look at the code, but know how the programming language it's coded in works itself. Those would be first and second order arithmetic, whilst playing the game is just regular math.

[@mike1024.]

@@Z3nt4 hmm fair enough. I'm confident I can understand a real explanation, but if it would be exceptionally long winded and too hard for most, that might explain why he didn't put it in. I can read your statement of the transitive property by the way. :-) do you know of a video or paper that explains it properly?

[@Z3nt4]

@@mike1024. A proper explanation (which I'm not privy to) requires some deeper undestanding of graph theory, in which I am no expert and don't necessarily know of any readily available resources on the topic. However, if you're set on going down the rabbithole I guess you could start by looking up Kruskal's tree theorem and working your way back from there (which is NOT trivial by any means). The massive TL;DR is that under graph theory you can prove that any tree (the mathematical object 'tree') of the same type as the ones built through the TREE function must be finite. How one would go about proving that in the first place is beyond me, but that's the tool for the job.

[@mike1024.]

@@Z3nt4 I'll play around with it! I've taken a couple of graph theory classes and seen some tree based proofs. Thank you.

[@cindyegweh7559]

Here's a way to put it in scale, brak an atom in half and get a hydrogen quark, an unbelievably small substance, fill the entire observable universe with those quarks and were about 0.0000000000000000000000000000000000000000001% of grahams number, lets shrink this quark filled universe to the size of a quark, then fill the universe up with it, repeat this roughly a million times and chances are, your number is still smaller the tree(3) by ALOT, when i say alot, I mean you can divide tree(3) by the amount of atoms in this universe and itll still be higher than the extremely densely packed universe This probably didn't help

[@mohankrishna2442]

Less than a minute into the video and things got out of hand!! Amazing video and explanation.

[@aprilbrandon3441]

I feel like I don’t know anything now

[@caspermadlener4191]

The general way to construct enormous numbers like this is: 1. Pick a HUGE ordinal. You have to be able to prove that it is well-ordered, so it can't be infinite, it should be recursive. 2. Make a function based on thay ordinal. Ordinals are complex subjects, but necessary at the "basis" of mathematics (which is not an important part). For every concrete rule to create ordinals, there is a unique ordinal that can't be created with this rule. Graham's function doesn't use that big of an ordinal, since its definition is very straightforwards. The ordinal should even be described by Peano axioms. But the TREE function uses the small Veblen ordinal. That is quite a big ordinal.

[@MyOneFiftiethOfADollar]

Please do a video over your fascinating comment.

[@0x6a09]

doesn't tree use buchholz's ordinal?

[@ser_igel]

@@0x6a09 i thought it used ackerman's ordinal..

[@tabainsiddiquee7611]

@@MyOneFiftiethOfADollar I recommend this series: kzbin.info/www/bejne/gqS0g2WdfbaMi8U Specifically Part 9 goes over the Veblen ordinals

[@caspermadlener4191]

@@0x6a09 Wikipedia says that the small Veblen ordinal is used, on both the page about Krustal's tree theorem (the reason why TREE exists as a function) and on the page about the small Veblen ordinal.

[@samjohnston1887]

Took a test one year that had a question about a card game and it asked about the number of possibilities. My answer ended up being 2 tetrated up 100 times. I’d never seen tetration before but I was super proud of finding the answer.

[@deltaspace0]

What is the card game that has that humongous amount of possibilities?

[@kunalkashelani585]

Please share the question

[@mike1024.]

Thirded, would love to see the question!

[@anonymouspersonthefake]

fourthed

[@azurezzz]

graham's numbered

[@AyarPortugal]

Crazy stuff, thank you so much for sharing, very insightful and interesting.

[@Alvin_ketchup]

3 hexation 3 is a mathematical operation that belongs to the hyperoperation sequence. It is also known as hexation and is the sixth operation in the sequence. The hyperoperation sequence is an infinite sequence of arithmetic operations that starts with a unary operation (the successor function with n = 0) and continues with the binary operations of addition (n = 1), multiplication (n = 2), exponentiation (n = 3), tetration (n = 4), pentation (n = 5), hexation (n = 6), and so on. The hexation operation can be defined recursively in terms of the previous operation, pentation, as follows: a ↑↑↑↑↑↑ b = a ↑↑↑↑ (a ↑↑↑↑↑↑(b-1)) where a and b are positive integers. For example, 3 hexation 3 can be calculated as follows: 3 ↑↑↑↑↑↑ 3 = 3 ↑↑↑↑ (3 ↑↑↑↑↑↑(2)) = 3 ↑↑↑ (3 ↑↑(3 ↑↑(3 ↑↑(3 ↑ 3)))) = 3 ↑↑ (3 ↑^(4) 27) = 3 ↑^(5) 7,625,597,484,987 Therefore, 3 hexation 3 is equal to 7,625,597,484,987.

[@Alvin_ketchup]

Sure! As I mentioned earlier, Graham’s number G63 is equal to 3 ↑↑↑… (with 63 arrows). To express this number in scientific notation, we can use the following steps: Convert the number to decimal notation by writing it as a power tower of 3’s: 3 ↑↑↑... (with 63 arrows) = 3^(3^(3^(3^(3^(... (with 63 threes) ... ))))) Count the number of threes in the power tower. In this case, there are 63 threes. Subtract 1 from the number of threes to get the exponent of the scientific notation. In this case, the exponent is 62. Write the significand or mantissa by dividing the original number by 3 raised to the power of the exponent: 3 ↑↑↑... (with 63 arrows) / (3^62) = 1.611... × 10^19728 Therefore, Graham’s number G63 expressed in scientific notation is approximately 1.611 × 10^19728. I hope this helps! Let me know if you have any other questions.

[@Orestekoa]

The highest one actually is called penetration but I doubt any mathematician's ever experienced it or used it

[@handtomouth4690]

Damn

[@mellborry]

Good one

[@jakeb_playz7079]

How does "penetration" work?

[@JonCombo]

All log, no pi.

[@coverscrowes4560]

​@@JonCombolmao

[@hellowow4631]

I don't think that we would even have colours for the seeds remaining for TREE(TREE(3))

[@JustAHuman-gb5go]

TREE(TREE(TREE(3))

[@MatthewConnellan-xc3oj]

After TREE(3,600,000), We would run out of humanly distinguishable colors. After TREE(16,777,000), We would run out of RGB 32-bit colors.

[@kepler_22b83]

The limited resources of this universe can not accomodate a representation of this number... But, although colors in the visible spectrum are finite, there may be no ceiling to how much energy a photon can pack... Neither a lower limit on how low the photon's frequency is possible. So, whether we'll run out of colors is questionable, we would run out of energy faster. P.S: if you want to destroy the universe, task an AI singularity with calculating every TREE(TREE(3)) tree. Tell it not to stop until it got the answer.

[@bicksinormus]

@@MatthewConnellan-xc3ojSSCG(3.6m) perhaps?

[@paolarei4418]

​@@MatthewConnellan-xc3ojAfter TREE(TREE(TREE(....... we would run out of TREES cause we used too much paper to write them on papers

[@newsgo1876]

This is the first time I heard about the operation of level >=4. Thank you for enlightening me.

[@markosskace514]

Nicely explained tetration and higher operations. I always get confused thinking about them.

[@ionic7777]

I like your explaination of the TREE function, much more easy to understand on a basic level!

[@danielxdvioletaxd]

nice BTC pfp

[@dimitrinotfound]

The fact that the number of real numbers between 0 and 1 is way laaaarger than any of the numbers discussed here is creepy

[@JordanMetroidManiac]

Seems like you found a good way to imagine infinity, if it’s giving you the creeps.

[@pyropulseIXXI]

how stupid; there is no finite amount of real numbers between 0 and 1, so this is utterly obvious and not creepy at all

[@zbz5505]

​@@pyropulseIXXI< I have to be an asshole on the Internet for no reason whatsoever.

[@Nivleknosnhoj]

Good thing that maths is a close imitation but fundamentally an imitation of reality irrespective of it's unreasonable utility in bits and bobs and things that make you go hmmmm. 😊

[@Nivleknosnhoj]

But maybe yeah I'm more than likely wrong maybe maths is the only thing that's real and it's reality that's the charade

[@gravysamich]

i gotta be honest... i finished your video and thought, "thats it?" i will give you credit, you are the first person to explain arrow notation that actually made sense to me. i just felt like all your video was is just saying, "hey there are some big numbers!" maybe next time explain the numbers significance a little better. grahams number in particular is very interesting because it relates to describing higher dimensional objects.

[@tinotino8349]

I cant wait for the octation update!

[@St2ele]

Thank you for taking these concepts and editing a video with visual proof with examples for all of them. This is some of the best work I've seen! Keep it up!

[@charredUtensil]

There was a great thread on the XKCD fora back around 2010 where a bunch of nerds tried to outcompete each other for largest number without just incrementing previous numbers. The forums are gone now but I think TREE showed up by the third page and by the fifth someone had a number that exceeded the "largest number" yet discovered. I wasn't able to follow along at the time but this definitely helps. Now if only I could find that thread and try to understand some of the larger numbers...

[@MatthewConnellan-xc3oj]

E:NN(x) is x^^^^^…x with x+1 up arrows. I just thought of it.

[@MatthewConnellan-xc3oj]

And then you can just go on with E:NN(E:NN(E:NN(E:NN… to insane lengths.

[@charredUtensil]

Yeah I think the largest numbers used some technique where they turned infinite numbers into mind bogglingly big non-infinite numbers somehow

[@worldprops333]

@@MatthewConnellan-xc3oj E:NN(x) = x^...x+1...x E:NM(x) = E:NN(E:NN...x)))...) with x nestations of the E:NN function onto x E:NM2(x,y) = same nestation on E:NM, with y nestations.

[@Lexivor]

I remember that thread from XKCD, it was an epic thread with hundreds of posts. After studying the math for quite a while I was able to understand most of it. That thread got me deeply into large numbers for a couple of years. I made about fifty pages of notes on large numbers, including a couple dozen of the numbers listed on that thread.

[@josephdouglas6482]

This is an insanely good way to describe these things. I was an accounting major in college, I did a Business Calculus class and several other courses based around statistics and predictions, and when you show how you got to the limit of n as x approaches thing, it's amazing that you built it up from just simple succession and addition.

[@QuentinStephens]

There's one thing I don't understand about tetration: the exponents do not follow the power of a power law (a^m^n = a^mn). At 1:03 we have 3 tetrated to the 4th which is equated to 3^3^3^3, but by the power of a power law that latter value is equal to 3^(3*3*3)

[@TheSpotify95]

That's because doing something like (a^m^n) using the power law means you're just bundling it into a single exponentiation term. With tetration of a number, you have to start at the top of the tower and work your way down - that's how the larger numbers are built as you're defining a new concept/sequence. So 3^3^3^3 = 3^(3^(3^3)), noting the brackets to determine order. This then works out to be 3^(3^27), or 3^(7.62x10^12), or three to the power of 7.62 trillion.

[@QuentinStephens]

@@TheSpotify95 Thanks.

[@Seyleine_]

This is so interesting, thinking that such big no.s could exist is mind boggling.also I was super excited to hear cipher here 😅.

[@bergnerm]

This is a good video, but one problem I have is that whenever anyone is explaining how big TREE(3) is, they explain the rules of how it's generated, but they never say how they know it's so huge. It basically boils down to "trust us... it's REALLY big". How do they know it's bigger than Graham's Number? What kind of mathematics do you use to show this--obviously not "trust me"!

[@astralphoenix69]

i think the same

[@yxx_chris_xxy]

The finiteness follows from Kruskal's tree theorem. It's not something that can be explained in a youtube comments section.

[@bergnerm]

@@yxx_chris_xxy I figured it was something quite complex, but all of the youtube videos I see on it have dumbed it down too much. Maybe you could do a video at least explaining Kruskail's tree theorem, in simplified terms, and at least allude to the techniques used to compare two gargantuan numbers like g(64) and TREE(3).

[@user-ct8rh2re4d]

Graham's Number is G64 not G63. G1 is 3^^^^3 not G0.

[@rickb_NYC]

I'd love more treatment of the tree function. I don't quite understand how it can get so big. Maybe going further with many examples of how it can grow. Also, is there an equation for it? (I assume there is, and bet it has factorials.)

[@samcertified7178]

x! grows slower than x^x which doesn't even come close to tetration. The levels of recursion required to represent a number as large as Graham's number (let alone TREE(3)) go well beyond factorial.

[@Danpg79Roblox]

​@@samcertified7178and (x!)! Grows unfathomably fast... 1!! = 1! = 1 2!!, same thing 3!! Though... 3!! =6! 6! = 720

[@denshi_lives29]

Factorial world be great great great great grandkid when compared to those Pappas

[@gareth2736]

I struggle to get my head both around that and also hoe if tree 3 is so big tree 4 or tree 1000 are still finite.

[@pi_man3]

@@gareth2736eventually you will run out of trees because of previous trees

[@dante7228]

Wrong video at 5 o'clock after waking up. It just obliterated my brain...

[@Excrecity]

Its crazy to think that no matter how big a number u can think of its still closer to 0 than it is closer to infinity

[@cardndmch]

disagree, you should just put the number 5

[@stone5401]

Infinity isnt a number

[@Excrecity]

@@stone5401 i didn’t say infinity is a number

[@IAmNumber4000]

It’s fascinating that these numbers are so big that computation with them is impossible, since even ^4 3 is greater than the number of Planck volumes in the observable universe.

[@Amphy2k]

One of these days I pray to see someone finally explain Large Number Garden Number. It’s the current largest number and no matter how much I read about it, I still feel like I don’t understand it fully.

[@big_numbers]

It’s best to think of uncomputable numbers as diagonalizing over the process of creating functions itself.

[@megubin9449]

its not exactly the largest number, but it is the largest well-defined number.

[@ieatcarsyum8248]

I just made a larger number: Large Number Garden Number + 1

[@TomFoster-en5uc]

@@ieatcarsyum8248hahaha large garden number+2

[@TheUnovanZorua]

What's that? I don't study complex math, so I've never heard of it.

[@vibecheck663]

Love the LEMMiNO music

[@eyeballdoorknob2330]

I’m having trouble sleeping thx for showing me this because it really helps me sleep

[@Farfocele]

This video blew up - and for good reason! This explains giant numbers very well. Thanks for the video!

[@The_NSeven]

Great video, one of the best I've seen this week! Love big numbers

[@mathisr.v3627]

Your video is awesome ! It’s very well done in the details !

[@paolarei4418]

Why im seeing ya in all vids now LOL

[@yeochxd]

hi mathis, found ya

[@gopalsamykannan2964]

Thanks for your explanation !

[@tabularasa_br]

Inifity always seemed magical to me. When I was a little child, I used to cry when trying to conceptualize the fact that the Universe (might) be infinite, or the sheer fact that there is not a "final number", and that things can always be bigger. I was overwhelmed by this as if I were an old archeologist beholding a non-euclidean Eldritch abomination from a parallel dimension in a Lovecraftian tome. As of today, mathematics is one of my favorite subjects, even though I was terrible at it at school. Finding this channel was like finding a precious gem!

[@PanthereaLeonis]

I had come to terms with infinity, that there is no end. I had not come to terms with how insanely large finity could be!!

[@gazabo-gam463]

When I was little I also thought things like that, about how we are the only thing that exists. There is nothing after death there is no way to escape since this is the only thing of the only thing. I was a weird 8 year old kid.

[@apollyon1]

same. I think there should be maths appreciation at school where we get taught cool shit about what maths can do but don't actually have to do any sums. like I can appreciate sports without having to jog you know!?

[@angularpy]

Wow, this was a super clear explanation. Thanks for sharing this knowledge! 🧠💡

[@rahumor7556]

So I like incremental idle games, they give big numbers and oftentimes feel trivial when you look at the next milestone. That is what Tree(tree(g63)) feels like. Its what silliness do I have to accomplish to reach that number. Love the video keep up the good work.

[@GdEnglishGalaxer]

7:48 Even though it is incredibly massive, It doesn't come close to SSCG(3), SSCG(4), SSCG(5), and SSCG(SSCG(3)).

[@weeblordgaming6062]

When u have completed 3 semesters of calculus but are still very scared right now

[@Jonasz314]

Minor nit - on the last slide, the Greater signs you use are inverted, you mean to say that Tree(3) is greater than g(1000) but it shows g(1000) > Tree(3), and than tree(3) > tree(4). I think it's clear when you listen to the audio, but someone watching it with no audio will be very confused.

[@jyto87yo987]

Amazing videos my man

[@huseynmmmdov9593]

Loved and subscribed!

[@TaranVaranYT]

This is how I learned about hyperoperations. My symbols that I use are right isosceles triangles that split down from the 90° angle right in between each. My symbol for Succession is an outline circle.

[@pyropulseIXXI]

such inefficient notation

[@TaranVaranYT]

@@pyropulseIXXI I know

[@madladam]

I've been asking this question for a year. Love the style and narration. Instant Sub

[@pyropulseIXXI]

it took you a year to not learn, yet keep asking, a question that a literal 6 year old could figure out on their own in less than 12 seconds

[@madladam]

@@pyropulseIXXI I discovered power towers on my own, essentially tetration and I learned of Graham's number, but didn't understand it. I never knew it was actually called tetration until now, nor the official notation. But you sir, have had a difficult day, to be sure. I am deeply sorry for any stress in your life, and I want you to know that there are so many people who love you; and they still love you, even if you write snarky comments on KZbin.

[@sounakbhattacharya9646]

Mind blown! Thank you Sir!!!!!🙏🙏🙏🙏🙏

[@Sebastian-gf2fk]

Underrated channel !!!!

[@jezze419]

Small critique, at the end you use the greater-than symbol > wrong which can lead to confusion

[@digitalgenius111]

Yes, they should be ">". Sorry for the mistake.

[@ChrisMMaster0]

@@digitalgenius111 you did it again XD

[@FailedAtNNN]

@@digitalgenius111

[@redgrengrumbholdt2671]

​@@ChrisMMaster0Chat-GPT be like

[@Kris_with_Banana]

You can see here, the limitless possibilities of math, otherwise known to mathematicians as "fuck it, more"

[@gosnooky]

Mind blowing when you consider that it's not possible to even store such a number physically, even if each digit only took up a single Planck unit of space.

[@EdithKFrost]

Math teacher: Please find the next term of the sequence: 1,3,… People who know the game of trees: 😢

[@pikaboy2dios841]

this is mind boggling in a good way

[@Weird_Jae]

Mind got blown again, just realized these operations can probably be done inversely. So then, Super-roots and Super-logarithm would exist.

[@bijipeter1471]

Thank you, so much

[@yaboijosephhh]

me up at 12am watching this when i don’t even fully understand basic algebra

[@football_cr7_fan10]

Great video ❤

[@generichuman_]

It's really difficult to get an intuition for how big TREE(3) is if you only have Knuth up arrow notation in your tool box. In the fast growing hierarchy, grahams number is on the order of f_omega+1, and if we continue to build larger ordinals to stick into the fast growing hierarchy, we exhaust omega by reaching an infinite tower of omegas which is epsilon naught, an infinite tower of that is epsilon 1, we can continue this and have other ordinals in the subscript of epsilon like epsilon sub omega, or epsilon sub epsilon naught, or even an infinite nesting of epsilons which is zeta naught. We can continue with an infinite nesting of zetas which is eta naught, and to avoid exhausting the greek alphabet we can move on to veblen notation in which epsilon naught is phi 1, zeta naught is phi 2 etc. We can create veblen functions with other ordinals as the argument like phi sub omega, and we can even have infinite nestings of veblen functions which is gamma naught, it then moves on to extended veblen notation which is messy so I switch to using infinite collapsing functions. Infinite collapsing functions define a very large ordinal that "collapses" to a well defined one when put into a function. We have a set that contains {0,1,omega, Omega} where Omega is our large ordinal. We define an ordinal that is the smallest ordinal that can't be constructed using this set using addition, multiplication and exponentiation, which turns out to be an infinite tower of omegas which is epsilon naught. This is Phi(0). We then add epsilon naught to the set and ask what the next ordinal is that can't be created using the set which is epsilon one, so Phi(1) = epsilon one. This continues on, but the function gets stuck at an infinite nesting of epsilons. To bail us out, we can plug Omega into the function and get zeta naught. We continue in this way bailing out the function with constructions of Omega when it gets stuck to reach larger and larger ordinals. Psi(Omega) = zeta naught, Psi(Omega^2) = Eta naught, Psi(Omega^x) = Phi sub x, Psi(Omega^Omega) = Gamma naught, and Psi(Omega^Omega^omega) which is the small veblen ordinal, is roughly on the scale of TREE(3). If you want an in depth deconstruction of this, it's on my channel, just search Giroux Studios.

[@kishorejuki5450]

Dang bro

[@xxUrek]

ah, yes, i know some of those words.

[@handtomouth4690]

Sir, this is a youtube comment section.

[@gpt-jcommentbot4759]

People on the internet are not going to understand bro don't bother explaining FGH to them.

[@seejoshrun1761]

You know all that, but you don't know how to use paragraphs

[@user-xr6xi5ym6e]

For heptation and up, we use {} symbol. This means number like 10^^^^^10, will written as 10{5}10 to represent it instead of using too many arrows

[@alansmithee419]

1:32 Expressions like this are usually said to be undefined since the only realistic way to get an infinity is to be calculating a limit (infinity not being a number), so the 1 may be a limit as well, in which case the way you got to one would determine the result - it is not always 1. e.g. (1+1/n)^n as n --> inf gives e, not 1 as the expression "1^inf=1" would imply, even though the exponent tends to infinity and the base tends to 1. So in this regard we cannot define 1^inf=1, and you'll run into similar problems with tetration.

[@alexandremenino2006]

anime vilains explaining how much times stronger they are from the protagonist

[@DeterminedCharcoalEater]

wow. you made addition sound complex. HOLY CRAP

[@JOLT_x3]

this is very comprehensive.

[@idontknowmusictheory532]

Very interesting. Awesome job!

[@mrsillytacos]

4:18 it goes to g64, not g63...

[@itzashham797]

Since he started it off with 3↑↑↑↑3 as G0 it makes sense in this video If he had started off with 3↑↑↑↑3 as G1 then it would be G64 instead of G63

[@sander_bouwhuis]

Personally, I'm a big fan of tetration because it is easy to explain to laymen, yet already leads to incredibly large numbers.

[@Tenissor]

The HPD(n) function is as follows: HPD(n) = n^^^^(7^20) (Using exponent symbol in the place of Knuth's Up Arrow). HPD(8) is greater than TREE(g64)

[@tetramur8969]

No, it isn't. 8^^^^(7^20) < 8^^^^(8^20)

[@JordanMetroidManiac]

This has me wondering. How does Tree(3) compare to the busy beavers function? After revisiting the busy beaver function, of course the busy beaver function grows faster than Tree(n), lol. And that’s because it is possible to write an algorithm that computes Tree(n), which means eventually BBF(n) > Tree(n).

[@shaunnotsean4308]

Isn't graham's number g64? Either way, it's huge. You made a difficult concept somewhat easy to understand. Great video!

[@PeaceTheBall]

it's g64 if you define 3^^^^3 as g1, but in this video it's defined as g0

[@philiphaseldine1135]

Yes, it is - see the Numberphile videos on the subject

[@TheSpotify95]

The only difference in this video was that g1 (hexation) was defined here as g0, and g2 (the thing with so many arrows we can't write it down) was defined as g1. The principles are still the same.

[@rodjacksonx]

@@TheSpotify95 - The principle's the same, but it still matters if it's literally wrong, especially THAT wrong. That's like saying that a google is 1 followed by 99 zeroes. Sure, you're close conceptually, and may still make your point, but you're literally giving the wrong definition of something with a very clearly stated and well-known definition, and you would fail by putting that answer on any test.

[@Instructor876]

​@@rodjacksonxGoogol. Be precise when criticizing others about precision. 😉

[@DavidHT]

I love how the characters are drawn. I'd love to know how this was done.

[@DoFliesCallUsWalks]

makes inaccessible infinity seem even more unimaginable.

[@MyOneFiftiethOfADollar]

This is a little too easy, but I have ask "what is the smallest large number that ONE can imagine"?

[@datguy3333]

The same thing but negative

[@talkysassis]

Well, what is a large number?

[@Crazytesseract]

It totally depends upon the person.

[@MyOneFiftiethOfADollar]

@@datguy3333 problem with that reasoning is -1 > than all other negative integers

[@xX-JQBY-Xx]

I\♾️

[@janftacnik1207]

I have a theory for an insane number like these (probably not as big though): Take the size of the observable universe and shape it into a cube, then fill the surface of that cube with quark sized squares and paint those squares on each side a different color, make the whole thing work like a rubik's cube. Our number is the number of possible combinations of that hypothetical rubik's cube.

[@pics2299]

Just ran the math, it turns out the number is not that big: With a volume of 4.3x10^32 ly^3, if the observable universe were a cube, the length of its side would be 7.6x10^10 ly = 7.2x10^44 quark lengths. Using the general formula for the number of permutations of nxnxn sized rubik's cubes (from Christopher Mowla on Quora), we get a number with approximately 2.0x10^90 digits (it's still quite big, but nothing like the numbers in the video, it's even smaller than the googolplex 10^(10^100))

[@rickb_NYC]

@@pics2299 That is interesting, and leads to the question of if there is any way to get a visualization or an intuition for these huge numbers. Because at least what @janftacnik1207 has a narrative behind it. You can say, "imagine if we...".

[@janftacnik1207]

@pics2299 Thank you for doing the math, this has been going through my head for the past few months and now i know how big it is. It’s a little underwhelming, but still huge. You just made my day.

[@CheeseSummoner]

3:18 really good way to show 2 hexation 2 would be 4

[@holo673]

Maths like this reminds me of a game i played when i was younger called tasty planet, where you started out as a microscopic organism that eats bacteria, before ending the game by eating galaxies and eventually just the fabric of the universe

[@VaraNiN]

What's the music starting @ 3:48 called? Ah, nvm, should have read the description, lol. It's "Lemmino - Cipher"

[@AlphaPizzadog]

Now what is the inverted function. Addition has subtraction, multiplication has division, exponents have square root (for x^2) and logarithms (for 2^x), what does tetration and above have?

[@big_numbers]

Super root and super logarithm

[@Luigicat11]

@@big_numbers What comes next after those? Super super root/logarithm? Ultra root/logarithm?

[@gsilva877]

Very well explained.

[@thewaver8]

Really awesome video - just an itsy bitsy sad that you didn't give kudos to Knut for the arrow notation once you got to Tetration :)

[@tomokokuroki2506]

Then there's Vexation, where you repeatedly hit yourself in the head.

[@worldprops333]

vexation is the 1,005 level hyperoperation (number derived from hebrew gematria of vex)

[@briangronberg6507]

I don’t understand the visual illustrations for TREE(3) at 6:53. Any of the two blue seeds in the 4th example have a green seed as a common ancestor but in the 3rd example two blue seeds have a green seed as a common ancestor. In the 5th example two blue seeds are connected, but we see not one, but two cases where two blue seeds are connected in the 4th example. In the 6th example there are three blue seeds linked together in the same way as we see in the 4th example and the blue-blue-green-blue pattern on the right is exactly the same as the right pattern in the 5th example. What is it that I’m missing?

[@joshuagetusername4779]

There's two things: First off, an entire tree must be contained. Having a part of a previous tree contained is not the issue so just because one tree contains two connected blue seeds doesn't mean any future tree can't contain two connected blue seeds. Second, it is about the NEAREST common ancestor. Yes any pair of blue seeds in the fourth example have a green seed as an ancestor. But that wouldn't be the nearest one, instead it would be the lower one of the two blue seeds themself as that one is already an ancestor to the other one. This might be a bit hard to put into text form for a YT comment and for a relatively short video it would of course be difficult to go into a lot of detail on this. Though if you want Numberphile went into some more detail during their video on Tree(3).

[@a.h.ss8316]

The last 20 seconds literally killed me I was laughing out of madness

[@spieagentl]

Am I mistaken, or are the greater than signage in the last section flipped? Regardless, this was a very informative and well-made video! Thank you for the lesson!

[@MaharetS]

they are

[@dutonic]

I would not have suspected that tree(3)>grahams number. That was genuinely surprising to me