Tetration: The operation you were (probably) never taught

We've all done addition and multiplication, and maybe even exponentiation. But what comes after? Is there more?

In this episode, we look into Tetration, a fun example of how a pattern that had been sitting right under your nose for a very long time might be explored!

Note: This is a re-uploaded video after fixing an issue pointed out by a user. Thanks again for the correction!

Yet another note: Holy mackrel I love you guys. There's yet ANOTHER minor detail I goofed up; it should be 1.9*10^2184, not 1.9*10^2185. Typos are a thing (see: the dangling parenthesis hidden somewhere in the video).

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Confused? Maybe one of the following tidbits will help! (This list is a work in progress; there will be more to come later)

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The Taylor Series aims to teach math. Whether you're a student in a math class or someone who has ever asked the question, "What was the big deal about Calculus?" there's something here for you.

I am aiming to have one new video up every month. I hope you enjoy them!

Also, I named this the Taylor Series after the mathematical construct created by James Gregory and formally introduced by Brook Taylor ... and I picked this one in particular because my name is Derrick Taylor. :)

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Music Credits (from AudioBlocks):

Intro music: Thinking by Patrick Smith
Main background music: Island Fun by Neil Cross
Sad piano music: Beautiful Piano by an uncredited musician on AudioBlocks
Outro music: Midnight City Lights by Neil Cross

[auxchar]

Protip: youtube actually has a frame-advance feature. You can use, and . to skip forward and backward by individual frames in a video. Super useful for the one-frame gag youtubers like to use all the time.

[Wecoc]

Here in Spain first we learn about Graham's number, then tetration, exponentiation, multiplication, addition and finally counting

[Monothefox]

And that's how you get Graham's number.

[Debg]

I actually thought about this at school, but didn't found anything about this operations anywhere (I had got no internet back then), so I just assumed it was of no interest.
What I later realized, however, is that the number 2 yields the the same value for all orders:
2 + 2 = 4
2 * 2 = 4
2 ^ 2 = 4
2 ^^ 2 = 4
...

[Kirbykradle]

3^7.6 trillion is trivial to calculate! what are you talking about?


It's 10 in base 3^7.6 trillion. ezpz

[blakeswensson]

I had heard of tetration before, but even after looking it up I didn't understand it. After watching this video I do, and I understand just how utterly insane the numbers you can calculate with it (and pentration etc) is.

[morgan]

at first i thought this was like titration, a chemistry thing, when i saw the title

[RalphInRalphWorld]

Woa, insanely good production value! Good luck on this channel

[ccm_priv]

since i was a kid i knew there was possible and had to be a "next level" operation a.k.a more and bigger hyperoperations than the potentiation or exponentiation. i was so wondered when i saw there was a theory behind it and that it had it's own symboles (i had to invent ones when i didn't knew the most used ones) Thank you for explaining this to more people!

[larryboi]

16 500 subscribers is a rather small number for this type of quality content

[Yotanido]

3^^5, also known as 3↑↑5

The end of this video would have so nicely flown into Knuth's arrows...

[blakelarkin]

Many computer scientists call this the “tower function, ” while the “super log” is referred to as the inverse-tower function. Some complex algorithms can be shown to take place in inverse-tower of n time. Note that if such an algorithm was run using all of the atoms in the universe as n, the inverse tower would be roughly 5

[ariztrad]

according to wolfram alpha the answer to 3^^5 is

[cpuwrite]

As a computer science major, I was taught in college that these operations are described using something called Ackerman's Function.
It is written A(x, y, z)
Where x is the type of operation and y and, if needed, z are the arguments.
Counting up to 5 would be A(1, 5) = 5. 1 for counting and 5 for what you're counting up to. No z argument needed here.
Adding 3 and 4 would be A(2, 3, 4) = 7 -- 2 for addition and 3 and 4 for what's being added.
Multiplication would be A(2, 3, 5) = 15 -- 3 for multiplication and 3 and 5 for what's being multiplied.
A(4, y, z) is, of course, tetration.

[RobotProctor]

1. Is there a log equivalent for tetration?
2. What sort of problems does tetration help solve?
3. What algebraic properties does tetration have? Commutation is out of the question, but if these numbers are there and we can't count them in our universe it'd be interesting to work with them in a different way.

[aDumbHorse]

I thought it was fascinating how you explained all those operations. Some of the schools I know just fail really hard to even explain multiplication, they just do some examples and tell "roll with it". Exponentiation sometimes is just "grab a calc, do this". With the really easy and didactic way you showed, I feel I don't need a calc anymore lmao.
Amazing video

[tz]

Here's a cool identity - the "pseudo-distributive property" of tetration over exponentiation:

(a^^b) ^ (a^^c) = (a^^(c+1)) ^ (a^^(b-1))

[ryanknight]

Wow the quality of this video is amazing, you deserve so many more subscribers

[murilovsilva]

It's past 1 am, I have no idea how I got here, and holy goodness, I freaking loved it.

[Nawmps]

3:25

I commend you man. I can see your dedication to the subject you love!