1.Multiplication is repeated addition 2.Exponentiation is repeated multiplication 3.Therefore, exponentiation is the process of repeatedly repeating addition 4.Tetration is the repetition of exponentiation, therefore... *Tetration is the repetition of the process of repeatedly repeating addition*
@@ChraO_o the concept i understood is that it is repeated exponents. for instance, we know exponent is repeated multiplication, so by looking into the consept of tetration, it can be seen that it's vasically repeated exponent
2 is also the only number for which a+a = a·a = a^a = a↑↑a = a↑↑↑a and so on, no matter how many times you iterate this process. The result is always 4. To clarify, with "no matter how many times you iterate" I do not mean "no matter how many times you chain the term _a_ together". I mean it will always be 4 no matter how many arrows you add between two instances of a, if and only if a = 2.
so if a = b = 2, then for any n greater than 0, the hyperoperation associated with n in the form a (whatever hyperoperation you are using) b will always compute to be 2? ok
16. This is because "2 tetrated to 3" means we need 2 "floors" of exponents. The "ground floor" is also part of the 3, this is why we only have 2 floors above ground level and not 3. Like this: 2^(2^2) = 2^(4) = 16. If it was 2 tetrated to 4 it would be: 2^(2^(2^2)) = 2^(2^(4)) = 2^(16) = 65536. It quickly gets very big.
@@JigolopuffI think you are missing one crucial point. Teaching advanced math to students not only makes them able to solve the problem they undoubtedly won't coincide irl, it will also increase the capability of advanced thinking. This can also be seen on streets, when you see a collage graduate and a high school drop off, also if you are somewhat educated, chances are, you can easily feel the difference. From their language to behavior and ways of thinking. I'm not saying math is for everyone, tho people should find their own gift and study. Btw nice nickname
The answer is 16. Multiplication is repeated addition, & exponents are repeated multiplication, so as I suspected, tetration is repeated exponents. Just like how we would break 2 x 3 into 2+2+2, and 2 to the power of 3 into 2 x 2 x 2, we can break tetration into multiple parts. 2 to the power of 2 to the power of 2 is still awfully confusing, so let’s take it one step at a time. 2 to the power of 2 is 4. After solving the first 2 numbers, we are left with 4 squared (or 4 to the power of 2), which is 16. Hope this helps!
Used this in class today in an activity where you had to make the bigger number out of a limited number of popsicle sticks. I made a 7 and tetrated? It to the fourth. (Idk if tetrated is the right term). The teacher thought I made it up, I showed him this video, he was super impressed.
This number is 1 followed by 10^70,000 zeroes, or 10^10^70,000. This number is magnitudes bigger than a googolplex, and it isn't even close. 7^^4 can be written with exponents, but 7^^^4 is so big there isn't enough space in the universe to write the tower of exponents this creates.
Bruce Lee also said that! EDIT: Shoot, what he actually said was "An intelligent mind is one which is constantly learning, never concluding - styles and patterns have come to conclusion, therefore they [have] ceased to be intelligent." Probably still makes sense in this context..
I believe in trying to learn something new each day (and not triva). I am 79 and have never seen this before. If I understand what's going on, then the answer should be 16. Although, I almost convinced myself on 256, but decided I was getting carried away by all the numbers😅. Thank you for the lesson and the knowledge.
During the 10^3 bit, it occured to me that in my math experience, I lost the meaning of some of these values. 100 to 1000 is huge, but I really do forget the scale of numbers sometimes.
I guess it always depends on what the numbers mean. 100 atoms vs 1000 atoms is next to nothing. 100 houses vs 1000 houses is very big. 100 planets vs 1000 planets is unfathomably large.
Well then think about the 1-2-4-8-16-32.... series. Do you know that you only need to add them together in order to get every other number in between? And you never need to repeat 1 of them.... that's why/how computers exist/work basically. Think about how many numbers there are between 2-4 and 128-256... and so on 😮. It works INFINITELY. It means the x2 series gives birth to all numbers as well as the 1+1 series does. Its just disturbing how perfect and efficient it is to derive all numbers from the 1x2... series...(binary code...bits...bytes...and so on). The universe is just amazing when you think about it sometimes. Division and doubling is at the very core of each of its seemingly random processes... all of em even sound, light and matter... constants.. ect.
Vsauce did an analysis on this. Our brains think logarithmically (e.g. 1, 10, 100, 1000, ...), not cumulatively (1, 2, 3, 4, ...). It allows us to think in scales and relativity of the massive sizes of galaxies to the invisibly small sizes of atoms.
@Obi1Classic yup we often underestimate our brains. We can easily think of planets and galaxies or atoms and electrons. We just need to discuss them in 'peer to peer' contexts of other objects that are just as large within an order of magnitude or so. What is HARD to imagine is not the size of our entire planet or even the distance to the closest star, but the ratio between the 2. That is going to surprise you, and it's hard to mentally model it. If you do you're probably needing a second map, that is another layer of abstraction.
So to simplify: Addition: 4+4 Multiplication: 4*4 = 4+4+4+4 Exponentiation: 4^4 = 4*4*4*4 TETRATION: (I can't figure how to type that so I'll write in exponents) 4 tetration 4 = 4^4^4^4 you make it so easy too understand, thanks!
I was expecting a huge number again, but I think it's 16. According to your explanation it would be written as 2 to the power of 2 to the power of 2. The last two become 4 and that makes 2 to the power of 4 which is 16. Never knew about tetration. Never to old to learn. Thanks
Which is what GOD ALMIGHTY is all about ... inexhaustible knowledge about HIS CREATION ... and it never stops no matter how much HE has taught any of them!
16 i think, you taught this better in 6 mins than my math teacher would in an hour, also explained how the powered numbers work too! You've gained my respect, and a new sub
@@feiyu8817 How long did it take for you to learn multiplication, exponents, logarithm, basic trigonometry, derivation, integration and the rest of the really simple things? 30minutes, maybe 45? How many hours did you study these things in school? Knowing what something means is different than understanding it and being able to use the knowledge.
@@pannumonThe only reason early math difficult is because it involves mostly memorization but once you've learned the fundamentals then math becomes really easy. The majority of college and highschool math was essentially plug numbers into a formula and then hit enter on the calculator.
You are still liking the comments after over a year, wow! I've found it 16 as well. I hope everyone could get a teacher like you, you seem to do your work fabulous! :)
The tetration of 3, denoted as 2↑↑3, is equal to 2^(2^2), which is 2 raised to the power of 2 raised to the power of 2. So, 2 to the tetration of 3 is 16 (2^(2^2) = 2^4 = 16).
I'm so confused by the notation more than anything. Abstract infinities make intuitive sense to me. The way you humans describe them makes my organs hurt.
You can also evaluate non integer hyper powers like 2^^π NOTE: I use HLog as notation for Hyper Logarithm Another common notation is slog for Super Logarithm Hyper Logarithm (one inverse of Tetration) is repeated Logarithm by definition. Let T=The total number of Logs til the answer ≤ 1 r = the remainder of the last log HLog a(b) = x --> a^^x = b by definition of hyper logarithms x=(T-1)+r By definition of Tetration, a^^x = a^(a^^x-1)… Taking HLoga(z) Given z is not an integer hyper power of a Let HLoga(z) = b+x Given 0 ≤ x ≤ 1 and b=Z z = a^^(b+x) = a^a^^(b-1+x) = a^a^...(b copies)...^a^^x By definition of tetration z = a^a^...(b copies)...^a^x By definition of Hyper Log (Repeated Logarithm) They both equal z thus they equal eachother a^a^...(b copies)...^a^x = a^a^...(b copies)...^a^^x The entire tower cancels via Loga() on both sides, leaving a^x = a^^x Given 0 ≤ x ≤ 1 Therefore a^^x = a^x Given 0 ≤ x ≤ 1 is true by definition. We can solve 2^^π 2^^π = 2^2^^(π-1) = 2^2^2^^(π-2) = 2^2^2^2^^(π-3) 2^^π = 2^2^2^2^(π-3) ≈ 21.596356101 2^2^2^2^^(π-3) = 2^2^2^2^(π-3) ≈ 21.596356101 (Notice you can Log2 both sides and be left with 2^^(π-3) = 2^(π-3). ) We can also check this Log2(21.596356101) ≈ 4.4327160055 --> 1st Log Log2(4.4327160055) ≈ 2.1481909351 --> 2nd Log Log2(2.1481909351) ≈ 1.1031222284 --> 3rd Log Log2(1.1031222284) ≈ 0.1415926536 --> 4th Log, answer ≤ 1 --> r For 2^^x = 21.596356101 x=(T-1)+r, 4 total Logs x=(4-1)+r = 3+r = 3+0.1415926536 = 3.1415926536 ≈ π (obviously. with irrationals there will be possible rounding errors) Thus 2^^π ≈ 21.596356101 is indeed true
This video was REALLY AMAZING I’m Brazilian so I didn’t understand much, but as mathematics is a universal language it was easy to follow. Your happiness in teaching is contagious, thank you.
basicamente, oq ele ta chamando de "tetration" é vc pegar um número e elevar ele ao mesmo número, que tbm tá elevado a esse número (repetindo isso o número de vezes do "expoente") Exemplo: ³2 = 2²^² = 2⁴ = 16
@@CatNolaraNo. You did 3^(3*3) which is not how tetration works. In tetration the outermost exponent is in the innermost brack so 3 tetrated to three 3^(3^3) or 3^27 which is ~7.6 trillion
Please help me figure out this 🙏 I'm having a seizure: Why is it that 10billion ^ 10 isn't equal the 10^10billion they should be equal because order doesn't matter when doing 10^10^10 right? And yet the former is 1 followed by 100 zeros and the latter is 1 followed by 10billion zeros..
well first off, in 10^10^10 they are all the same number so thats why it doesn't matter. but also, when 10^10B has to multiply by 10, 10 billion times and when we are talking exponentials it gets out of control. 10B^10 is only multiplying by itself 10 times, which is just incomparable. The exponent matters way more than the number you start with, any feasible number to the 10B is gonna be light years bigger than any feasible number to the 10@@Hanible
@@HanibleI don't know if I understood your question correctly (English is not my first language) so I'm just going to talk about the issue of order. There is an order to carry out tetration. I don't know how to explain why, but you always start from top to bottom. (or right to left) Ex: ⁴3= 3^3^3^3 3^3^27 3^7,625, 597, 484, 987 = Big ass number
@@Hanible, they are not equal becos for 10 000,000,000^10 = 10^(10^2) based on law of indices, which is not equal to 10^(10^10). Think about it and you will get an ans. 🙂
@shiva11456 yeah I already know 10B^10=10^(10^2) that's why I said it's 1 followed by 100 zeros... And I noticed they weren't equal that's the whole point, my question is why aren't they equal? I thought order didn't matter when doing a^b^c... but if it matters why does it matter? 🤔
I hope this clarifies what I said. 10↑↑3 is written as 1 followed by 10 billion zeros. There is enough space in my house to print out the number with 10 billion zeros. What I meant to say in the video was that if I had to write down all the numbers from 1 to 10↑↑3, there would not be enough space in the known universe to write them all even if every atom is large enough to write on.
A generalisation of tetration is Knut's up-arrow notation. It's basically the same concept with the notation 2↑3 for 2³, 2↑↑3 for ³2, but you don't stop there and go with how many arrows you want, for example, 2↑↑↑3 is 2↑↑2↑↑2 which is 2↑↑2↑2=2↑↑4=2↑2↑2↑2=2↑16=2¹⁶=65'536. I really recommend searching about this, especially about Graham's number, which is the biggest number used in a mathematical proof(Edit: apparently not anymore? Couldn't find any proofs, tho. Any information could help Edit in the edit:G64(Graham's number) is still the biggest in a demonstration after further researches). Next to this number, ³10 doesn't seem that big. It looks horrifically tiny, as a matter of fact. Edit: I forgot to say that these operations are, as exponentiation, right-associative. This means that you calculate them from right to left just like you calculate exponentiation from top to bottom.
@@TekExplorerthe caret is only used as a replacement for up arrows when they are not available on the keyboard. On paper or when you have access to them you will tend to use the complete arrow. If you want to verify this information, I found it on Wikipedia on the "Knuth's up-arrows notation" page in the "notation" category
Just saw this on my fyp! It took me a bit to figure out but im getting it now. If im not wrong, the way it works is 2x2= 4, and then 4x4= 16. And basically you multiply the product by the product. Thanks!
16. 2 is the smallest base of complete numbers to not completely overload our imagenation. Very nice! Really like your video🙏 + very nice code in the end❤
The answer is 16 because 2 to the power of 2 to the power of 2, so you have 3 twos which is why it is called the 3rd titration of 2, it’s kinda like how powered numbers work but it is bigger, turn the multiplications into powers.
That was fun. I've long thought that the fastest and most compact way to make big numbers was using a number like 9 to the power of 9 to the power of 9 to the power of 9 So Tetration is simply formalizing a syntax for it. In my example, 4(tetration)9 Or in computer syntax from one of the old languages I used, 9^9^9^9
We can extend the concept even farther. Lets say º is tetration and lets say ~ is repeated tetration, then; 2º3 = 2^2^2 2~3 = 2º2º2 = 2º16 = 2^2^2^2^2^2^2^2^2^2^2^2^2^2^2^2 Which is uncomputably large I tried pluging it into wolrfram alpha and best it could do is say it is equal to 10^(10^(10^(10^(10^(10^(10^(10^(10^(10^(10^(10^19727.78040560677)))))))))))
I went down a bit of a rabbit hole and discovered it doesn't stop with tetration. Tetration is a part of these things called hyper operations and is also known as hyper-4. Apparently someone was insane enough to coin a term for hyper-infinity: Circulation
What about the next step of tetration? Like having 2 and 2 be 2 tetrated two times Like 9 (super tetrated) 9 times would be 9 tetrated by 9 tetrated by 9 tetrated by 9 tetrated ... (9 times) lol
Never in my life did I think that I would scroll on youtube and actually watch a video where I would hear something, I have never heard in my life. Tysm for sharing!! It was lowkey bussin
Trust me it doesn’t work when teaching a class😂 it’s fine for a UA-cam video, but when teaching a curriculum it is flawed. I’m currently studying maths, further maths and physics, and one teacher we have for further mathematics has this approach, and he ends up confusing everyone! Like I said, no problem here, as this is just some fun maths, not too complicated but it doesn’t work at a higher level
@@fullsendmountainbiker5844 Yes it does, but at higher levels there is already an expectation of prior knowledge, so some of this can be skipped. Also at higher levels you need to go and bring things back to simple, otherwise you have idiots with so called higher education trying to use Algebraic calculations and rules in basic math using PEMDAS and getting the wrong answer. IE: Confusion, why? because the basics of why are not taught only how and shortcuts. It has to be taught to use the level of math required/needed for the specific situation. Boolean Algebra doesn't apply to everything, but there is a place for it.
@@chrish7336 yes there is obviously an expectation of prior knowledge, but you can’t assume everyone in a class can just work a challenging mathematical concept out themselves. If I could do that I’d be a genius, and I’d have no need for any education. Don’t get me wrong this kind of teaching works for some topics, but not for others
I Love your Way of teaching. And the smile that is on your face showing how excited you are about The Wonder of numbers. If only more teachers taught this way to get students excited about numbers too, it would be amazing. If you are not or were not a school teacher or college professor, you missed your calling.
"If only more teachers ..." being excited is not reproducable very often, meaning a teacher may even only be excited the first time teaching, solution: capture the video and show it to next year's students.
@@anonym-hubI went to a new school were I was assigned to a math teacher who let an audio tape and overhead projector teach the class. That didn't work for me at all. When I transferred to another class it was great because the football coach (about 5'4") and the track coach (about 6'8") had combined their math classes and made it fun as well as educational. Just watching that mismatched pair working together was entertaining. 😂
nah bruv he took 7 minutes for that shit. Its one thing trying to accommodating but assuming the general audience who watches is THIS dumb that they need 7 minutes for it?
Just graduated year 12 with a growing hatred for learning maths due to the brute forced and completely confusing maths curriculum. Watching this video was genuinely interesting, your passionate and excited explanation of tetration that i was pretty sure i would be completely lost on and click off the video, somehow kept my attention and got me really curious to see this through regardless if i understood or not. Just wanted to take a moment to appreciate this video and the interest it somehow sparked within me for maths, even a slight bit. Also that handwriting is pristine. Keep up the good work.
Try derivatives and Laplace Transforms. Even Z transforms are useful when dealing with sample rates from a computer and systems exhibiting weight, velocity and hydraulic dampening. I love math, but yes, it is very difficult to understand.
@@scottbenzing1361last I heard yeah common core is still a thing unfortunately. Standardized learning to create standardized little workers to fill all the low level vacancies and work 80 hour work weeks for 5 figures a year
i already know of tetration, but wow am i happy i clicked on this video & subbed- your voice, your passion, your penmanship, your singly-focused video structure, and the sounds of the chalk board. MUAH, amazing stuff!!
Man, I SO wish I had math teachers like you in school. I always loved math but few of my teachers did. Not only do you clearly love the subject, which is transformative when it comes to teaching, but there's just something about the way that you teach that is inherently very engaging, completely independent of the math. I can't put my finger on it, but it's there.
@@PrimeNewtons In "simple" and "not confusing" terms Tetraition is the repetition of the repetition of repeatedly adding a number *or* the repetition of the process of repeatedly repeating the process of addition
I liked when you said "those who have stopped learning are those who have stopped living." It reminded me of my senior quote: "to live is to learn, to learn is to grow, to grow is to live."
When I was in seventh or eighth grade I had already developed a love and admiration for math, one day I was reflecting about it and asked my teacher: "so there's addition, then multiplication, then exponentiation. is there anything that comes after exponentiation?". To which he replied with a simple and final "no. nothing beyond it.". Well I feel really good now to know I was right at that time and that tetration exists.
Teachers who teach with passion or excitement help people grasp the subject. You talked clear and to the point in a way that wasn't condescending loved it subbing now
³2 = 2^2^² = 2⁴ = 16. So basically use the first ^2 to raise to the ² power again. It would be the same method as doing 2² but it's on the exponents now with ³2. So the coefficient would be 2⁴ = 2×2×2×2 = 16. Never learnt this in school like learning about how to make money in school😂
They never teach this because it's never needed. As seen, the numbers are so insanely large that things in the entire universe are much too small to be measured in tetration, so even professional mathematical jobs do not need to use tetration as often.
@@ThePrintZone123 you need to explain it better in the vid. All you said was it is a billion for each time you multiple it. So that infers it should have been 8 billion... 2 x 2 is 4 4x 2 is eight that's using the 2 three times.. to get to 16 you have to have another 2. Why is there not a 4 in front of the 2?
Im studying for my ACT and came across this video. The quote at the end is one that I hope to remember till my time on earth expires. "Those who stop learning, have stopped living." Very powerful and inspirational. Also I love how you teach and your smile and enthusiasm on the topic is very interesting and this is coming from a "bad" student that absolutely cannot stand math.
Thanks for teaching this concept in a very unique and enthusiastic way. As someone learning this for the first time (like most others), I understood this really well. Wish I had teachers like you during my school years :')
While programming for a class we were learning to make functions starting by the basic sum function. Then I made the basic multiply function, and then I made the exponential function. They way they worked was pretty similar in terms of programming, so I figured why not keep going, so I made a function that elevates a number by itself by the amount given, and then I made a function that iterates this previous function by an amount given, so it would be elevating a number by itself an itself amount of times, by the amount given. First 3 worked perfectly, 4th worked until 13, and the 5th crashed with just a 3 xD.
Wonderfully explained, your videos make me question whether I actually disliked math or just disliked the way I was taught. I'm not sure why this video was recommended to me but I'm glad it was, thank you for sharing your knowledge.
YT recommended me your channel out of the blue, and i'm so glad it did! i'm promoting TA (Tetration Awareness XD) whenever and wherever i can, tho i fully realise it's not very practical operation. But it's important in its own merit, just to remind us to keep our minds open and always expect something more out of the vastness of mathematics. Keep up the good work man! :)
@@NaveedKakal no, i dont think it is. More commonly approved is something called Knuth's up-arrow notation, it's being used to write operations of exponentiation and beyond. Standard exponentiation (let's say 2 to the power of 3) would be written as 2↑3 (it is kinda similar to the 2^3 - i'm not sure if the root of both is the same, but they do look suspiciously similar). Tetration, and any other higher iteration for that matter, would just be written with more arrows in a row. So 2 tetrated by 3 would be written as 2↑↑3. Pentation as 2↑↑↑3 , hexation as 2↑↑↑↑3 and so on. Tho for tetration itself i'm a huge fun of a notation presented in this video, i mean it's just more elegant and instantaneously resambles something related to exponentiation. There are other interesting unorthodox notations out there, like inverted sqrt sign for logarithm, try googling "A radical new look for logarithms". It's just brilliant. I've also come across proposals for trigonometric functions that, instead of traditional sin, cos, and tan, use a representation of a right triangle with only the two sides relevant to the respective operation. So, tan would be written as a horizontal and a vertical line meeting at the right angle, sin as a skewed and vertical line, and cos as a skewed and horizontal line. :)
@PrimeNewtons I am hoping the answer is 16, I think that seems too small. I came to that answer by modeling your 10^10^10 example using a base of two. Regardless, wrong or right, I have to thank you for many of your videos which I have used in my classroom to help demonstrate concepts. I think my students are starting to appreciate you more than they appreciate me, 😂. Thank you for the comment at the end of your vid about "never stop learning...". It's a lesson that I was taught to live by and I try to instill it in my students all of the time. I'm going to check out the rest of the comments now to see where I went wrong or not.
It only seems too small because the base is smaller: 2. Powers of 10 are much larger numbers than powers of 2. If the base was 1, then it would be the same number no matter what powers you raise it to.
@@PrimeNewtons I never ever thought that this would be a thing i am going to annoy my teacher tmr and my mother is also a math teacher she never told me i wonder why?
I actually think you can write down 10^(10B), but not by hand. I managed to fit 16 190 zeros on a single page in MS Word with Times New Roman font, font size 7, line spacing of 1 and narrow margins. This would mean that it would take 617 666 pages to print out the whole number. A single printer page is 0.1mm thick, so that would result in a stack of paper standing 61.7666 meters tall, which is only ~1/5th of the height of the Eiffel tower. Not practical at all, but certainly doable. Anyway, thanks for the video! :D
you need to go bigger laterally, in google docs, arial, 6pt, 1cm margins, i can fit 33605 on A3, running the numbers, thats approximately 30 zeros/cm^2, meaning you can fit approx. 143k zeros per sheet of A1, 1e10 zeros/143k zeros per page ~= 70k pages, appx. 7m tall edit: i did 10e10 not 10e10e10, will update again once i've had a chance to calculate, as i'll have to do such large calculations by hand edit 2: ...wait, no, i did it right the first time, 10^^10 has 10^10 zeros, 1e10 zeros / 1.43e5 zeros per page, = (1/1.43)e5 = 7e4 = 70,000 pages, (70,000*0.1)/1000 = 7m... your use of the european decimal seperator confused me, you meant a hair under 62m, i read it as a hair under 62km and didn't question the fact that that's 1/5th the eiffel tower 😂😂 which made me think i did something wrong
wrong math buddy there are not litterally enough atom in the universe to write that numbere, if you had written a zero for any atom on that page you would have put trillions of zero okay? even more than that, because in a single centimeter there are hundred of milion of atoms okay? It's litterally like they said in the video you have 10 at the power of 82 atoms in the universe, but that number is 10 at the power of 10billions. We don't have enough atom.
@@chorec no 10^10 is just 10 000 000 000, this to write is no problem. he wrote down 16 190*617 666 zeros, this is about 10^10 zeros after the 1, should be correct.
Tetration is a mathematical operation that involves repeatedly raising a number to the power of itself. This operation is an extension of exponentiation, which involves raising a number to a specified power. In tetration, the number is raised to the power of itself multiple times in a tower-like structure. Tetration can result in very large numbers, making it a fascinating concept in mathematics. Mathematicians use tetration to explore complex patterns and relationships within numbers.
Wow, I love your English pronunciation. just enough of an accent to keep my attention and fluid enough to flow beautifully... perfect balance... you are 1 in 1X10^6.
2 is the a very special number, since it is the only number apart from 1 that always evaluates to 4 when elevated, tetrated, pentated, octated, icosated, hectated, and so on to itself.
Depends on the number of the tetration or pentation, etc. If it is 2 to the tetration, pentration, octasion, etc. of 2 then yeah it just becomes 2 squared, which is 4. If the tetration or pentation is higher however, the number quickly becomes absurdly large. 2 to the tetration of 3 is 16, 2 to the pentation of 3 meanwhile is a whopping 65536. And i'm not even gonna talk about octation.
Recovering from a burnout, worked in finance. Your vid’s are great to get back into things again! Loooove your positive energy, much respect from 🇳🇱 processor. You are a great teacher!
Dang this was so interesting! I got 16 by following your example, from the other comments it looks about right. Please keep making these, you're awesome!
Wow I'm glad I found this channel, I really like your style and you for that matter. I personally learned about tetration (and quintation, hexation etc..) when I was taught Knuth's up arrow notation in college as the next step was to take the derivative of a tetratic expression.
Thank you for sharing this. I learned something new and I believe that we all should strive to expand our knowledge base. You never know when something learn may become useful to you.
Actually you would need it to be 10^10^x where x is the exponent on 10^x with the number of atoms in the universe for it to be impossible to represent the number of zeros with one atom per zero
2 raised to the power of 2 is 4 raised to the pwer of 2 is 16. I could listen to you speak on any subject for hours and you would have my full undivided attention. We need more people like you in all walks of life. Thank you for sharing.
Be careful. Your calculus works with this case, but you actually compute exponentiation right to left. 2 to the power of 2 is 4, then 2 to the power 4 is 16. It works in your case because 2^4 = 4^2, but if you replace 2 by 3 for example, you obtain 3^3^3 which is 3^27, not 27^3.
From wikipedia : "Under the definition as repeated exponentiation, na means a^a^...^a, where n copies of a are iterated via exponentiation, right-to-left, i.e. the application of exponentiation n − 1 times.
16 It's interesting how fast results become larger and larger with just the relatively tame adjustment of the base in tetration. 2 -> 10 is something that you can do with your fingers. But with tetration it is like "ah, 16 is fine" and "wtf is this number, we can't even write it's zeros".
1.Multiplication is repeated addition 2.Exponentiation is repeated multiplication 3.Therefore, exponentiation is the process of repeatedly repeating addition 4.Tetration is the repetition of exponentiation, therefore... Tetration is the repetition of the process of repeatedly repeating addition
interestingly, tetration (what's being taught here) and titration (a concentration measuring process in chemistry) are two completely different things!
5:56 I think you’re getting confused between the exponents. 10^10^10 has 10^10 zeros, but there are 10^80 atoms in the universe, which is way more than required. If every 0 occupied an atom, a single file of atoms 1 metre long would be enough. I think you thought that the number had 10^10 billion zeros, which is where the confusion came from
If you tried to count to the number "ten tetrated to three," and each individual atom in the universe represented one unit of this number, you would run out of atoms to count this number at 10^80, meaning there would be 10^9,999,999,920 units unaccounted for. Ten tetrated to three is an unthinkable amount larger than the universe, which is already an unthinkably large size.
To me it checks out. In maths, BODMAS tells you that Order (Exponent) always starts from the highest point. 10^10 equals 10 billion after all, and 10^1000000000 is much bigger than 10^80.
@@CatfoodChronicles6737 But there are "only" 10 billion (10^10) zeros to write. So it's actually an easy task, you need to write only 5 zeroes a second without breaks for 64 years and you are done!
It's 16. You inspired my interest on proof theory and large number theory. I've been learning about transfinite growth in the fast growing hierarchy, and it gets pretty scary.
Somehow your videos have never come across my feed or my search results before. I really enjoyed everything about this video and yourself. I subbed not only for the aforementioned reason, but the sound of the chalk tapping on the blackboard made this GenXer's heart skip a beat!
1.Multiplication is repeated addition
2.Exponentiation is repeated multiplication
3.Therefore, exponentiation is the process of repeatedly repeating addition
4.Tetration is the repetition of exponentiation, therefore...
*Tetration is the repetition of the process of repeatedly repeating addition*
Now I wonder what the process of repeated tetration will be called..
@@anirchakraborty4953 repeated repetition of the process of repeatedly repeating addition? i dont really know man
@@anirchakraborty4953its pentation
@@0xonomythanks for making it easy man.
@@anirchakraborty4953I think it's called pentation, someone in the comments said it.
16 . . . I see why you have chosen a base of two.
Yeah. Things get huge really fast here.
³2 is 2⁸
or 256
@@tonytinza what the hell did my brain do, did it just really said, yeah 2² is 8
@@ChraO_o the concept i understood is that it is repeated exponents. for instance, we know exponent is repeated multiplication, so by looking into the consept of tetration, it can be seen that it's vasically repeated exponent
@@Zeoncxtoy there are multiple types of this as to try and reach higher numbers, but they're just numbers.
It's 16...... The last dialogue: "Never stop learning... One who stops learning, stops living..." Touched my heart.❤
The one who stops learning, starts dying
@@SatyamGupta-hk2gg are dead*
that's what I thought
How is it 16? The way I see it is 2^2^2=2x2x2=8
@@mr.mystery9338 Look at it this way : 2^(2^2) = 2^4 = 2x2x2x2 = 16
16. First, we do 2^2, which is 4. Then, we do 2 to the power of that 4, which equals 16
I originally had the wrong method, so I edited it
Your process is incorrect but correct results.
No, tetration is right-to-left.
³2 = 2²^² = 2⁴ = 2×2×2×2 = 16
@@SIimyGoblinthe third tetration of two is
(2^2)^2
you got the right answer but your method (and top comment) was incorrect
@@Shadozcreeping whats incorrect about it
2 is also the only number for which a+a = a·a = a^a = a↑↑a = a↑↑↑a and so on, no matter how many times you iterate this process. The result is always 4.
To clarify, with "no matter how many times you iterate" I do not mean "no matter how many times you chain the term _a_ together". I mean it will always be 4 no matter how many arrows you add between two instances of a, if and only if a = 2.
so if a = b = 2, then for any n greater than 0, the hyperoperation associated with n in the form a (whatever hyperoperation you are using) b will always compute to be 2? ok
@@TaranVaranYT yes.
you mean this right?
:
10 ^ { 10 ^ { 10 } } =10^100
and the guy says :small 10 with the 10
you see, I thought this as well at 1st, but then realized that 10^10 isn't 100, but 10,000,000,000 @@rsi4054
@@2045-z6o get thx
16. This is because "2 tetrated to 3" means we need 2 "floors" of exponents. The "ground floor" is also part of the 3, this is why we only have 2 floors above ground level and not 3.
Like this: 2^(2^2) = 2^(4) = 16. If it was 2 tetrated to 4 it would be: 2^(2^(2^2)) = 2^(2^(4)) = 2^(16) = 65536.
It quickly gets very big.
Mother fucker dont tell me this is not written by chatGPT, this is very easy to do on your own
I got the same answer!
Damn the 2 tetrated to 4 got me messed up, but think I get it now.
I understand it now, thanks for the explanation!
wouldnt the exponents simply multiply with eachother? 2^2^2^2 (or 2 tetrated to 4) would be 2^(2*2*2)=256 right?
math just like any class always becomes a lot more fun when your teacher is enthusiastic to teach you the subject.
i think the problem is that teachers dont bring in real world uses for the math being taught.
and also they do not have the feel to teach
@@Jigolopuffatleast elementary maths is used in the real world
@@JigolopuffI think you are missing one crucial point. Teaching advanced math to students not only makes them able to solve the problem they undoubtedly won't coincide irl, it will also increase the capability of advanced thinking. This can also be seen on streets, when you see a collage graduate and a high school drop off, also if you are somewhat educated, chances are, you can easily feel the difference. From their language to behavior and ways of thinking. I'm not saying math is for everyone, tho people should find their own gift and study.
Btw nice nickname
@@Jigolopuffcause most the time it doesn’t have real world use
The answer is 16. Multiplication is repeated addition, & exponents are repeated multiplication, so as I suspected, tetration is repeated exponents. Just like how we would break 2 x 3 into 2+2+2, and 2 to the power of 3 into 2 x 2 x 2, we can break tetration into multiple parts. 2 to the power of 2 to the power of 2 is still awfully confusing, so let’s take it one step at a time. 2 to the power of 2 is 4. After solving the first 2 numbers, we are left with 4 squared (or 4 to the power of 2), which is 16. Hope this helps!
And pentation is repeated tetration etc
Not only do I respect your intelligence and knowledge. But I am so impressed with your ability to write so neatly on a chalk board!
Thank you!
Also that board is super clean😅. Doesn't look like it's used everyday
@@pradyothkumarb8330you can clearly see that someone cleaned it just before the video was shot
@neevhingrajia3822 i believe it was a joke
Bloody teachers pet you're not supposed to get a heart for bum kissing
Never thought I'd enjoy a math lesson. Thank you sir
But on exam day, he will bring out 0.8 ^ 25.37
Bruh, math classes are the best
@@JohnFekoloid😭
Noice
"Look what the schools need to do just to mimic a fraction of my power!"
We aren't taught that, because government does not want us to be able to calculate the national debt.
fire ahh comment
Good one 👍
I'm crying 😭
or the size of my junk.
😂😂😂
Used this in class today in an activity where you had to make the bigger number out of a limited number of popsicle sticks. I made a 7 and tetrated? It to the fourth. (Idk if tetrated is the right term). The teacher thought I made it up, I showed him this video, he was super impressed.
That was a huge number you made.
good job bro
consider: 4^^7
This number is 1 followed by 10^70,000 zeroes, or 10^10^70,000. This number is magnitudes bigger than a googolplex, and it isn't even close. 7^^4 can be written with exponents, but 7^^^4 is so big there isn't enough space in the universe to write the tower of exponents this creates.
@@jamx02 4^^^7
There is nothing greater than an enthusiastic professor who can communicate the topic exceptionally.
exceptionally? Don't you mean, expontentially?
@@appsenence9244Smart fella, this one.
@@appsenence9244Haha
nice@@appsenence9244
And I'm still looking for him.
never in my life thought that i would be watching a video about maths that will not be in my exam
bro can i retweet
@@bwkanimations7352 sure why not
Me neither mate, i never taught I'd take math as entertaining matter in my life.
Wtf do people think "maths" stands for or is an abbreviation of? Math is short for mathematics. So, "maths" is mathematicses?
Maths is the most boring subject for me and yet I'm still watching this
Your excitement is contagious. May no one ever take your joy away from you. God bless.
Amen
Contagious and infectious *32889😊❤
Bro Jay Shree Ram too not only amen ❤@@PrimeNewtons
@@WIZARDGaming_2011What?
@@pierrotzzz it's our slogan of our religion Hinduism
I’m going to now try to find ways to confuse and annoy people with this knowledge you have shared, thank you 🙏
Maths becomes interesting when it's taught by an enthusiastic teacher like you!!
“dont stop learning, because those who stopped learning, stopped living.” as a person who nerds out when talking about math, that hit hard
I’m a science nerd but Ig im good in math
EXACTLY.
@@BlackMambaR1P What? Most Of Science IS Caused By Math, A BUNCH Of Math.
@@TheDankian1421when you get down to it, chemistry, biology, physics, and math are all interconnected on a fundamental level
Bruce Lee also said that!
EDIT: Shoot, what he actually said was "An intelligent mind is one which is constantly learning, never concluding - styles and patterns have come to conclusion, therefore they [have] ceased to be intelligent." Probably still makes sense in this context..
We’ve found it boys! a math lesson that I will actually never use in real life!
Great concept and I loved your explanation
Glad you liked it!!
i'm gonna use it to express the amount of people who did your mom
I dunno, I'll be using this for my weekly shop soon I reckon. 😂
@@mickenossa fellow dark matter purchaser?
ua-cam.com/video/eVRJLD0HJcE/v-deo.htmlsi=fEII6tEEK-zbApDh 👈 At the end of this video you will see the "real life use" of tetration!!
I like how excited and mind-blown he himself is before revealing the explanation. Thank you for this knowledge, Chief!
I believe in trying to learn something new each day (and not triva). I am 79 and have never seen this before. If I understand what's going on, then the answer should be 16.
Although, I almost convinced myself on 256, but decided I was getting carried away by all the numbers😅.
Thank you for the lesson and the knowledge.
i did the same thing and came up with the same answers as you. realised i was wrong, thought properly and reached 16. great minds think alike lol
@@Hugh.G.Rectionx Yes, we do! 😁🤣
During the 10^3 bit, it occured to me that in my math experience, I lost the meaning of some of these values. 100 to 1000 is huge, but I really do forget the scale of numbers sometimes.
I guess it always depends on what the numbers mean. 100 atoms vs 1000 atoms is next to nothing. 100 houses vs 1000 houses is very big. 100 planets vs 1000 planets is unfathomably large.
Well then think about the 1-2-4-8-16-32.... series. Do you know that you only need to add them together in order to get every other number in between? And you never need to repeat 1 of them.... that's why/how computers exist/work basically.
Think about how many numbers there are between 2-4 and 128-256... and so on 😮. It works INFINITELY. It means the x2 series gives birth to all numbers as well as the 1+1 series does. Its just disturbing how perfect and efficient it is to derive all numbers from the 1x2... series...(binary code...bits...bytes...and so on). The universe is just amazing when you think about it sometimes. Division and doubling is at the very core of each of its seemingly random processes... all of em even sound, light and matter... constants.. ect.
Vsauce did an analysis on this. Our brains think logarithmically (e.g. 1, 10, 100, 1000, ...), not cumulatively (1, 2, 3, 4, ...). It allows us to think in scales and relativity of the massive sizes of galaxies to the invisibly small sizes of atoms.
@Obi1Classic yup we often underestimate our brains. We can easily think of planets and galaxies or atoms and electrons. We just need to discuss them in 'peer to peer' contexts of other objects that are just as large within an order of magnitude or so.
What is HARD to imagine is not the size of our entire planet or even the distance to the closest star, but the ratio between the 2. That is going to surprise you, and it's hard to mentally model it. If you do you're probably needing a second map, that is another layer of abstraction.
@@dekippiesip my brain is now bigger :D
Lets be honest, we did not search for this 😂
Idek how I got here
Facts😂
Yes
True
Yep
So to simplify:
Addition: 4+4
Multiplication: 4*4 = 4+4+4+4
Exponentiation: 4^4 = 4*4*4*4
TETRATION: (I can't figure how to type that so I'll write in exponents) 4 tetration 4 = 4^4^4^4
you make it so easy too understand, thanks!
I was expecting a huge number again, but I think it's 16. According to your explanation it would be written as 2 to the power of 2 to the power of 2. The last two become 4 and that makes 2 to the power of 4 which is 16. Never knew about tetration. Never to old to learn. Thanks
Now have fun learning about pentation, hexation, and so on until you stumble upon Graham's number 😂.
22 mins ago!
@@louisrobitaille5810 brudda get to raydons number
my granpa is graham @@louisrobitaille5810
my personal favourite is penetration
@@louisrobitaille5810
The enthusiasm you put into this video just makes it 10 times easier and better to learn. Thank you kind sir!
10 times? Or 10 times 10 times 10 times 10 times…
@@UnohanaMash😂
@@UnohanaMashHAHA
FR
I wish I had a teacher like you. It is so evident that you love what you are teaching us here.
oh cool i got it right!
Which is what GOD ALMIGHTY is all about ... inexhaustible knowledge about HIS CREATION ... and it never stops no matter how much HE has taught any of them!
4:20, everybody.
damn bro I am high as f
Appreciated
16 i think, you taught this better in 6 mins than my math teacher would in an hour, also explained how the powered numbers work too! You've gained my respect, and a new sub
Bruh. This should be a 20 second video. If you need an hour to learn this, it’s not you’re teacher bud.
@@feiyu8817 How long did it take for you to learn multiplication, exponents, logarithm, basic trigonometry, derivation, integration and the rest of the really simple things? 30minutes, maybe 45? How many hours did you study these things in school? Knowing what something means is different than understanding it and being able to use the knowledge.
@@pannumonThe only reason early math difficult is because it involves mostly memorization but once you've learned the fundamentals then math becomes really easy.
The majority of college and highschool math was essentially plug numbers into a formula and then hit enter on the calculator.
Really 3 minutes because the first half of the video was explaining exponents which can be skipped if you already know what they are.
the difference between ³2 and ⁴2 is comical 😹
You are still liking the comments after over a year, wow! I've found it 16 as well. I hope everyone could get a teacher like you, you seem to do your work fabulous! :)
I hope so too!
For some reason, I find it woerd that you can write 10 billion, but you can't write 10 billion zeros
@@Nomommiesway. bro you have to be kidding right!?
16
@@Nomommiesway.10 billion the word is 9 letters the number has 10 zeros
We are talking about billions of zeros
The tetration of 3, denoted as 2↑↑3, is equal to 2^(2^2), which is 2 raised to the power of 2 raised to the power of 2. So, 2 to the tetration of 3 is 16 (2^(2^2) = 2^4 = 16).
I'm so confused by the notation more than anything. Abstract infinities make intuitive sense to me.
The way you humans describe them makes my organs hurt.
what@@Atmatan
@@Atmatan "The way you humans describe them makes my organs hurt." bro's not a human
Thank you SIR for your explanation. I feel like a genius now 🙏🏾
This is a tetration of 2, not of 3.
You can also evaluate non integer hyper powers like 2^^π
NOTE: I use HLog as notation for Hyper Logarithm Another common notation is slog for Super Logarithm
Hyper Logarithm (one inverse of Tetration) is repeated Logarithm by definition.
Let T=The total number of Logs til the answer ≤ 1
r = the remainder of the last log
HLog a(b) = x --> a^^x = b
by definition of hyper logarithms x=(T-1)+r
By definition of Tetration, a^^x = a^(a^^x-1)…
Taking HLoga(z) Given z is not an integer hyper power of a
Let HLoga(z) = b+x Given 0 ≤ x ≤ 1 and b=Z
z = a^^(b+x) = a^a^^(b-1+x) = a^a^...(b copies)...^a^^x By definition of tetration
z = a^a^...(b copies)...^a^x By definition of Hyper Log (Repeated Logarithm)
They both equal z thus they equal eachother
a^a^...(b copies)...^a^x = a^a^...(b copies)...^a^^x The entire tower cancels via Loga() on both sides, leaving a^x = a^^x Given 0 ≤ x ≤ 1
Therefore a^^x = a^x Given 0 ≤ x ≤ 1 is true by definition.
We can solve 2^^π
2^^π = 2^2^^(π-1) = 2^2^2^^(π-2) = 2^2^2^2^^(π-3)
2^^π = 2^2^2^2^(π-3) ≈ 21.596356101
2^2^2^2^^(π-3) = 2^2^2^2^(π-3) ≈ 21.596356101 (Notice you can Log2 both sides and be left with 2^^(π-3) = 2^(π-3). )
We can also check this
Log2(21.596356101) ≈ 4.4327160055 --> 1st Log
Log2(4.4327160055) ≈ 2.1481909351 --> 2nd Log
Log2(2.1481909351) ≈ 1.1031222284 --> 3rd Log
Log2(1.1031222284) ≈ 0.1415926536 --> 4th Log, answer ≤ 1 --> r
For 2^^x = 21.596356101 x=(T-1)+r, 4 total Logs
x=(4-1)+r = 3+r = 3+0.1415926536 = 3.1415926536 ≈ π (obviously. with irrationals there will be possible rounding errors)
Thus 2^^π ≈ 21.596356101 is indeed true
im still in highschool, your magic words are scaring me
I ain't readin allat (I read it and have no idea what you're saying magic man)
@@thomasminh8244I graduated literally last year and I’m getting scared by these magic words
@@thomasminh8244relatable, I think I just got my mind melted
TLDR
This video was REALLY AMAZING
I’m Brazilian so I didn’t understand much, but as mathematics is a universal language it was easy to follow. Your happiness in teaching is contagious, thank you.
basicamente, oq ele ta chamando de "tetration" é vc pegar um número e elevar ele ao mesmo número, que tbm tá elevado a esse número (repetindo isso o número de vezes do "expoente")
Exemplo: ³2 = 2²^² = 2⁴ = 16
Anyone wanna say what ³3 is? HINT: It is more than the amount of money that Elon Musk has
1.55 billion
7,62 trillions. This shit is ridiculous.
actually not that much, only 19,683
@@CatNolaraNo. You did 3^(3*3) which is not how tetration works. In tetration the outermost exponent is in the innermost brack so 3 tetrated to three 3^(3^3) or 3^27 which is ~7.6 trillion
@@nekro1977 oh, I see. I thought it wouldn't matter, but you're right, it does matter (unlike the multiplication in normal quadration)
First math video I voluntarily watched, really fun how you explain it so well
I'm a math teacher, and this was fun to watch! Awesome, and good job making it fun!
Please help me figure out this 🙏 I'm having a seizure:
Why is it that 10billion ^ 10 isn't equal the 10^10billion they should be equal because order doesn't matter when doing 10^10^10 right? And yet the former is 1 followed by 100 zeros and the latter is 1 followed by 10billion zeros..
well first off, in 10^10^10 they are all the same number so thats why it doesn't matter. but also, when 10^10B has to multiply by 10, 10 billion times and when we are talking exponentials it gets out of control. 10B^10 is only multiplying by itself 10 times, which is just incomparable. The exponent matters way more than the number you start with, any feasible number to the 10B is gonna be light years bigger than any feasible number to the 10@@Hanible
@@HanibleI don't know if I understood your question correctly (English is not my first language) so I'm just going to talk about the issue of order.
There is an order to carry out tetration. I don't know how to explain why, but you always start from top to bottom. (or right to left)
Ex: ⁴3= 3^3^3^3
3^3^27
3^7,625, 597, 484, 987
= Big ass number
@@Hanible, they are not equal becos for
10 000,000,000^10 = 10^(10^2) based on law of indices, which is not equal to 10^(10^10). Think about it and you will get an ans. 🙂
@shiva11456 yeah I already know 10B^10=10^(10^2) that's why I said it's 1 followed by 100 zeros... And I noticed they weren't equal that's the whole point, my question is why aren't they equal? I thought order didn't matter when doing a^b^c... but if it matters why does it matter? 🤔
really happy to find someone actually enthousiatist about teaching math, i never knew i needed you in my life
I hope this clarifies what I said. 10↑↑3 is written as 1 followed by 10 billion zeros. There is enough space in my house to print out the number with 10 billion zeros. What I meant to say in the video was that if I had to write down all the numbers from 1 to 10↑↑3, there would not be enough space in the known universe to write them all even if every atom is large enough to write on.
i was about to ask about that. it would probably take about 1,000 years to write it, or 2,000 maybe.
no, more like 700 years i think
10↑3 = 1000. As a single arrow is exponentiation. 10↑↑3 is 1 followed by 10 billion zeros.
Really? It maked sense to me in the video now I’m confused
Yeah, I thought that was an error, but just a misinterpretation.
This is the ultimate ASMR experience... Ever!
I'm experiencing learning and relaxation at once.
You are amazing.
I love the enthusiasm and simplicity of your explanations. Thank you.
“Those who stop learning, stop living” great quote and great conclusion
A generalisation of tetration is Knut's up-arrow notation. It's basically the same concept with the notation 2↑3 for 2³, 2↑↑3 for ³2, but you don't stop there and go with how many arrows you want, for example, 2↑↑↑3 is 2↑↑2↑↑2 which is 2↑↑2↑2=2↑↑4=2↑2↑2↑2=2↑16=2¹⁶=65'536. I really recommend searching about this, especially about Graham's number, which is the biggest number used in a mathematical proof(Edit: apparently not anymore? Couldn't find any proofs, tho. Any information could help Edit in the edit:G64(Graham's number) is still the biggest in a demonstration after further researches). Next to this number, ³10 doesn't seem that big. It looks horrifically tiny, as a matter of fact.
Edit: I forgot to say that these operations are, as exponentiation, right-associative. This means that you calculate them from right to left just like you calculate exponentiation from top to bottom.
Not sure where you got "up arrow" from - the character you mean is actually on your keyboard: "^"
@@TekExplorernope, that particular notation uses up arrows, as dictated by the name..
@ He's clearly talking about the lesser-known Knuth's Caret Notation.
@@TekExplorerthe caret is only used as a replacement for up arrows when they are not available on the keyboard. On paper or when you have access to them you will tend to use the complete arrow. If you want to verify this information, I found it on Wikipedia on the "Knuth's up-arrows notation" page in the "notation" category
Well then what's 3↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑3?
Just saw this on my fyp! It took me a bit to figure out but im getting it now. If im not wrong, the way it works is 2x2= 4, and then 4x4= 16. And basically you multiply the product by the product. Thanks!
16. 2 is the smallest base of complete numbers to not completely overload our imagenation. Very nice! Really like your video🙏 + very nice code in the end❤
Incorrect, 1 would be the smallest base to not overload us. 1 raised to the 1 raised to the 1 is still 1. Wrecked.
no he's right. if you go past 2 (3 for example) it quickly becomes unimaginable but with 2 as the base it you still can.@@Dyanosis
@@milanhaver3915 no he's wrong if he said largest then he would be right
@@astromacheand what of 0?
@@Toast_Sandwich if 0 is the number in supertext would it not just be 1? however the other way around I have no idea.
The answer is 16 because 2 to the power of 2 to the power of 2, so you have 3 twos which is why it is called the 3rd titration of 2, it’s kinda like how powered numbers work but it is bigger, turn the multiplications into powers.
this is the first time i've stuck around for a six minute video of a 10 second explanation, his demeanor and voice are just that likable.
Multiplication comes after addition exponents comes after multiplication tetration comes after exponents
That was fun. I've long thought that the fastest and most compact way to make big numbers was using a number like 9 to the power of 9 to the power of 9 to the power of 9 So Tetration is simply formalizing a syntax for it. In my example, 4(tetration)9 Or in computer syntax from one of the old languages I used, 9^9^9^9
And the number of your base can increase too, so 9(tetration)99 is unimanginably bigger than 9(tetration)9
We can extend the concept even farther. Lets say º is tetration and lets say ~ is repeated tetration, then;
2º3 = 2^2^2
2~3 = 2º2º2 = 2º16 = 2^2^2^2^2^2^2^2^2^2^2^2^2^2^2^2
Which is uncomputably large I tried pluging it into wolrfram alpha and best it could do is say it is equal to 10^(10^(10^(10^(10^(10^(10^(10^(10^(10^(10^(10^19727.78040560677)))))))))))
Then search grahams number hahahah
I went down a bit of a rabbit hole and discovered it doesn't stop with tetration. Tetration is a part of these things called hyper operations and is also known as hyper-4. Apparently someone was insane enough to coin a term for hyper-infinity: Circulation
What about the next step of tetration? Like having 2 and 2 be 2 tetrated two times
Like 9 (super tetrated) 9 times would be 9 tetrated by 9 tetrated by 9 tetrated by 9 tetrated ... (9 times)
lol
Never in my life did I think that I would scroll on youtube and actually watch a video where I would hear something, I have never heard in my life. Tysm for sharing!!
It was lowkey bussin
I would expect a person like yourself to hear something everyday that you have never heard in your life.
@@MyOneFiftiethOfADollar And that's why you're single
You ratioed him 🔥
I wish teachers did this more often, getting students to figure out how concepts work by providing just the steps grants better understanding
Trust me it doesn’t work when teaching a class😂 it’s fine for a UA-cam video, but when teaching a curriculum it is flawed. I’m currently studying maths, further maths and physics, and one teacher we have for further mathematics has this approach, and he ends up confusing everyone! Like I said, no problem here, as this is just some fun maths, not too complicated but it doesn’t work at a higher level
@@fullsendmountainbiker5844 Yes it does, but at higher levels there is already an expectation of prior knowledge, so some of this can be skipped.
Also at higher levels you need to go and bring things back to simple, otherwise you have idiots with so called higher education trying to use Algebraic calculations and rules in basic math using PEMDAS and getting the wrong answer. IE: Confusion, why? because the basics of why are not taught only how and shortcuts.
It has to be taught to use the level of math required/needed for the specific situation. Boolean Algebra doesn't apply to everything, but there is a place for it.
@@chrish7336 yes there is obviously an expectation of prior knowledge, but you can’t assume everyone in a class can just work a challenging mathematical concept out themselves. If I could do that I’d be a genius, and I’d have no need for any education. Don’t get me wrong this kind of teaching works for some topics, but not for others
Your channel is amazing. Every time there is something new to learn.
I Love your Way of teaching. And the smile that is on your face showing how excited you are about The Wonder of numbers. If only more teachers taught this way to get students excited about numbers too, it would be amazing. If you are not or were not a school teacher or college professor, you missed your calling.
"If only more teachers ..." being excited is not reproducable very often, meaning a teacher may even only be excited the first time teaching, solution: capture the video and show it to next year's students.
I would love watching years old video that contains an excited teacher, instead of watching live attitude of most teachers.
GOD ALMIGHTY calls everyone where HE needs him/her most!
@@anonym-hubI went to a new school were I was assigned to a math teacher who let an audio tape and overhead projector teach the class. That didn't work for me at all. When I transferred to another class it was great because the football coach (about 5'4") and the track coach (about 6'8") had combined their math classes and made it fun as well as educational. Just watching that mismatched pair working together was entertaining. 😂
nah bruv he took 7 minutes for that shit. Its one thing trying to accommodating but assuming the general audience who watches is THIS dumb that they need 7 minutes for it?
16 is the answer. I like how you put passion in what you do; meaning you like what you're doing.
Just graduated year 12 with a growing hatred for learning maths due to the brute forced and completely confusing maths curriculum. Watching this video was genuinely interesting, your passionate and excited explanation of tetration that i was pretty sure i would be completely lost on and click off the video, somehow kept my attention and got me really curious to see this through regardless if i understood or not. Just wanted to take a moment to appreciate this video and the interest it somehow sparked within me for maths, even a slight bit. Also that handwriting is pristine. Keep up the good work.
Try derivatives and Laplace Transforms. Even Z transforms are useful when dealing with sample rates from a computer and systems exhibiting weight, velocity and hydraulic dampening.
I love math, but yes, it is very difficult to understand.
Do they still teach common core? If so, that's a big part of it and it's designed to hamper people in their learning.
@@scottbenzing1361last I heard yeah common core is still a thing unfortunately. Standardized learning to create standardized little workers to fill all the low level vacancies and work 80 hour work weeks for 5 figures a year
Do you people still exist?@@scottbenzing1361
i already know of tetration, but wow am i happy i clicked on this video & subbed- your voice, your passion, your penmanship, your singly-focused video structure, and the sounds of the chalk board. MUAH, amazing stuff!!
sameee
Man, I SO wish I had math teachers like you in school. I always loved math but few of my teachers did. Not only do you clearly love the subject, which is transformative when it comes to teaching, but there's just something about the way that you teach that is inherently very engaging, completely independent of the math. I can't put my finger on it, but it's there.
bro explained 20 second thing in 7 minutes, what a legend.
This is my problem with the school system and 99% of youtube tutorials
That's why I left the system too.
@@PrimeNewtons In "simple" and "not confusing" terms Tetraition is the repetition of the repetition of repeatedly adding a number *or* the repetition of the process of repeatedly repeating the process of addition
I don't have 7 minutes to spare, please explain it in 20 seconds
@@NilsMueller tetration = bigger numbers scaled up by its scaler
I liked when you said "those who have stopped learning are those who have stopped living." It reminded me of my senior quote: "to live is to learn, to learn is to grow, to grow is to live."
And…”Knowledge breeds enthusiasm!” When students say a subject is boring, I tell them it is because they don’t know enough about the subject.
This was really cool. I’ve never heard of this type of calculation before.
Thank you for sharing.
I’ve learned something new!
When I was in seventh or eighth grade I had already developed a love and admiration for math, one day I was reflecting about it and asked my teacher: "so there's addition, then multiplication, then exponentiation. is there anything that comes after exponentiation?". To which he replied with a simple and final "no. nothing beyond it.". Well I feel really good now to know I was right at that time and that tetration exists.
Well if you just define it it "exists"...
@@rjtimmerman2861 that's technically right. but who was I to claim having invented anything in math
@@nowherenearby9461 the same as all inventors, a person with an idea :)
Why isn't this hearted ❤
So now figure out what's after tetrarion.
Teachers who teach with passion or excitement help people grasp the subject. You talked clear and to the point in a way that wasn't condescending loved it subbing now
5:13 understatement of the year
Hyperoperations
Repeated zeration - addition.
Repeated addition - multiplication.
Repeated multiplication - exponentiation.
Repeated exponentiation - tetration.
Repeated tetration - pentation.
Repeated pentation - hexation.
Repeated hexation - heptation.
Repeated octation - nonation.
Repeated nonation - decitation.
Repeated decitation - undecation.
It continues till infinity.
Pls pin
³2 = 2^2^² = 2⁴ = 16. So basically use the first ^2 to raise to the ² power again. It would be the same method as doing 2² but it's on the exponents now with ³2. So the coefficient would be 2⁴ = 2×2×2×2 = 16. Never learnt this in school like learning about how to make money in school😂
How'd you make a small 4?
Oooh... I never noticed you could make more than superscript 2 and 3.
On a android, if you hold down a number, say 2, you'll get an option for ², and other characters. (⅔ and ⅖, apparently)
@@colonelquackyou are right 😮
They never teach this because it's never needed. As seen, the numbers are so insanely large that things in the entire universe are much too small to be measured in tetration, so even professional mathematical jobs do not need to use tetration as often.
you never learned how to make money at school because if teachers knew how to make money they wouldn't be teachers 😳
This man is a legend… still liking comments to this day
(Btw the answer is 16)
16, 2x2 = 4, 4 x 4 = 16
Why is it not 2x2=4x2=8?
that would be 2 to the 3rd power this is the third tetrate of 2@@FreeAmericanSpirit
@@ThePrintZone123 you need to explain it better in the vid. All you said was it is a billion for each time you multiple it. So that infers it should have been 8 billion... 2 x 2 is 4 4x 2 is eight that's using the 2 three times.. to get to 16 you have to have another 2. Why is there not a 4 in front of the 2?
@@FreeAmericanSpirit because the 2*2 is 2^2.
So the 4 is the exponent with base 2.
2^2^2= 2^(2*2)= 2^4= 2*2*2*2= 16
@@FreeAmericanSpirit
Step 1: 2 to the 2nd power = 4.
Step 2: 4 to the 2nd power = 16
It means 2 to the power of 2 to the power of 2, which equals to 2 to the power of 4, which equals to 2×2×2×2=4×4=16.
0:16 is the answer
Smort
31
This is so clever 👌🏼
Im studying for my ACT and came across this video. The quote at the end is one that I hope to remember till my time on earth expires. "Those who stop learning, have stopped living." Very powerful and inspirational. Also I love how you teach and your smile and enthusiasm on the topic is very interesting and this is coming from a "bad" student that absolutely cannot stand math.
If you hate math, you might have been abused. Hint: math is a language, and you're clearly doing just fine with English.
Thanks for teaching this concept in a very unique and enthusiastic way. As someone learning this for the first time (like most others), I understood this really well.
Wish I had teachers like you during my school years :')
While programming for a class we were learning to make functions starting by the basic sum function.
Then I made the basic multiply function, and then I made the exponential function. They way they worked was pretty similar in terms of programming, so I figured why not keep going, so I made a function that elevates a number by itself by the amount given, and then I made a function that iterates this previous function by an amount given, so it would be elevating a number by itself an itself amount of times, by the amount given.
First 3 worked perfectly, 4th worked until 13, and the 5th crashed with just a 3 xD.
c, python or java ??
@@mckillua97 it's electronics so C
nice :o i use it every day lol@@martinrodriguez1329
You mean x^x?
@@tanveshkaviskar442 I think it's.
Scanf("%d",&n)
For(I=0; I
Wonderfully explained, your videos make me question whether I actually disliked math or just disliked the way I was taught. I'm not sure why this video was recommended to me but I'm glad it was, thank you for sharing your knowledge.
I didn't need to learn this, but I don't regret learning this.
YT recommended me your channel out of the blue, and i'm so glad it did! i'm promoting TA (Tetration Awareness XD) whenever and wherever i can, tho i fully realise it's not very practical operation. But it's important in its own merit, just to remind us to keep our minds open and always expect something more out of the vastness of mathematics.
Keep up the good work man! :)
Is this a widely accepted notation or is there something spicy like a minority trying to push it into acceptance
@@NaveedKakal no, i dont think it is. More commonly approved is something called Knuth's up-arrow notation, it's being used to write operations of exponentiation and beyond. Standard exponentiation (let's say 2 to the power of 3) would be written as 2↑3 (it is kinda similar to the 2^3 - i'm not sure if the root of both is the same, but they do look suspiciously similar). Tetration, and any other higher iteration for that matter, would just be written with more arrows in a row. So 2 tetrated by 3 would be written as 2↑↑3. Pentation as 2↑↑↑3 , hexation as 2↑↑↑↑3 and so on. Tho for tetration itself i'm a huge fun of a notation presented in this video, i mean it's just more elegant and instantaneously resambles something related to exponentiation.
There are other interesting unorthodox notations out there, like inverted sqrt sign for logarithm, try googling "A radical new look for logarithms". It's just brilliant. I've also come across proposals for trigonometric functions that, instead of traditional sin, cos, and tan, use a representation of a right triangle with only the two sides relevant to the respective operation. So, tan would be written as a horizontal and a vertical line meeting at the right angle, sin as a skewed and vertical line, and cos as a skewed and horizontal line. :)
@PrimeNewtons I am hoping the answer is 16, I think that seems too small. I came to that answer by modeling your 10^10^10 example using a base of two. Regardless, wrong or right, I have to thank you for many of your videos which I have used in my classroom to help demonstrate concepts. I think my students are starting to appreciate you more than they appreciate me, 😂. Thank you for the comment at the end of your vid about "never stop learning...". It's a lesson that I was taught to live by and I try to instill it in my students all of the time. I'm going to check out the rest of the comments now to see where I went wrong or not.
16 is correct. Thank you for the feedback 😊
It only seems too small because the base is smaller: 2. Powers of 10 are much larger numbers than powers of 2. If the base was 1, then it would be the same number no matter what powers you raise it to.
@@surferdudemiyeah, if you raise it to 3 it would the be 3^3^3 which would then be 3^27 and you can see how it then balloons from there
@@PrimeNewtons I never ever thought that this would be a thing i am going to annoy my teacher tmr and my mother is also a math teacher she never told me i wonder why?
Agreed 16
This is great! Thanks for the lecture. Answer is 16. Can you explain how would you write the googol or the googolplex with Tetration?
Nice handwriting
He really does
I actually think you can write down 10^(10B), but not by hand. I managed to fit 16 190 zeros on a single page in MS Word with Times New Roman font, font size 7, line spacing of 1 and narrow margins. This would mean that it would take 617 666 pages to print out the whole number. A single printer page is 0.1mm thick, so that would result in a stack of paper standing 61.7666 meters tall, which is only ~1/5th of the height of the Eiffel tower. Not practical at all, but certainly doable. Anyway, thanks for the video! :D
That way with no exponent, you could write only 10^10, not 10^10^10. You probably ment it written this way: 10^100000....
you need to go bigger laterally, in google docs, arial, 6pt, 1cm margins, i can fit 33605 on A3, running the numbers, thats approximately 30 zeros/cm^2, meaning you can fit approx. 143k zeros per sheet of A1, 1e10 zeros/143k zeros per page ~= 70k pages, appx. 7m tall
edit: i did 10e10 not 10e10e10, will update again once i've had a chance to calculate, as i'll have to do such large calculations by hand
edit 2: ...wait, no, i did it right the first time, 10^^10 has 10^10 zeros, 1e10 zeros / 1.43e5 zeros per page, = (1/1.43)e5 = 7e4 = 70,000 pages, (70,000*0.1)/1000 = 7m... your use of the european decimal seperator confused me, you meant a hair under 62m, i read it as a hair under 62km and didn't question the fact that that's 1/5th the eiffel tower 😂😂 which made me think i did something wrong
Wrong math 😁
wrong math buddy there are not litterally enough atom in the universe to write that numbere, if you had written a zero for any atom on that page you would have put trillions of zero okay? even more than that, because in a single centimeter there are hundred of milion of atoms okay? It's litterally like they said in the video you have 10 at the power of 82 atoms in the universe, but that number is 10 at the power of 10billions. We don't have enough atom.
@@chorec no 10^10 is just 10 000 000 000, this to write is no problem. he wrote down 16 190*617 666 zeros, this is about 10^10 zeros after the 1, should be correct.
Tetration is a mathematical operation that involves repeatedly raising a number to the power of itself. This operation is an extension of exponentiation, which involves raising a number to a specified power. In tetration, the number is raised to the power of itself multiple times in a tower-like structure. Tetration can result in very large numbers, making it a fascinating concept in mathematics. Mathematicians use tetration to explore complex patterns and relationships within numbers.
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Wow, I love your English pronunciation. just enough of an accent to keep my attention and fluid enough to flow beautifully... perfect balance... you are 1 in 1X10^6.
2 is the a very special number, since it is the only number apart from 1 that always evaluates to 4 when elevated, tetrated, pentated, octated, icosated, hectated, and so on to itself.
Depends on the number of the tetration or pentation, etc. If it is 2 to the tetration, pentration, octasion, etc. of 2 then yeah it just becomes 2 squared, which is 4. If the tetration or pentation is higher however, the number quickly becomes absurdly large. 2 to the tetration of 3 is 16, 2 to the pentation of 3 meanwhile is a whopping 65536. And i'm not even gonna talk about octation.
Now isolate it.. 😂😂😂
@@Aglassofwater67 I know that, that's it is precisely why it is a fun fact
Who penetrates it
Also 2 is both a composite and a prime number.
Composite - we can divide it by 2
Prime - It can only be divided by itself and 1
I haven't used math since they tried to teach me and yet here I am
Lol 😆
I am truly amazed at your abilities to express and explain so lucidly.
WE NEED TEACHERS LIKE YOU IN OUR SCHOOLS IN MISSOURI. 🎉🎉❤❤
I agree
*everywhere
Seriously boring teachers like him
@musemamo : Looks like you are in the wrong class...
You are looking for HOW TO WHILE AWAY MY LIFE CLASS.
Tetration!
³2 = 2↑↑3 (using Knuth's up-arrow notation)
Tetration is a repeated exponentiation:
³2 = 2^(2^(2))
= 2^4
= 2 × 2 × 2 × 2
= 16
So, ³2 = 16.
Tetration examples:
- ²3 = 2^(2^3) = 2^8 = 256
- ⁴2 = 2^(2^(2^2)) = 2^16 = 65,536
That’s correct except to 2 to the 3rd tetration=16 not 256 and 3 to the 2nd tetration=27
Recovering from a burnout, worked in finance. Your vid’s are great to get back into things again! Loooove your positive energy, much respect from 🇳🇱 processor. You are a great teacher!
Dang this was so interesting! I got 16 by following your example, from the other comments it looks about right. Please keep making these, you're awesome!
Wow I'm glad I found this channel, I really like your style and you for that matter. I personally learned about tetration (and quintation, hexation etc..) when I was taught Knuth's up arrow notation in college as the next step was to take the derivative of a tetratic expression.
Thank you for sharing this. I learned something new and I believe that we all should strive to expand our knowledge base. You never know when something learn may become useful to you.
This thing has increased my curiosity in the world of maths.
You don't need 10^10billion, 10^100 is already more than number of atoms in the known universe.
i used my calculator it answered 1
@ThatOneProFloppaTheBest you did something wrong then. The universe has 10^84 to 10^90 atoms.
@@Perrigon no bro, it said 1.00000000E+84
What r u on
Actually you would need it to be 10^10^x where x is the exponent on 10^x with the number of atoms in the universe for it to be impossible to represent the number of zeros with one atom per zero
³2 = 2^2^2 = 2^4 = 16. Thank you so much! I learnt something new today!
ur extremely underrated sir, i hope you get the recognition u deserve 😇😇😇
Tldr + ratio + suck deez nuts
Congratulations; you found out the answer!! 🎉🎉🎉🎉
2 raised to the power of 2 is 4 raised to the pwer of 2 is 16. I could listen to you speak on any subject for hours and you would have my full undivided attention. We need more people like you in all walks of life. Thank you for sharing.
Be careful. Your calculus works with this case, but you actually compute exponentiation right to left. 2 to the power of 2 is 4, then 2 to the power 4 is 16. It works in your case because 2^4 = 4^2, but if you replace 2 by 3 for example, you obtain 3^3^3 which is 3^27, not 27^3.
From wikipedia : "Under the definition as repeated exponentiation, na means a^a^...^a, where n copies of a are iterated via exponentiation, right-to-left, i.e. the application of exponentiation n − 1 times.
16 is right answer, where's his HEART?
16
It's interesting how fast results become larger and larger with just the relatively tame adjustment of the base in tetration. 2 -> 10 is something that you can do with your fingers.
But with tetration it is like "ah, 16 is fine" and "wtf is this number, we can't even write it's zeros".
No wait. 2*2*2=8, 8*2*2=32
@tomvano nope lol it's an elevation not moltiplication, it's 2^2^2, which is 2^(2^2) which is 2^4=16
1.Multiplication is repeated addition
2.Exponentiation is repeated multiplication
3.Therefore, exponentiation is the process of repeatedly repeating addition
4.Tetration is the repetition of exponentiation, therefore...
Tetration is the repetition of the process of repeatedly repeating addition
Thank you that was extremely interesting. The answer is 16 I’ve never heard of titration I’m 82 years old and I found this absolutely fascinating
interestingly, tetration (what's being taught here) and titration (a concentration measuring process in chemistry) are two completely different things!
@@k.whatever9046bruh
It is 16. Really impressed. Saw this concept for the first time in my life. 2 raised to 2 raised to 2 or 2^4 or 16.
5:56 I think you’re getting confused between the exponents. 10^10^10 has 10^10 zeros, but there are 10^80 atoms in the universe, which is way more than required. If every 0 occupied an atom, a single file of atoms 1 metre long would be enough. I think you thought that the number had 10^10 billion zeros, which is where the confusion came from
I think so, too. Numbers are not my strength 😀
If you tried to count to the number "ten tetrated to three," and each individual atom in the universe represented one unit of this number, you would run out of atoms to count this number at 10^80, meaning there would be 10^9,999,999,920 units unaccounted for. Ten tetrated to three is an unthinkable amount larger than the universe, which is already an unthinkably large size.
To me it checks out. In maths, BODMAS tells you that Order (Exponent) always starts from the highest point. 10^10 equals 10 billion after all, and 10^1000000000 is much bigger than 10^80.
@@CatfoodChronicles6737 But there are "only" 10 billion (10^10) zeros to write. So it's actually an easy task, you need to write only 5 zeroes a second without breaks for 64 years and you are done!
But 10^10^10 is 10^10b because 10^10 is ten billion
It's 16. You inspired my interest on proof theory and large number theory. I've been learning about transfinite growth in the fast growing hierarchy, and it gets pretty scary.
Somehow your videos have never come across my feed or my search results before. I really enjoyed everything about this video and yourself. I subbed not only for the aforementioned reason, but the sound of the chalk tapping on the blackboard made this GenXer's heart skip a beat!