Counting in a Complex Base

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Күн бұрын

Пікірлер: 38

@haavind 8 ай бұрын

I did some study of base 1+i last fall. If you allow for p-adic values, the "infinite repeating ones" (...1111111) is equivalent to "i" and you can start from there and get a second interlocking spiral, rotated by 90 degrees, that completes the plane.

@TheGrayCuber 8 ай бұрын

That is really cool! How does one demonstrate that …111 = i? Something like -1 = (…111)*(1+i) - (…111) ?

@haavind 8 ай бұрын

@@TheGrayCuber I worked out some carry rules of addition and subtraction i.e. how to move a '2' or '-1' upwards in the digits. Then you can show that ...222 is equal to 100 which is equal to 2i. It's similar to how ...999+1 equals 0 in normal 10-adics. The same carry rules can also show that 'any' integer can be written in base 1+i, thus proving completeness. Intrigued to learn that i-1 does not have that "bijection" problem though, maybe this will solve something for me

@angeldude101 6 ай бұрын

@@TheGrayCuber ...1111 is basically 1b⁰ + 1b¹ + 1b² + ..., which is a well-known geometric series that can be reduced to 1/(1 - b) when |b| < 1. |1+i| is not less than 1, but p-adics use a different notion of distance that essentially flips everything around. While I'm not sure how to properly use it with complex p-adics, I'm going to just ignore that requirement and assume it probably converges in the (1+i)-adics. In that case, we have 1/(1 - (1 + i)) = 1/(1 - 1 - i) = 1/-i = i.

@40watt53 6 ай бұрын

@@haavind man math is so cool

@StentorCoeruleus 6 ай бұрын

Ratio is always 1+I therefore all 1 according to the formula of geometric series we get 1/(1-(1+i))=1/(-i)=I so that’s proof

@benpuzzles Жыл бұрын

Cool concept! This can be represented as a complex iterated function system with f1(z)=b*z, f2(z)=b*z+1. b is the complex base you're "counting" with, you start with z=0 and you choose either one of the two functions to get the next iterate. What you're looking at is all possible iterates up to a certain cutoff (the max number of digits). It turns out the magnitude and phase angle of b are related to the dynamics of this, and parameterizing the base in polar coordinates would show how the patterns form a bit more clearly. (I haven't watched the original video, so maybe this was already mentioned)

@dylan7476 Жыл бұрын

Fascinating! Cool web tool as well :)) Out of curiosity, what sort of industry/role do you work in? I'd assume a maths-intensive field?

@TheGrayCuber Жыл бұрын

I’m in data analytics, fairly maths-intensive

@MDNQ-ud1ty 6 ай бұрын

These give the dragon curve so there is likely some deeper connection to that which connects to IFS's and fractals more generally. Essentially sum(d_k*b^k) is a value in the base b where d is some element from the base ring. Writing b = re^(it) gives sum(d_k*r^k*e^(ikt)) One can define F_n(d_n) = d_n*r^n*e^(int) + F_(n-1)(d_(n-1)) The idea is that for whatever base you choose you end up with a pattern plus some extra term. When you let d_n vary over the possible digits you get the pattern but it is both scaled and rotated by the iteration depth which corresponds to the digit place. That is, the "extra term" is a set of points d_n*r^n*e^(int) which once understood then describes the entire process. Also you can then add any set of digits(they don't have to have any special properties. One could also use basis such as matrices or even functor categories.

@wyattstevens8574 8 күн бұрын

"Magnitude" and "argument" for the numbers relates to the famous re^it= cos(t)+i*sin(t). Argument is the value of t, magnitude the value of r.

@Dent42 5 ай бұрын

“We can’t have a complex amount of digits-that doesn’t really make any sense” Someone hasn’t been paying attention in class. The problem you have is not with math, but with its notation

@neologicalgamer3437 5 ай бұрын

Where did 0.69 come from 💀

@DavyCDiamondback 6 ай бұрын

Do rational numbers in base -1+I map to all complex rationals?

@TheGrayCuber 6 ай бұрын

yes!

@DavyCDiamondback 6 ай бұрын

@@TheGrayCuber Now... Prove it for the reals 🤪

@TheGrayCuber 6 ай бұрын

I believe if you dont just use integers but also allow points after a decimal, this base will give you all complex numbers

@0ans4ar-mu 6 ай бұрын

is there a way to get the sandbox with the sliders being smooth instead of having to load each time? itd be interesting to mentally map the space the two variables sweep out

@TheGrayCuber 6 ай бұрын

Unfortunately I don't think that would be easy to do with Tableau, it woudl need to be made using a different software

@nadn1399 8 ай бұрын

This is the most beautiful gift ive ever been given. Are there any resources/literature you'd recommend to read?

@TheGrayCuber 8 ай бұрын

Elementary Theory of Numbers by LeVeque is pretty good! It covers continued fractions, complex integers, and goes into the idea of complex primes which is really cool. Richard Borcherds has a lot of lectures in various topics: www.youtube.com/@richarde.borcherds7998 Michael Penn has videos on some fun topics that walk through calculations kzbin.info/www/bejne/jZeug5KDjtqSi6Msi=znpauTW-0se2uwgr