FINALLY someone explained to me that there IS a repeated exponentiation called tetration. i literally asked all my math teachers about this and they all said "such a function does not exist" and only when i asked the smartest math nerd in the class, he told me it was like exponentiation but in front. but when i told that to the math teachers, they said it was nonsense. im glad someone finally explained to me all this. you earned yourself a subscriber
you would be surprised how much math teacher dont know or at least pretend not to. I once told a classmate about imaginary numbers and he asked the teacher and the teacher said no such thing. There was this joke going around that i was schizophrenic and this occasion did not help it lol
@TannerJ07 Maybe your calc teacher really didn't know until the time came for your teacher to teach integrals. Teachers sometimes need to teach stuff they're not familiar teaching because they're assigned by the powers that be to teach a subject that they are not familiar with. Schools can be lousy that way.
[Break the 4th wall?] "Yeah uh... World Machine, please put me back in Doki Doki Literature Club please..." [No no, the only 4th wall that exists, is a giant mass of dead Deadpools, obviously nobody cares about OneShot.] "Okay..."
This is a great example of how math is all interconnected, and why 2² (two squared) literally makes a square. Formula for area is A². You literally turn it into a square... hole.
07:24 At which point everyone says to themselves, _"Oh, of course. 65,536"_ and no one's joking because we all know that number the same way we know 256
@ I know, but factorial is an operation with a single argument (in this case, a single 2), while all the other operations use two arguments (both being 2s)
Actually I also have heard of this theorem before, it is really interesting to read about! I highly suggest everyone to google it when you have some spare time
This is... this is the perfect youtube thumbnail. The second i glanced at it, i knew what the subject was. And before i even watch it, this was something that drove me nuts in Jr High, and i pesterd so many math teachers about 2+2=2*2=2^2. I didn't know about the other operations, or i would likely have been even more insane about it. That was the only math thing that ever caught me tho.
these hyperoperators are actually part of a niche field of mathematics called googology, and i highly recommend people to dig deeper into it! there are many other notations such as bird's array notation and other functions that give weirdly crazy numbers (although it has been a long time, so i might be inaccurate). also, slight correction at 10:08 - 2 -> 2 -> 4 should be 2^^^^2 (4 arrows) instead of 2^^^2 (only 3 arrows)
I love that this implies that there exists a function whose output sequence hops straight from 4 (hyperoperations based on two) all the way to Graham's Number (hyperoperations based on three) ... I'm probably wrong, but the thought is still pretty funny :)
yeah this is how fast growing functions tend to go. they give a few normal numbers, then you get a number that's more than the atoms in the observable universe.
@liam.28 It is based on the up arrow notation and 3. It's just that the number of up arrows used is inconceivably greater than the number of planck volumes in the observable universe. And even that is massively understating how large it is.
This is both hilarious and also may be strangely useful if we ever have to establish communication with aliens via radio or whatever (as in, not immediately face to face). Numbers and mathematical operations and constants are among the first things you always read about in hard science fiction where human and alien starships unexpectedly meet for the first time - because it makes sense and should be a fairly universal touchstone to show others how we represent information and communicate about it. Demonstrations of what the symbols (or transmitted codes) for operators (and even _the equal sign_ ) mean would probably be easier if you have several operations for which the same inputs give the same answers (so each line is identical except for the listed operation), and _then_ you can go through the results of varying one of the inputs for each operation to a series of different values, allowing them to pin down exactly what operation is associated with each. And a notation which generalizes these operations, like you showed at the end, could help with making them more orderly and understandable.
I love coming here from Reddit, about to perform a cumbersome search of the internet in an attempt to find out what this meme was about, when you so conveniently made a video about it (: thank you
Ok, but does this pattern also extend to non-integer hyperoperators (like halfation and sesquation)? That's something I've always wondered, because it's PROBABLY true but I'm not sure if it's FORCED to be true
I remember discovering this in math class years and years ago, but I was unaware that there were higher iterations than exponents. This was really neat. I really enjoyed it. I find it humorous how 2 does this.
so glad we changed successor to increment. It just makes more sense. Also, a better explanation of multiplication (and what it's ACTUALLY telling you to do): Start at 0. You first term is what you add to 0, the second term is how many TIMES you add it to 0 (hence, x times y). 2x2 = 0+2+2 = 4. Doesn't make sense to write it that way, due to our (correct) habit of eliminating useless terms, but it demonstrates why we call it times. This also makes its inverse operation easier to explain - division is the process of starting at a number, x, and taking y from it until we get as close to 0 as we can without going under, resulting in the amount of times we can do so (quotient) and what's left over (if not 0, the remainder). What would really be nice is explanations of the inverse of exponents like that, so people could once again actually learn to do math without a calculator on things like square roots without that awful guess and check crap I was taught (which was like 1 day of math, followed by the teacher going "from now on, we'll use calculators for that"). Knowing _why_ things work is required for things like math and programming, and it's a shame that one spends TOO MUCH time on the _why_ (programming), while the other completely ignores it for the most part (math in school).
tbh this information is irrelevant in the age of calculators. I'm sure 100 years ago knowing how division and multiplication worked was a mandatory skill for any academic/scholar. But nowadays it simply isn't. Outside of doing some basic every day addition/substraction like calculating the time, you don't need to know math for anything. Anything remotely difficult you have a calculator in your pocket. So why would schools spend more time than necesary to teach that
I follow Alison Burke on UA-cam, and that "square hole" meme has certainly been the viral moment of her career! Following on from memes, I'm thinking of the one where Omni-Man points to the fighter jet, and here he's tan(theta) as theta -> tau/4, and the fighter jet is Conway's notation.
Speaking of the Omni-Man meme, I've seen one where the jets are labelled "floating point numbers" and he's saying "Look what they need to mimic a fraction o̶f̶ ̶o̶u̶r̶ ̶p̶o̶w̶e̶r"
Someone should 3D print toys outta these, just the shapes of the mathematical operations on one side, then turn it around in the y-axis, z-axis on top, then it's the shape of the number 4.
It has always fascinated me this phenomenon where no matter the how high you go, if you have 2 in the first and second slot of the function, you get 4.
the value of k isn't valid for Knuth's up-arrow notation if it's less than 1. specifically, the notation requires that a, n, and k are integers, a >= 0, n >= 0, and k >= 1.
What nice about this, is 1 also shares this property (except addition). This means we know that any number between 1 and 2, with any of these operators, must also be between 1 and 4, by implication. Banger videos as always!
I'd never seen conway's notation before, so after you explained it, I thought it should be 2 ➡️2➡️♾️ but then I thought about it and realized "n" is far worse!
Finally a good explanation of hyperoperations. The arrow notation came up as early as Numberphile's video on Graham's number like 10 years ago, but it was never as clear as in this video.
Proof by induction that all 2 ^k 2 = 4: For k=0 this is because 2 ^0 2 = 2 + 2 = 4, and assuming the claim is true for k, we see that 2 ^(k+1) 2 = 2 ^k 2 = 4.
I was familiar with the tiktok video, and could guess all the solutions would be "4". What I was unfamiliar with was the exponent being in front and those up arrows and sideways arrows, so thank you. But once you started explaining, I was like, those numbers are gonna get big fucking fast even if you just do 3 like (small 3)3 or even (small4)2 So thank you, learned something new, pentation and other terminology
Yeah so 3^^^^3 is number known as graham’s number, which is famous for being extremely big and you can see that only uses 3s, and it could easily get huge
Yea. Im gonna follow you now. I used to love math but due to a really traumatic experience and teacher in my childhood, I went on to “hate math” and since math builds upon itself it led to me being (or just saying which led to being) terrible at math. But this explanation never really lost me and just made sense. Of course because I’m behind it’s not like I really “learned” much but it made me remember that the way math works is why I liked it in the first place. I’m a logical thinkin that tends to overthink so the fact that math has rules and structure is why there isn’t really much to overthink. Only apply. Basically this kinda reignited my love for math a bit. Thank you. Take my sub.
9:45 I had to look up Conway's notation here because it seemed odd to me that it'd be a->n->4 for pentation and not a->n->3 to match the other arrow notation. At least the wikipedia page seems to think a->n->3 would be pentation, since it lists 2->4->3 as 2^^^4 (so final 3=pentation), so that'd be a minor nit in the video if I'm not just misunderstanding something.
I think Wikipedia might be wrong as iirc Conway's Chain Arrow Notation starts at length 2 for exponentiation, which makes a→n→3 tetration. The Wikipedia editor probably just got confused because the behaviour is different from the other Arrow Notations which start at 1 for exponentiation.
I had a math prof who had a similar stile to this video. He always picked some ridiculous example like this so explain the a concept. This had a lot of advantages. It was simple and memorable. It also served to show a concept at it's extremes. One of his core ideas of teaching was, that if you understand and remember the extrem behaviours of what ever you are looking at, you have an easier time understanding the stuff in the middle. Worked great for me.
10:05 But why 4 for pentation? I thought it was 3 as stated in the introduction of this new notation...? 1 for exponentiation 2 for tetration 3 for pentation ... Guess it's just a small error :) Thank you for this video!
You use natural numbers here, can the hyper-operators be extended to all reals and beyond, and are there any logarithms associated with the hyperoperators?
I mean they Can be, but between impractically and the difficulty of calculating decimal hyper-operations, non-integer hyper-operations aren't exactly well documented. You pretty much have to just reduce them to exponentiation operations, which will usually be irrational and consequently prone to imprecision as k increases. But so long as you have a valid k value, any values for a & n are at least theoretically calculable, if not practically so (2.5^π is a great practise example for decimal hyper-operations).
humor me for a moment, what if we want it to be commutative turning exponentiation into a commutative operation can be done, in the following way: we can make an observation recall that ln(a) + ln(b) = ln(a*b) so applying a logarithm transforms addition into multiplication if we apply it again ln(a)*ln(b) = ln(a^ln(b)) so applying a logarithm transforms multiplication into our 'commutative' exponentiation so assume we have some 'commutative' hyperoperation % and the next such operation $ ln(a)%ln(b) = ln(a$b), therefore a$b = exp(ln(a)%ln(b)) therefore, going 'up' a 'commutative' hyperoperation is simply the application of logarithm to its inputs and the exponential function (exp, e^.) to its output. that means it's possible to go 'down' indefinitely (something not true for regular hyperoperations) by applying the exponential function to its inputs and the logarithm to its output ln(exp(a)$exp(b)) = ln(exp ( ln(exp (a) ) % ln(exp (b) ) = a%b due to being able to go 'up' and 'down' by applying two inverse operations, that means they form an isomorphism we can use to generate groups from additive and multiplicative groups, corresponding to this operation. The usefulness of this is being able to identify the identity elements. e.g. the additive identity is 0, so exp(0) gives the next identity, which is 1 the multiplicative identity, so exp(1) gives the next identity element, which is e itself, then e^e, e^e^e, etc. let's call this identity E(?) for some operation ?, so for successive 'commutative' hyperoperations E($) = exp(E(%)) and E(%) = ln(E($)) that also means means we can define inverses with respect to these operations, since they are commutative (otherwise there should be both a left and right inverse that are distinct) the inverses for successive 'commutative' hyperoperations are given by: (x$'y)$y = x ln(x$'y)%ln(y) = ln(x) ln(x$'y) = ln(x)%'ln(y) x$'y = exp(ln(x)%'ln(y)) so E($)$'y is the inverse element, let's write this as $'y for short, so $'y = exp(ln(E($))%'ln(y)) = exp(E(%)%'ln(y)) = exp(%'ln(y)) the inverses probably can't be considered 'logarithms' since they don't even include the usual logarithm. so can we create a 3-parameter operation (x,y,z) that computes the z'th 'commutative' hyperoperation of x and y? how expansive can we make the domain? I will call this 'uniformly parameterised' if we can break the domain down into a cartesian product X . Y . Z. (not merely a subset thereof). Of course, as it's commutative, X = Y. the natural logarithm is defined for all values in the complex plane except for 0, however it is multivalued. without using complex numbers, we have a single valued map from R+ to R choosing a single branch gives us a single valued map from C* to R U i[-pi,pi] or if we instead take the riemann surface of ln, called S, then we have a single valued map from S to C in the reals, we have that ln is a map from R+ to R, which means it can only work on a limited domain ergo a 'uniformly parameterised' 'commutative' hyperoperation cannot exist (at least with a non-trivial domain), as the domain must shrink indefinitely to cover all cases. but in the complex numbers, we have that ln is a more complicated map, I don't know what to conclude from that. At the very least we know the domain must always change, which is probably enough to say the same as in the real case. one can probably make an argument without introducing the 'commutative' property but I am not able to make it, although maybe they truly are categorically different. even if it's not uniformly parameterised it's still a fun idea to try to define, but we're going to have to figure out the constraints on the inputs and without uniformity it's pretty hard to say that it could be continuously extendable to intermediate values like rational or real 'z' values. in the 'commutative' case, the isomorphism between the sucessive operations means we must find a way to find the continuous 'endofunction extent' of the exponential function, which very well could be arbitrary in the same way there are arbitrary continuous extensions of the factorial in the (0,1) interval (the gamma function is not the only useful one)
2>2>n is just the expression defining the whole chain, 2>2>4 is 2^^^2, but everything else is correct I think Edit: reading some other comments, 2>2>3 might be 2^^^2, but I'm not sure as I'm finding differing thoughts on this
Just asked my teacher, he laughed a lot, said that he didnt knew about it but then remembered a few things and we agreed on solutions and stuuf. Great video
Yes, in the same way multiplication is a repeated addition, division is a repeated subtraction. The result of a division is how many times you can subtract a number until you can't (you then get the reminder that is smaller than the divider)
also following what jeromesnail said, the exponential function turns addition into multiplication, and in exactly the same manner it turns subtraction into division.
Subtraction is opposite of addition Repeated addition is multiplying Multiplying is opposite of division So division is opposite of repeated opposite of subtraction
I realized this awhile ago and have thought a lot about it in the past, very cool to actually see it in a video. It'd be interesting to analyze the behavior of numbers very close to 2 for each hyperoperation, similar to taking a limit.
Hey thanks for using my video in such a cool and informative way!
Thank you for making such a funny video! It's ironic that this old Tiktok video should be used in a math lesson that has nothing to do with geometry!
It is really cool :3
I’ve seen it used with an orthopedic surgeon ordering antibiotics for different infections. 😅 It’s a wonderful video!
The video so good you know where it goes?
That's right it goes into the square hole
Thank you both! This video was so fun and insightful. ❤
Guess what shape of hole has 4 sides?
That's right! The square hole!
That's right, it's the quadrilateral hole!
Guess what the *2nd* digit of the *square* root of 4 is?
That’s right, it’s 4
Guess how many legs a quadruped has?
That's right, it's 4
@@noober52614 Wait, I never thought of that. Weird. I has to be a coincidence... right?
@@AbacusMan The square root of 4 literally is just 2. 😂
now i can put 2 and 2 together
ba dum tsss
This shit is too stupid for me to laugh at 😂😂😂😂
I love this
@@AliAli-hg9njIt's so awesome but also such a terrible joke that I wanna fight him 😂
Wow...😮
Despite everything, it’s still 4
Damm it, even on a math video, UNDERTALE still follows me.
@@yahshua1073 Undertale is life
Those who know:
BFDI:
4 really is determined to stick around
FINALLY someone explained to me that there IS a repeated exponentiation called tetration. i literally asked all my math teachers about this and they all said "such a function does not exist" and only when i asked the smartest math nerd in the class, he told me it was like exponentiation but in front. but when i told that to the math teachers, they said it was nonsense. im glad someone finally explained to me all this. you earned yourself a subscriber
a^^^^n should be called penetration
you would be surprised how much math teacher dont know or at least pretend not to. I once told a classmate about imaginary numbers and he asked the teacher and the teacher said no such thing. There was this joke going around that i was schizophrenic and this occasion did not help it lol
It's rarely ever useful, so it makes sense that no one knows what it is
My calculus teacher pretended not to know about integrals until January
@TannerJ07 Maybe your calc teacher really didn't know until the time came for your teacher to teach integrals. Teachers sometimes need to teach stuff they're not familiar teaching because they're assigned by the powers that be to teach a subject that they are not familiar with. Schools can be lousy that way.
this is surely... the power of two
ba dum tss
BFDI reference!?!?
@ yes friend. yes.
those who know
didn't expect a BFDI reference on a math video
breaking the *4th* wall
no, no - to do that, we need to find an operation that we can put two twos into and not get four. such as, for example, an nth root…
@@somethingforsenro 2-2=0. 2/2=1. root{2}(2)=1.414... log_{2}(2)=1. You get the point. All "negative" operators done on 2 give non-4 answers.
[Break the 4th wall?]
"Yeah uh... World Machine, please put me back in Doki Doki Literature Club please..."
[No no, the only 4th wall that exists, is a giant mass of dead Deadpools, obviously nobody cares about OneShot.]
"Okay..."
that's right! the theater goes in the square hole!
Where do the humans go?
. . .
That's right, the four wall square whole
The fact that squares have 4 sides makes this even better
This is a great example of how math is all interconnected, and why 2² (two squared) literally makes a square. Formula for area is A². You literally turn it into a square... hole.
07:24 At which point everyone says to themselves, _"Oh, of course. 65,536"_ and no one's joking because we all know that number the same way we know 256
my two favorite numbers
There are still people out there who have never found a mysterious number in a program/game that's almost -2.14 billion. The rest of us, we know.
i know the powers of 2 for many reasons
Heck yeah more 65536 fans other than me :D
@@tristanridley1601 yay, a 32-bit signed integer overflow.
She’s gonna be happy when she finds out about “2!”
that's only one 2 though
@@gwgwgwgwgwgwgwgwgw I think they're referring to 2! literally, as in the factorial of 2
@ I know, but factorial is an operation with a single argument (in this case, a single 2), while all the other operations use two arguments (both being 2s)
Fun Fact: You can replace any 2 in any of these operations with 2! and it won't change anything other than making it more exciting.
@@gwgwgwgwgwgwgwgwgw its literally one*2
This is surely a result of the fourier transform, which SMBC taught us makes things four-ier
SMBC mention!
take my angry upvote
i love super mario bros... cocaine...
@@crazgon7261 did you post this after scrolling on reddit or smth lol
HAPPY CAKE DAY
1:53 absolutely the most pedantic nitpickiest of nitpicks, but I’d argue the most basic operation is the identity function
That is most definitely correct
@@sofer2230 it definitely was not the implication, as the successor function also only takes one input lol
Negation?
Well, we need the 2nd most basic function. For the identity applied no matter how many times will not result in a different number.
@sasho_b. yeah that’s what makes it so nitpick-y haha
I expected this video to be like 3 years old, not an hour lol
😂😂
Stickerbush symphony vibes
SAME
Old numberphile vibes
Same. It has 2020 math vid vibes for me (in a good way)
This is all part of the theorem that in all spacer lower than 3 dimensions the limits tend to go towards 4 - that is also known as the "rule 34 hole"
Don't you dare.
Genius
Actually I also have heard of this theorem before, it is really interesting to read about! I highly suggest everyone to google it when you have some spare time
This is 200iq in many ways ahahah
I thought this was real interesting till I saw that last sentence 😭
This is... this is the perfect youtube thumbnail. The second i glanced at it, i knew what the subject was. And before i even watch it, this was something that drove me nuts in Jr High, and i pesterd so many math teachers about 2+2=2*2=2^2. I didn't know about the other operations, or i would likely have been even more insane about it.
That was the only math thing that ever caught me tho.
I would be so happy if I had a student that got this into a piece of math
9:03 her distressed face popping in and out of frame with the backdrop of a serious mathematical explanation is hilarious
Basic program:
LET A=4.
Then all go in the A hole.
these hyperoperators are actually part of a niche field of mathematics called googology, and i highly recommend people to dig deeper into it! there are many other notations such as bird's array notation and other functions that give weirdly crazy numbers (although it has been a long time, so i might be inaccurate).
also, slight correction at 10:08 - 2 -> 2 -> 4 should be 2^^^^2 (4 arrows) instead of 2^^^2 (only 3 arrows)
Actually the 3 arrows is correct, he's writing the pentation represented by 2 -> 2 -> 4, not just changing notation
YEAHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH
Googology mentioned
@ YEAHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH (2x)
mmm, googology my beloved
I love that this implies that there exists a function whose output sequence hops straight from 4 (hyperoperations based on two) all the way to Graham's Number (hyperoperations based on three)
... I'm probably wrong, but the thought is still pretty funny :)
grahams number is not hyperoperators based on three
@liam.28on the same ballpark
yeah this is how fast growing functions tend to go.
they give a few normal numbers, then you get a number that's more than the atoms in the observable universe.
@@binarycat1237tree()
@liam.28 It is based on the up arrow notation and 3. It's just that the number of up arrows used is inconceivably greater than the number of planck volumes in the observable universe. And even that is massively understating how large it is.
1:06 So you’re telling me that it happens to equal to four like that? No! They orchestrated it! Four!
You could say… it was FOURchestrated…
no not four! 24
AND THEY GET TO BE FOUR? WHAT A SICK JOKE.
The moment I heard the word "chicanery," I knew someone would comment this
Mista is gonna be mad
I rapped 17,000 digits of pi: ua-cam.com/video/Qt4pdX-w2JA/v-deo.html
Thanks for watching!
looks like ack(2,2,n) = ack(2,2,n+1) = 4 :D
When Iife gives you two, it onIy ever means to give you four
Sometimes I forget how okay Origami King is.
@@loganpowers3875 wdym
bro really posted a vip restricted link
Ah yes hyoer operations, 2 pentated to 2 is 4, 2 pentated to 3 is 65536, beauty
and then 2 pentrated to 4 is probably too big to write in a youtube comment
@@MyNameIsSalo 1.1579209e+77 (I think, no clue if that’s actually right. 2^256)
1 penetrated inside 0 back and forth is aah binary number sèx.
This is both hilarious and also may be strangely useful if we ever have to establish communication with aliens via radio or whatever (as in, not immediately face to face). Numbers and mathematical operations and constants are among the first things you always read about in hard science fiction where human and alien starships unexpectedly meet for the first time - because it makes sense and should be a fairly universal touchstone to show others how we represent information and communicate about it.
Demonstrations of what the symbols (or transmitted codes) for operators (and even _the equal sign_ ) mean would probably be easier if you have several operations for which the same inputs give the same answers (so each line is identical except for the listed operation), and _then_ you can go through the results of varying one of the inputs for each operation to a series of different values, allowing them to pin down exactly what operation is associated with each. And a notation which generalizes these operations, like you showed at the end, could help with making them more orderly and understandable.
I love coming here from Reddit, about to perform a cumbersome search of the internet in an attempt to find out what this meme was about, when you so conveniently made a video about it (: thank you
Reddit or spotted
@Boselaphus Reddit. I don’t know of a platform called spotted? XDXD Sorry I had to
1 also has this unique property in all operations higher than addition.
Thus the property is not unique.
@@gamesofsteffen9159if you exclude addition
@gamesofsteffen9159 You got me there.
The difference is that x -> 1 -> n = x (I think)
And 0 absolutely fucks everything up
Ok, but does this pattern also extend to non-integer hyperoperators (like halfation and sesquation)? That's something I've always wondered, because it's PROBABLY true but I'm not sure if it's FORCED to be true
The only agreed upon definition is in the complex set of numbers
What is halfation? Never heard of it.
Same@@HugoHabicht12
@@HugoHabicht12 Halfation is halfway below addition. Sesquation is half between addition and multiplication.
@@PatashuCan you define what "half between addition and multiplication" means?
I remember discovering this in math class years and years ago, but I was unaware that there were higher iterations than exponents.
This was really neat. I really enjoyed it. I find it humorous how 2 does this.
"Wait, it's all four?"
"Always had been"
2:13 you made addition look hard
so glad we changed successor to increment. It just makes more sense.
Also, a better explanation of multiplication (and what it's ACTUALLY telling you to do): Start at 0. You first term is what you add to 0, the second term is how many TIMES you add it to 0 (hence, x times y). 2x2 = 0+2+2 = 4. Doesn't make sense to write it that way, due to our (correct) habit of eliminating useless terms, but it demonstrates why we call it times. This also makes its inverse operation easier to explain - division is the process of starting at a number, x, and taking y from it until we get as close to 0 as we can without going under, resulting in the amount of times we can do so (quotient) and what's left over (if not 0, the remainder).
What would really be nice is explanations of the inverse of exponents like that, so people could once again actually learn to do math without a calculator on things like square roots without that awful guess and check crap I was taught (which was like 1 day of math, followed by the teacher going "from now on, we'll use calculators for that"). Knowing _why_ things work is required for things like math and programming, and it's a shame that one spends TOO MUCH time on the _why_ (programming), while the other completely ignores it for the most part (math in school).
tbh this information is irrelevant in the age of calculators. I'm sure 100 years ago knowing how division and multiplication worked was a mandatory skill for any academic/scholar. But nowadays it simply isn't. Outside of doing some basic every day addition/substraction like calculating the time, you don't need to know math for anything. Anything remotely difficult you have a calculator in your pocket. So why would schools spend more time than necesary to teach that
I follow Alison Burke on UA-cam, and that "square hole" meme has certainly been the viral moment of her career!
Following on from memes, I'm thinking of the one where Omni-Man points to the fighter jet, and here he's tan(theta) as theta -> tau/4, and the fighter jet is Conway's notation.
Speaking of the Omni-Man meme, I've seen one where the jets are labelled "floating point numbers" and he's saying "Look what they need to mimic a fraction o̶f̶ ̶o̶u̶r̶ ̶p̶o̶w̶e̶r"
Someone should 3D print toys outta these, just the shapes of the mathematical operations on one side, then turn it around in the y-axis, z-axis on top, then it's the shape of the number 4.
It has always fascinated me this phenomenon where no matter the how high you go, if you have 2 in the first and second slot of the function, you get 4.
The cut in the middle of writing 65,536 was perfectly placed, great edit
They also all go in the square hole
Guess how many sides a square has?
the 2 squared hole
let me remind you what 2 squared is
Guess how many letters are in *four*
0:32 you can clearly see the square (hole) is a 2x2 square.
8:43 would a↑⁰n be multiplication, and if so would a↑⁻¹n be addition?
yes
the value of k isn't valid for Knuth's up-arrow notation if it's less than 1. specifically, the notation requires that a, n, and k are integers, a >= 0, n >= 0, and k >= 1.
WITCHCRAFT!
@@iamc24 I think conway might let it slide.
Then would a(^(^(-2))n be succession?
Wonderful video.
I don't think I've ever seen Conway's chained arrow notation. Neat.
Thank you!
11:12 so what you’re telling me, is it is the way it is because it is the way it is?
it is what it is 🙂↕️
All math is what it is. Its all defined by some axioms
Arrows gives me goosebumps because they look like a secret language to control walls at least that’s what my dream was about.
Im not going to lie, i failed highschool mathematics and this is making my brain leak. Gonna try this again later.
A 2x2 block preserves itself in the Game of Life, and the usual Look-and-Say function sends 22 to 22.
What nice about this, is 1 also shares this property (except addition). This means we know that any number between 1 and 2, with any of these operators, must also be between 1 and 4, by implication.
Banger videos as always!
actually.... for tetration and onward, there isn't yet a proper way to raise something to a decimal
This is what makes addition the "weird" one among these iterated operations
1+1=2
1*1=1
1^1=1
because 1*1 isnt 1+1 or something
@ read: > (except addition)
@Axcyantolwell, 1+1 is 1*2 as it's two terms, while 1*1 is +1 because it's a single instance of addition of 1, hence the exception.
I'd never seen conway's notation before, so after you explained it, I thought it should be 2 ➡️2➡️♾️ but then I thought about it and realized "n" is far worse!
Finally a good explanation of hyperoperations. The arrow notation came up as early as Numberphile's video on Graham's number like 10 years ago, but it was never as clear as in this video.
What does 2+2 go in? That’s right! The 4 hole! 2x2? Right! The 4 hole! 2²? The 4 hole! ² 2? The 4 hole! 2>2>n? The 4 hole!
“You know what else is arbitrary? FOUR!…O FOURK! ITS EVEN ON MY FOURARM!” -Micheal Stevens
I just saw the thumbnail! I've literally been waiting for you to make this video! Hell yeah! Let's go!
good video 👍🏻
5:27 penatration
AYOOOO
I thought exactly that!
😂😂😂
AYO-
😂😂😂
Proof by induction that all 2 ^k 2 = 4: For k=0 this is because 2 ^0 2 = 2 + 2 = 4, and assuming the claim is true for k, we see that 2 ^(k+1) 2 = 2 ^k 2 = 4.
I always wondered if this pattern continued as a child
Everything goes in the "squared" hole
I really appreciate the simplicity. I get confused easily :)
I was familiar with the tiktok video, and could guess all the solutions would be "4".
What I was unfamiliar with was the exponent being in front and those up arrows and sideways arrows, so thank you.
But once you started explaining, I was like, those numbers are gonna get big fucking fast even if you just do 3 like (small 3)3 or even (small4)2
So thank you, learned something new, pentation and other terminology
Yeah so 3^^^^3 is number known as graham’s number, which is famous for being extremely big
and you can see that only uses 3s, and it could easily get huge
Explaining funny math jokes! Great job. Power two you man.
Yea. Im gonna follow you now. I used to love math but due to a really traumatic experience and teacher in my childhood, I went on to “hate math” and since math builds upon itself it led to me being (or just saying which led to being) terrible at math. But this explanation never really lost me and just made sense. Of course because I’m behind it’s not like I really “learned” much but it made me remember that the way math works is why I liked it in the first place. I’m a logical thinkin that tends to overthink so the fact that math has rules and structure is why there isn’t really much to overthink. Only apply.
Basically this kinda reignited my love for math a bit. Thank you. Take my sub.
12:01 is 3 fours plus 1 (4 numbers) and 1+2+1 is also 4.
9:45 I had to look up Conway's notation here because it seemed odd to me that it'd be a->n->4 for pentation and not a->n->3 to match the other arrow notation. At least the wikipedia page seems to think a->n->3 would be pentation, since it lists 2->4->3 as 2^^^4 (so final 3=pentation), so that'd be a minor nit in the video if I'm not just misunderstanding something.
I think Wikipedia might be wrong as iirc Conway's Chain Arrow Notation starts at length 2 for exponentiation, which makes a→n→3 tetration. The Wikipedia editor probably just got confused because the behaviour is different from the other Arrow Notations which start at 1 for exponentiation.
The meme made learning about these new operations entertaining, thanks!
8:41 Is this the explanation for the x^n notation for exponentiation I’ve been using in programming for the last 30 years? Was it Knuth all the time?
random recommendations by yt is sure amazing. who could've thought that I'd watch the whole video of "math" with full concentration! Amazing!
4:06 Free tooth? Yay!
I had a math prof who had a similar stile to this video. He always picked some ridiculous example like this so explain the a concept. This had a lot of advantages. It was simple and memorable. It also served to show a concept at it's extremes. One of his core ideas of teaching was, that if you understand and remember the extrem behaviours of what ever you are looking at, you have an easier time understanding the stuff in the middle. Worked great for me.
Chicannery
Slipping squares!!
i think hijinx would have been a better word - but oh well, chicanery it is! or rather, chicanery it was
Shenanigans!
Dread it, run from it, 4 arrives all the same
10:05
But why 4 for pentation? I thought it was 3 as stated in the introduction of this new notation...?
1 for exponentiation
2 for tetration
3 for pentation
... Guess it's just a small error :) Thank you for this video!
This is like the carcinization of mathematics
You use natural numbers here, can the hyper-operators be extended to all reals and beyond, and are there any logarithms associated with the hyperoperators?
I mean they Can be, but between impractically and the difficulty of calculating decimal hyper-operations, non-integer hyper-operations aren't exactly well documented. You pretty much have to just reduce them to exponentiation operations, which will usually be irrational and consequently prone to imprecision as k increases. But so long as you have a valid k value, any values for a & n are at least theoretically calculable, if not practically so (2.5^π is a great practise example for decimal hyper-operations).
humor me for a moment, what if we want it to be commutative
turning exponentiation into a commutative operation can be done, in the following way:
we can make an observation
recall that
ln(a) + ln(b) = ln(a*b)
so applying a logarithm transforms addition into multiplication
if we apply it again
ln(a)*ln(b) = ln(a^ln(b))
so applying a logarithm transforms multiplication into our 'commutative' exponentiation
so assume we have some 'commutative' hyperoperation % and the next such operation $
ln(a)%ln(b) = ln(a$b),
therefore a$b = exp(ln(a)%ln(b))
therefore, going 'up' a 'commutative' hyperoperation is simply the application of logarithm to its inputs and the exponential function (exp, e^.) to its output. that means it's possible to go 'down' indefinitely (something not true for regular hyperoperations) by applying the exponential function to its inputs and the logarithm to its output
ln(exp(a)$exp(b)) = ln(exp ( ln(exp (a) ) % ln(exp (b) ) = a%b
due to being able to go 'up' and 'down' by applying two inverse operations, that means they form an isomorphism we can use to generate groups from additive and multiplicative groups, corresponding to this operation. The usefulness of this is being able to identify the identity elements.
e.g. the additive identity is 0, so exp(0) gives the next identity, which is 1 the multiplicative identity, so exp(1) gives the next identity element, which is e itself, then e^e, e^e^e, etc.
let's call this identity E(?) for some operation ?, so for successive 'commutative' hyperoperations E($) = exp(E(%)) and E(%) = ln(E($))
that also means means we can define inverses with respect to these operations, since they are commutative (otherwise there should be both a left and right inverse that are distinct)
the inverses for successive 'commutative' hyperoperations are given by:
(x$'y)$y = x ln(x$'y)%ln(y) = ln(x) ln(x$'y) = ln(x)%'ln(y) x$'y = exp(ln(x)%'ln(y))
so E($)$'y is the inverse element, let's write this as $'y for short, so $'y = exp(ln(E($))%'ln(y)) = exp(E(%)%'ln(y)) = exp(%'ln(y))
the inverses probably can't be considered 'logarithms' since they don't even include the usual logarithm.
so can we create a 3-parameter operation (x,y,z) that computes the z'th 'commutative' hyperoperation of x and y? how expansive can we make the domain? I will call this 'uniformly parameterised' if we can break the domain down into a cartesian product X . Y . Z. (not merely a subset thereof). Of course, as it's commutative, X = Y.
the natural logarithm is defined for all values in the complex plane except for 0, however it is multivalued.
without using complex numbers, we have a single valued map from R+ to R
choosing a single branch gives us a single valued map from C* to R U i[-pi,pi]
or if we instead take the riemann surface of ln, called S, then we have a single valued map from S to C
in the reals, we have that ln is a map from R+ to R, which means it can only work on a limited domain
ergo a 'uniformly parameterised' 'commutative' hyperoperation cannot exist (at least with a non-trivial domain), as the domain must shrink indefinitely to cover all cases.
but in the complex numbers, we have that ln is a more complicated map, I don't know what to conclude from that. At the very least we know the domain must always change, which is probably enough to say the same as in the real case.
one can probably make an argument without introducing the 'commutative' property but I am not able to make it, although maybe they truly are categorically different.
even if it's not uniformly parameterised it's still a fun idea to try to define, but we're going to have to figure out the constraints on the inputs and without uniformity it's pretty hard to say that it could be continuously extendable to intermediate values like rational or real 'z' values.
in the 'commutative' case, the isomorphism between the sucessive operations means we must find a way to find the continuous 'endofunction extent' of the exponential function, which very well could be arbitrary in the same way there are arbitrary continuous extensions of the factorial in the (0,1) interval (the gamma function is not the only useful one)
I love the music you use here. Great video, love math :D
6:30 Isn't that Knuth's Up Arrow notation?
8:11 Yes
X was never the same after this
so basically,
S(S(2)) = 4
2 + 2 = S(S(2))
2 x 2 = 2 + 2
2 ^ 2 = 2 x 2
2 ^^ 2 = 2 ^ 2
2^^^2 = 2 ^^ 2
2>2>n = 2^^^2,
correct?
2>2>n is just the expression defining the whole chain, 2>2>4 is 2^^^2, but everything else is correct I think
Edit: reading some other comments, 2>2>3 might be 2^^^2, but I'm not sure as I'm finding differing thoughts on this
@TommyDog145 k
someone must have had a flash of inspiration to put two and two together like that. Desmond would be proud.
1:07 did you say... chicanery?
DO YOU THINK A BLOCK JUST HAPPENS TO GO INTO the square hole! 😊 LIKE THAT? NO. HE ORCHESTRATED IT. @WrathofMath!!
@@conabe HE DEFECATED THROUGH A SQUARE HOLE!
im glad somebody made a meme about this really cool concept
I Have never seen the right arrow notation
It's from _The Book of Numbers_ by Conway and Guy. Highly recommended.
Just asked my teacher, he laughed a lot, said that he didnt knew about it but then remembered a few things and we agreed on solutions and stuuf. Great video
Is subtraction related to division
Yes, in the same way multiplication is a repeated addition, division is a repeated subtraction.
The result of a division is how many times you can subtract a number until you can't (you then get the reminder that is smaller than the divider)
also following what jeromesnail said, the exponential function turns addition into multiplication, and in exactly the same manner it turns subtraction into division.
Yea
Subtraction is opposite of addition
Repeated addition is multiplying
Multiplying is opposite of division
So division is opposite of repeated opposite of subtraction
@@erenerbayare logarithms repeated division? I'm typically good at math but bad at logs.
missed opportunity to roll up the paper and stick it in the square hole
YES! A four video from Wrath of Math!!
it's the second video in my series of videos on four ua-cam.com/users/shortsW1KrfG_qzvU
@WrathofMath that's one eggscelent video of yours that I've never seen before
4 tier list when @@WrathofMath
@ When do you want it?
1:06 HE DEFECATED THROUGH A SUNROOF
edit: added timestamp
Are you telling me the shapes just happen to all fit into the square hole? No! He orchestrated it!
Wait shouldn't 2->2->3 be the pentation 2^^^2, instead of 2->2->4?
Props for linking the OG video
this video feels 5+ years old, not 10 hours old
i guess i'm behind the times
@WrathofMath lol
numberphile vibes
I realized this awhile ago and have thought a lot about it in the past, very cool to actually see it in a video. It'd be interesting to analyze the behavior of numbers very close to 2 for each hyperoperation, similar to taking a limit.
I've seen pentation represented as subscript y to the left of x
ᵧx
I could see that easily being confused with exponentiation, tho (unless you have some sort of paper rule in a notebook).
as a math nerd who figured out this property in middle school (and also found those odd notations), i'm glad to see this get some attention
The four hole. 🤭
Only 1 hole, 2 if you don't care about results, 3 if you don't mind choccy, 4 if you don't care.
I don’t normally understand or otherwise like math memes, but I liked this one. Took me a second to get it, but it was pretty funny once I did
6:38 for some reason I thought it would be represented as a exponent in the bottom right
it would make it a very awkward notation to use with a countably long list of unknowns that have been indexed via subscripts!
I have seen it represented in the bottom left
I'm a simple person
I hear Kirby's Epic Yarn music, I notice and appreciate
Starts at 03:08
Actually the video started two days ago
Fr
Your comment didnt show up till 3:08 ☹
@TooFarBeyond omg I just realized
I went back and checked and I only saw it when the mini comment window showed it at 3:08
This is truly the Wrath of Math
that outtro is actually fire though
I've seen the 3 up arrow notation in some videos about large numbers, but never really understood them, now I do, thanks!
What a well fricking made meme
This is Two,and thats Two.
4 is the cosmic number, after all.
It would be neat to see a graph of the values produced from applying these iterations to 1, 2 and 3, just to see how wildly different the results are.
What started as a "yes I know that 2² is also 4 lmao" video actually got pretty damn interesting
Wait till she realises what the 1 hole is