The Rare Levels Beyond Exponents

Addition, Multiplication, and Exponentiation are just Levels 1, 2, and 3 in a whole chain of operations. Many people don't know about Level 4: Tetration (or the further levels like Pentation and Hexation). Let me explain these rarer operations and show you some of their powerful secrets!

By the way, some comments are confused as to why 2 tetrated 2 (or 2 pentated 2) are just 4, so let me explain that more. Remember that these stacks include the bottom number, like how 2 to the 2nd power means "2 times itself with 2 total of them multiplied" and similarly 2 tetrated 2 means "a stack of 2^2 which is 2 tall (including the lower number) which is just 2^2. Same with 2 pentated 2, which is a tetration stack 2 tall (including the bottom number) which simplifies to the above example.. So yep, just like 2 to the 2nd power simplifies to the same as 2 times itself, so does 2 (any hyperoperation) 2.

Stay tuned for next episode, when we'll visit an amazing realm of infinite fractions.

Disclaimer: This episode is for educational purposes. Do not burn any clocks, and do not lean on any water-damaged desks.

Links:
Other channel - @Domotro

Combo Class, taught by Domotro, is an unconventional learning experience where anybody (whether they're a fan of normal school or not) can become excited to learn rare things about math, science, language, and more. Also check out the shorter videos on the Combo Class Bonus channel. Thanks for coming to Combo Class!

[raimondssmiltins]

Its not often you find such a small scale content creator who's as interesting and educational as you are.

[steelegagnon]

i love how low the barrier to entry is to this. your video makes it so i don’t have to have a phd and 87 years of theoretical math experience to have fun exploring this weird concept. thank you!

[RGC_animation]

Fun Fact: One of the largest number ever created, the Graham's Number, is defined using Up Arrow Notations, although many many many times bigger than a tetration, it's actually 64 layers deep.

[MatthewPiercey]

Domotro seems like the rare type of teacher who'll go on a wild-yet-coherent tangent when somebody asks a question and the textbook answer just isn't satisfying enough.

The kind of teacher who'd be liable to talk about stuff like the _forbidden fourth state of matter_ when somebody asks what's after gas 😁.

[ZarHakkar]

I remember figuring these out in the 10th grade. I was super into math and realized there were sequences of increasing numbers that couldn't be represented by any operations I knew. Then I figured out tetration on my own and realized there were basically an infinite amount of operations "above" addition, multiplication, and exponentation. The numbers get pretty huge and unreasonably whack pretty quick, so I figured that's why they're not used all that often.

[DeJay]

My man is ON THE GRIND! Please, keep it up, you're such a rare case, this was one of the most interesting, nice videos I've ever seen. Perfect energy, just right explanations, awesome. It's crazy how much effort you put into these videos for such a small community at the moment. Hope people find you and appreciate you.

[jackgreenearth]

Yes! I **knew** that this existed, but every time I tried to explain it - to my mother, to my maths teacher (with a similar method to you), they wouldn't understand, or wouldn't care. Thank you for showing me that this is a real thing!

[lued]

I sometimes say that we should update PEMDAS to STEPS. That's Sets (stuff in parentheses), Tetration, Exponents, Portioning (a word I picked to represent both multiplication and division) and Sliding (the same for addition and subtraction). A lot of people end up getting misled by M and D being separate letters and will do multiplication before division even when division is further toward the beginning of the question. And of course they'll do the same thing with A and S. That's why we always see those stupid "only 69/420 people can solve this" posts on social media where they've intentionally put a division in front of a multiplication so that people who rely on the acronym will get it wrong and argue in the comments. Plus, PEMDAS doesn't mean anything while STEPS has a very relevant meaning.

[sykoe]

You did a fantastic job explaining tetration by working up from the basics while also making me jam out and laugh throughout the video. Keep doing what you're doing and you'll have millions of subscribers in no time!

[spongebobbatteries]

@ 5:07, The "tetration tower: a to the a to the a to the a" slaps so hard with the beat and the piano melody. That was sick!!!

[okplay]

This is probably my favorite part of mathematics to explore, ever since we studied roots and logarithms at school, I've been wondering what comes after, and I was surprised by how little information there was out there. So I'm glad that I found this video.

[GringoLingo]

5:07 “A to the a to the a to the a” was so on beat with the music I loved it

[aogasd]

It's strangely comforting to 'sit in math class' as an adult. I missed learning about some obscure detail that gets confusing really fast and is very likely to never come up in any real practical scenario.
I'm not sure why it felt different to any other math videos I've seen on YouTube. I guess most math channels don't do video, or if they do it's not as conversational as this.

I should probably subscribe kek

[josiahtaylor]

I’m glad I found your channel again, stumbled across it a couple months ago and was really interested, glad to see you’ve kept the style and I made the right choice to subscribe this time you popped up in my feed

[somerandomcsgoplayerlol]

I like this content. as a 15 year old preparing to major in math, it's nice to watch a video that isn't just about manifolds. yes, you're also right about the first part. incrementation is crucial when it comes to addition, or defining the set of naturals and such, like in Peano's axioms

[badasson]

This gives me an intense 2012 YouTube nostalgia. Great work dude

[Ishsa]

This was a great video! I've never seen you before, but I love hyperoperations and large numbers. Thanks for exploring it in depth while keeping it engaging!

[jamesieza]

Love your videos, after watching a few I came back here to my first one to say: I hope you get to 1, 081, 080 subscribers while retaining your unique style, something about your enthusiasm and the minimal polish is awesome. The content and writing are perfect, refreshing to have hand written notes in a world of slick YouTube animations.

[Sinnester_GD_Player]

Tetration? Pentation? Hexation? I didn't even know these existed! Ty bro for the vid.

[matthiasbockbreder]

It’s 6:18 in Germany and I love your video.

Awesome! Your channel will grow fast.