Addition, multiplication, ... what comes next? (It's not exponents)

This video begins with a search for a binary operation satisfying certain properties, and ends with the creation of a new number system: the exponential numbers.

#SoME3 (finished late)

Although I came up with these concepts independently, a commenter has pointed out that the operations defined in the first part of the video already have a name: the "commutative hyperoperations."

[paulbloemen]

I’m 73 now. When 13 I asked my math teacher if there was an operation next in line in the series multiplication, addition, so in the other direction. I finally got my answer here.

[kjetil]

these are called the "commutative hyperoperations". i think you should add that name somewhere in the description or the title, so that this video shows up when people search for it, because this is definitely the best explanation i've seen of them.

i remember seeing another cool overview of them a few years ago, that went in a different direction than this video. unfortunately i'm no longer able to find it, because again, they didn't include the name anywhere.

[demonicdrn]

This was such a cool video. I cannot express how exciting it is to learn all of this; it feels like discovering the complex numbers again. You totally deserve to win #SOME3

[dtcovax]

This video feels like discovering math

[timseguine]

In case you didn't realize K[-1] is also known as the log semiring, and it comes up when talking about log probabilities which are all of the nonpositive reals. It is a convenient way of dealing with numerical issues while dealing with large categorical distributions, since it more efficiently uses the available floating point bits.

[Nonkel_Jef]

The vast majority of numbers have over 1.65 million digits, so it’s still practical.

[pocarski]

This is very interesting if we look at it in complex numbers. In C, log(z) is defined for all values except 0, so +_n can be used on anything except 0. If we define log(0) as inf, and exp(inf) as 0, then we get a consistent projective plane-like system without any weird points. The "number line jumping and/or sharing" described in the video fits very neatly into the multivalued nature of the complex log.

A natural idea is to try and find a continuous extension of +_n. This can probably be done by first deriving log^1/N and exp^1/N (functions that when applied N times become equal to log and exp) and then extrapolating onto the rest of the rationals and then the reals. I will look into this later and will edit this comment after I do

[musicarroll]

I wrote a paper on this back in 2001: " The Natural Chain of Binary Arithmetic Operations and Generalized Derivatives." Bennett also discussed this in a 1915 paper, "Note on the operation of the third grade."

[AJMansfield]

Your audio is mixed rather quiet. YT allows replacing audio tracks after the fact IIRC; if you can, see if you can replace the audio with a re-render with the master gain set 20 dB higher.
In general, when mixing audio for YouTube, your target should be mix to somewhere above -16 LUFS. Anything louder than that will be automatically normalized to the same volume as other videos when you upload, but YouTube doesn't automatically increase the volume of videos that are quieter than that the same way.

[VaradMahashabde]

Man, casually giving a polylog and polyexp classifying function as an exercise is too much.

Amazing video!

[sanes]

You made me feel as if everything i've learned in math classes has lead me to understand this video. Thank you <3

[cparks]

20:37 That escalated quickly. A wild category theory has appeared.

[juchemz]

I really loved this video. I can tell you understand this number system very deeply. It all clicked for me seeing each Kn with its own number line, really highlighting the isomorphisms with the real numbers. I appreciated the pacing too, covering all the fundamental ideas without over explaining. You deserve many more views. Thank you for sharing this.

[twixerclawford]

Trying to imagine how this all extends into the complex numbers makes my brain break

[luker.]

I love functional analysis focusing on identities like the ones you use. Great stuff.

[lock_ray]

That's funny... I had this exact train of thought at some point, to an uncanny degree. Even went as far as thinking about "subaddition" log(exp x + exp y), though this is where our tracks diverge - the exponential numbers system is new to me :)

An interesting tidbit I came across on my own track is: if you switch to base 2 rather than base e, you get a pretty neat relationship between operator n and operator n+2. Namely, in any base the "square" function x -> x •_n x has the form x-> x •_(n+1) c_n for some constant c_n depending on n. If the base is 2 then we get the nice property that c_n is the identity element for •_(n+2).

Example: if ° represents subaddition then x°x = x+1 so that c_(-1) = 1, or the multiplicative identity. Similarly x+x = x•2 so c_0 = 2 or the identity of •_2 and so on...

A way to express this fact formulaically is
x [n] (x [n+1] y) = x [n+1] (y [n-1] y)
where [k] is the kth operator.
In case n=0 this becomes
x + xy = x(y°y)

[rauljvila]

Love these "generalization of simple operations"! Great video, thanks!

[krakow]

I had the same thought like 10 years ago and came up with the log exp solution forwards and backwards, but I couldn't imagine how it could be useful so I stopped playing with it. You really carried the idea as far as you could without expectation for a reward, you must have true mathematicans' blood!

[flavertex]

THANK YOU! I got weirdly obsessed with this question and put a lot of effort into it, but had to give up before reaching a satisfying answer to completely generalizing evening. This is exactly the train of thought i was pursuing and you've pursued it to the horizon - thank you so much!

[a-h]

Absolutely incredible for your first ever maths video! This is way better quality than most well-established maths channels. Well done and thank you!