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Hold your horses... So your saying K1 means a☆b=Exp(Ln(a)•Ln(b)) and K-1 means a☆b=Ln(Exp(a)•Exp(b)). And (we're assuming "Log" means "natural logarithm" here), that all makes sense. But If Exp(x)=e^x, then isn't Ln(-Exp(x))= -x ? It's just a simple -/+ sign switch, right? So what is this stuff about ~x ? Where does it come from and what does it mean? And if "Log" refers to base ten then Exp(Log(x)•Log(y)= WTF are we even doing!?

2 ☆ 1 = 1

The curiosity I entered this video with was brutally raped by the deadening banality of your presentation.

Cool. Now calculate K-∞ for complex numbers

i never understood anything so little

maaan, i feel there is some things with eliptic curves group)

1984_B10 = 11111000000_B2 It's also abundant and odious.

28th (perfect). This goes hard.

Using a star isn't a good idea, because of how many computer languages use it for multiplication.

Your presentation is very good, and the video very informative. But i wish TH-cam wouldn't show me this stuff. I just need to pass calculus 2 and 3 then i never need to worry about fancy math stuff again! We are a society, we have specialists for this stuff! You!

I love this, I have to find a way to share it with someone who will find ot as interesting as I do. Good job with the title and thumbnail, had they been less interesting, because I found this right at the end of my lunch break, I wouldn't have saved it to my watch later, I would never have given a view to this master piece.

I ve been trying to find a formula for exp^r(0) for every r in the real number line

I understand that the math and logic are sound, but does this sound like gibberish to anyone else?

I'm surprised you didn't mention the complex number logarithm, which has values for negative real number inputs

If I were to put this on my college's radio station, how should I credit you?

idk man, wolfram says exp(log(x)*log(y)) = x^log(y)

Something very Hilbert Hotel like is happening here. If you didn't have infinite digits the isometry between addition and multiplication would fail for a simple reason. If you had two numbers represented with n bits of information each, adding them together at most carries over by one place, so the output number is represented with at most n+1 bits. Meaning at least n-1 bits was not retained on the output. However multiplying two n bit numbers produces at most a 2n bit output, so information isn't inherently missing from the output. The bits of information in the digits has to somehow be preserved by exp and log for this isomorphism trick to function, likely in a way that cannot work with any finite approximation of exp and log.

The problem with extending operations is that levels other than + and × relies on a constant (for this video, e is used)

I was looking for a playlist with such songs with piano to listen to while I'm studying. I found one. Now I need to find the rest! Great song

bluds inventing new numbers XD

They live!

Yo i wanted to ask if there any triatary function for division/multiplication. Basically my question is if its possible to divide 3 numbers at the same time (not doing 2 first and then that outcome with the third)

This is quite a good song to listen to while doing math

If you are curious about the line of repetition as the definition of how addition goes to multiplication instead of properties. Knuth’s arrows describes all operations after multiplication that is exponents, tetration, pentation, hexation, etc. Before addition is far more boring and is all defined by x * y = x + 1. The proof that this is true for operations before addition (in this different case than the video) is interesting in it of itself.

Thomas Donna Anderson Jennifer Anderson Steven

The swirl function is actually equal to x+i*pi A proof would be that ~x=ln(-exp(x))=ln(-1*exp(x))=ln(-1*e^x) and now lets simbolizise -1 as e^pi*i and that means that ~x=ln(e^(pi*i)*e^x)) and now we use the property that you explained to gey ~x=ln(e^(pi*i+x)) and ln and e simplify so we get pi*i +x and bc of addition property we will gey x+pi*i

Bro dropped a new number system and dipped for a year...

When you drew the lines showing the multiple representations of a number x, it seemed to imply we can construct fractional K's. Like K_(1/2) for example.

i bet you just called it exp( because actually doing the powers formatting on a computer is a deathly pain. if so good on ya. if not good on ya

it was never stated explicitly but i'm enjoying the fact that a^ln b = b^ln a, something i'd never thought about before

beautiful video

Continue posting please, Why all good math chanels only have 1 or 2 vidios (Except 3blue1brown)

Shouldn't we just extend to complex numbers? It solves everything

I remember discovering what you call K_{-1} just a bit over a year ago, but unfortunately it wasn't useful for my purposes since, as you note, this is just another copy of R. Unfortunately I was studying njmber theory but I had forgotten that exp and log may preserve relationships between R+ and R>0*, but Q+ and Q>0* are not isomorphic and that's the magic if number theory!

Bro I literally did same thing more then 2 years ago and uploaded to my channel

I remember asking my math teacher this exact thing.

me: wow what a neat and cool video my brain: *_WHAT ABOUT IMAGINARY VALUES OF K_*

oof my brain stopped working :( good video

In 1 at the end, f(x) is commonly called "slog", as the inverse of tetration (repeated exponentiation)

this is so good oh my god

If you think about it, maneuvering Euler's Identity you can find that ln(-1) is πi, so technically you don't have to create the exponential numbers, you can easily use complex numbers. But this affirmation comes with a question: immaginary numbers are smaller than minus infinity?

you could replace e in log and exp with something like 1.00001 so .n is usable for bigger n

6:20 A fun choice of symbols for the new operations would be to make the third operation in the chain (i.e. star in the video) use a triangle as the symbol, the fourth operation a square, the fifth a pentagon, and so on. (Or you could be a little more boring and use dot with numerical subscripts, it’s easier to write than say a 15-sided shape for the 15th member of the chain. But if you’re only doing the lowest few the polygons look cool at least. 😄)

isomorphism

It IS exponents.

8:46 I didn't expect Tetration in this video! Repeated logs are exactly like slogx(y)

for the series 0, 1, e, e^e... we can see that the next term is e^n=n2 >0, e^0, e^e^0, e^e^e^0... working backwards e^n=0 ln0 is the -1th term which equals -♾️ as for the -2 its ln(-♾️)= ln(♾️)+iπ(1+2k). {k€Z} ♾️+iπ(1+2k) as for the next and the rest in(♾️+iπ(1+2k))=in(♾️). ♾️ therefore the series should look something like this ... e^e^e^e^0 e^e^e^0 e^e^0 e^0 0 -♾️ ♾️+iπ ♾️ ♾️ ♾️ ... to calculate any term use tetration f(n)=ⁿe

I'm actually curious about what the inverses of some of these functions look and act like. Inverses of some of the other operations have unique effects when working with more restricted number systems, like how the inverse of addition on the naturals creates the integers with negatives, or how the multiplication inverse on the integers creates the rationals, and how exponenentiation has roots and logs that introduce the irrationals. Only need the trancendentals and you've got all the reals, and roots on the negatives brings into question the complex numbers. So, the question is, does this continue to occur for additional higher layers of these operations? Are there equations with them that don't have solutions that make sense within them? I mean, you DID bring up the point that they have a limited domain whose minimum increases as you increase n, but what exactly HAPPENS if you try to resolve values to the left of that? Perhaps that's where more unique number systems could lie.

After you laid out the basic premise, I watched the rest of the video pretending to pay attention and thereby understand it Are these sort of analogous to the conventional hyperoperations? And could you extend the log thing if you used complex numbers... or whatever You aren't even gonna read this, are you (→Great, not my problem then!)

I’ve been exploring these concepts in the context of wheel theory and how they apply to parallel addition. It’s all very interesting