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.

16 . . . I see why you have chosen a base of two.

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*

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! :)

now imagine ³(³2)

It's 16...... The last dialogue: "Never stop learning... One who stops learning, stops living..." Touched my heart.❤

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!

Your excitement is contagious. May no one ever take your joy away from you. God bless.

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.

math just like any class always becomes a lot more fun when your teacher is enthusiastic to teach you the subject.

16 is the answer. I like how you put passion in what you do; meaning you like what you're doing.

Thank you so much ! I was kinda exhausted learning the old things.... this new thing kinda lifted up my spirits !! Hope you'll continue presenting these new concepts !! 😃

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.

never in my life thought that i would be watching a video about maths that will not be in my exam

Nah i watched your eigenvalue video for my calc 3 class and found this gem, ngl you might be one of the few passionate teachers on youtube, u genuinely have fun with maths and that's i think is rare nowadays, hats off to you gentlemen.

I have studied math at uni for 5 years and never come across the definition this. As you said it is probably because it is a bit useless (as ³10 is basically infinity). Very interesting and a nugget of knowledge. Thanks!

Never thought I'd enjoy a math lesson. Thank you sir

There is nothing greater than an enthusiastic professor who can communicate the topic exceptionally.

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.

Found this in my recommendation, learned something new, was not disappointed. Good work.

“dont stop learning, because those who stopped learning, stopped living.” as a person who nerds out when talking about math, that hit hard

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 got 16. Love the commitment you show to this video by the way! I just want to thank you for showing your teaching so well, and explaining in such a clear fashion as to why tetration is not taught, and for teaching me another fact. Thanks a lot man!

Thank you so much! for teaching us this lesson. It is really something new that maybe very few people actually know and use also.

We’ve found it boys! a math lesson that I will actually never use in real life! Great concept and I loved your explanation

Maths becomes interesting when it's taught by an enthusiastic teacher like you!!

Thank you. You explain it very well. I have been working with numbers all my life. I am 82 now. I don’t expect to see something on KZfaq about arithmetics that I have not seen before. Thank you again.

Based on the well instructed lesson with hints that make you see, the answer cannot be missed. It can be said that raised value number is equal to the base number raised based operation twice, such that ^3 base 2 = (2^2)^2 = 16. So the example given, provided the key information to solve. Thus (10^10)^10 = 10^100. Prince Newton, you are amazing.

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.

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

Good work sir. More like you would not be any competition and your contribution is precious. All my respect, may God bless you, sir, and your precious two bits in. I say Thanks for all those who won't.

Amazing Video! They don't teach this in school but I had heard of it before, thanks for explaining how it works. "Don't stop learning, those who stop learning have stopped living" Such a moving thought! 🤩

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

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.

As a social studies teacher (who myself has a math disorder and alllllways struggled with it greatly) I appreciate your instruction! Being autistic, I got the NON math-genius variety, and instead got the “I make 3 right turns instead of one left turn and don’t realize there was an easier way until 4 weeks later” type, so math based subjects always trip me up, but your manner of explaining and visualization is fantastic, your students are very lucky!

That was very interesting to know, added with your calming voice!

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).

The enthusiasm you put into this video just makes it 10 times easier and better to learn. Thank you kind sir!

Awesome, it makes calculating our debt so much easier. Thank you.

This video suddenly recommended to me.. I am glad that i got it.. and the answer is 16.. i was confused first but then i understood.. Thanks for this!✨

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

I wish I had a teacher like you. It is so evident that you love what you are teaching us here.

Such a cool concept! Thank you for sharing.

16 because ³2 is like writing 2^2^2^2, where the base is used as an exponent, and the 3 in front indicates how many times the exponent is repeated. ³2 written in exponent form would be 2⁴

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❤

I'm a math teacher, and this was fun to watch! Awesome, and good job making it fun!

Tetration is a hyperoperation that involves iterated exponentiation. The notation \( ^n a \) denotes the \( n \)-times iterated exponentiation of \( a \). For example, \( ^3 2 \) means 2 raised to the power of 2, raised to the power of 2. For \( ^3 2 \), this can be calculated as: \[ ^3 2 = 2^{2^2} \] First, solve the innermost exponentiation: \[ 2^2 = 4 \] Then, use this result as the exponent for the base 2: \[ 2^4 = 2 \times 2 \times 2 \times 2 = 16 \] Therefore, \( ^3 2 = 16 \).

It took me a long time to justify that the answer is 16, for about 10 minutes, and I finally figured out why, really good job sir, now I have to get back to my studies, the mathematics I need to use so that I can pass the exams, wish me luck ❤

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.

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.

I feel so productive watching educational videos.

This is the first video of your chanel which i have seen. That shows that this channel is for knowledge.

Anyone wanna say what ³3 is? HINT: It is more than the amount of money that Elon Musk has

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."

OoOOooO my cerebral cortex is tingling!!! I love math, and I favored it more so when I had a patient and well spoken math teacher. You’re awesome 🥲

So 2 with the tetration of 3 is 16 2 with the tetration of 4 is 256 3 with the tetration of 2 is 27 3 with the tetration of 3 is 81 And 10 with the tetration of 3 is a Googol with the exponent of 10. Ngl, I WISH I could've learned this in school! This is just as fun as factorials! :D

“Those who stop learning, stop living” great quote and great conclusion

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.

I enjoyed the setup of comparing the infinitesimally large number to the test question. Nice trick.

A staircase of exponentiation... A tool so powerful we don't have many practical applications for it in our universe! Amazing thank you for sharing!

I wish teachers did this more often, getting students to figure out how concepts work by providing just the steps grants better understanding

really happy to find someone actually enthousiatist about teaching math, i never knew i needed you in my life

2^2^2 is not ambiguous, but 2^2^2^2 is. Use parentheses to differentiate between (2^2)^2)^2) = 256 and 2^(2^(2^2)) = 65536. The latter is what the lecture is about.

In short, tetration is just the base to the power of itself the value of the number in the "exponent". For example, if you do base 4 and the tetration exponent 3 you get 4 to the power of 4 to the power of 4 (4^4^4) which is 4x4x4x4x4x4x4x4x4x4x4x4 which gives you 4,294,967,296 as opposed to 4^3 which gives you 64.

My son insisted we leave a comment. He says the answer is 16! He’s got his mind wrapped up in exponentiation and tetration right now. He’s 9

bro explained 20 second thing in 7 minutes, what a legend.

Thanks for teaching us a new concept and for your awesome teaching performance.

Great point. Don't stop learning. Those who stop learning have stop living.

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.

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.

Such simple concept with such loooong explanation.

This makes me love maths more and more, truly amazing! ❤😊

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.

You don't need 10^10billion, 10^100 is already more than number of atoms in the known universe.

Teaching such an interesting topic to us with such enthusiasm is fabulous 🤩

you're a amazing teacher! love your energy! :)

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

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 split it up like this = 2^2 is 4, so i then added the remaining “2” exponent, onto the 4, which is 4^2. So 4^2, is 16. I can’t believe i actually learned a math concept from a KZfaq video this efficiently. Genuinely a wonderful teacher.

16, of course, since you marked that as correct. Factorials are another kind of number that I have found fascinating. They grow so fast, relating to the base. Have a good day!

Nice handwriting

This man is a legend… still liking comments to this day (Btw the answer is 16)

A beautiful concept explained in a beautiful way . ❤

I found 16 because according to the logic, the exponent before the number means it’s how many times you rise the number by, for instance 3^3 would be 3 times 3 times 3and then 27 times 27 times 27 so you would get 19,683

I love the enthusiasm and simplicity of your explanations. Thank you.

I haven't used math since they tried to teach me and yet here I am

Really loved this lesson, got to learn something new!

Really nice to see you hearting our comments even though there are so many. Great explanation btw.

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! :)

16, 2x2 = 4, 4 x 4 = 16

Very nice video, and I like your saying "those who've stopped learning, have stopped living"

Its 2^4 = 2×2×2×2 =16 I learned it in belgian college (to help calculate big numbers) but the notation was different , we used a special combination of exponents and accolades

I didn't need to learn this, but I don't regret learning this.

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 had to double check to see if this was uploaded on April fools

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

2^2^2 = 16. I feel like tetration will be used one day as quantum computers become reality.

Im actually watching during summer...cause i like your channel man 😎👍

16 is the answer. 2 squared- 4. 4 squared =16. Thank you for the new knowledge sir! Ive been alive 34 years and you just taught me about something i didnt know existed

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.