I Found a Simple Pattern That Encodes Different Bases

While I was trying to invent a card game recently, I stumbled into some questions/answers that I realized were describing patterns about other numeral bases! Let me show you about something I call "numberations":
0:00 - Part 0: Introduction
1:48 - Part 1: What Are Numberations?
4:43 - Part 2: Visualizations of Doubling
8:11 - Part 3: Visualizations of Tripling
10:37 - Part 10: Converting to Other Bases
13:15 - Part 11: Finding the Missing Numbers
15:48 - Part -1: Base Ten and Stranger Bases
20:31 - Part -2: Outroduction

(I'll add more applicable links to episodes I've made about bases here soon. If you're curious in the meantime, you can scroll through my Grade -1 and Grade -2 playlists for episodes with "base" in the name. And if you're curious about the livestream where I accidentally lit myself on fire, see the "live" tab on my @Domotro channel.)

Disclaimer: Do NOT copy any dangerous-seeming actions you may see in this video, such as any actions related to fire.

Note: although I did discover/realize all of the things in this episode myself, I'm sure other people have noted similar things throughout the history of mathematics. Since my description/visualization is slightly different than other ones I've encountered, I nicknamed the seeds of these patterns "numberations". I think it's a useful term; do you think we could ever get it into a dictionary?

Special thanks to all of my supporters on Patreon. Supporting the show not only helps me keep improving my content, but also lets me avoid putting any product placements in episodes. Brands (not the company that makes Connect 4 haha) have offered me money for me to advertise them during an episode on this channel but I want to avoid that unless/until necessary, so thank you to all of these supporters who keep our Grade -2 episodes from having any product placements in them!
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Domotro
1442 A Walnut Street, Box # 401
Berkeley, CA 94709

If you want to try to help with Combo Class in some way, or collaborate in some form, reach out at combouniversity(at)gmail(dot)com

In case anybody searches any of these terms to learn about them, some topics in this video include: number theory, whole numbers, integers, base 2 / base two / binary, base 3 / base three / ternary, balanced ternary / balanced base 3, base ten / decimal, other different numeral bases and number systems / representations of numbers, a pattern I nicknamed "numberations", more!

This episode was directed/edited/soundtracked by me (Domotro) and was filmed by Carlo Trappenberg.

[ComboClass]

Do you think I can ever get "numberation" in the dictionary (after I get "threeven" in there)? In any case, thanks for watching! Check the description for more info and timestamps of different chapters.

[shivamchouhan]

You are so underrated!
Even though your style is one of the most unique and still youtube doesn't recommend your videos.
I had to manually search your videos even I subscribed you over a year ago

[CaedmonOS]

You're a math channel and yet. Somehow i'm entirely not surprised that you set yourself on fire

[richardjacobson]

Fun fact: the double-and-add algorithm you described in this video is actually used all the time in cryptography! Specifically, it's used to efficiently compute elliptic curve point multiplication and its cousin, the square-and-multiply algorithm, is used in RSA to efficiently exponentiate numbers modulo another number.

[vcuberx]

for your efficiency question, I would look into "Hamming Weight". Also just a fun fact I discovered looking into this myself: if you have (x*2)+1+1, you can always rewrite it as (x+1)*2, leading to a nice recursive way to find the minimum number of moves.

[neologicalgamer]

I was planning on leaving halfway through the video because I could see the solution, but I'm so glad I stayed because that was so incredibly beautiful, how universal those charts were across all bases

[hkayakh]

0:25 Only Domotro is strong enough to be on fire and not notice it

[claytonhiggins]

Great vid! Reminds me of something I discovered: if you extended the idea of numberations to just functions in general: imagine starting at 1 and being able to multiply by 2 whenever, and being able to subtract 1 then divide by 3 when the result is an odd number. Asking whether or not all positive integers can be reached is equivalent to the collatz conjecture.

[Seedwreck]

This is really cool, I personally have been working on an encoder where ”i” is the index and “r” is the reducer value; it takes a binary value and adds it to the reducer, then we multiply the reducer by “i” + 1
This would allow you to then reverse the operator to get “rv” where “v” is the value (0/1) and “r” being that layer reducement.

Very neat work, very cool to learn about!

[aplcc]

A hidden gem on YouTube, thank you for the videos!

[allthe]

Setting junk on fire really enhances the performance

[Intelligenthumour]

Reminds me of the Collatz Conjecture but in reverse.

[ElusiveEllie]

Found your channel again recently and have been blown away by your base videos, the imaginary number based are my favorites. Stay safe, no more catching yourself on fire!

[alexanderlehner]

Some clever compilers do this when multiplying a variable with a constant, say
x = i * 129 same as x = i << 7 + i
It is often 'cheaper' to bit-shift, add, bit-shift etc. instead of performing the actual multiplication.
This is more or less O(n+m) vs O(n*m).
Anyway, nice thought experiment! I Was puzzled at the beginning, then came the eye opener ;)

[TheChefmike]

I love it. I'm glad I found your channel! Happy Holidays!

[logo]

The numberations remind me of generators from group theory. The tree diagrams remind me of bit shifting binary strings and incrementing them.

[SteinGauslaaStrindhaug]

18:25 That's one of my favourite experiences in my job as a programmer. ...And when watching maths youtubers. I'm not particularly good at maths in practice (except those problems that I can force into being a programming problem, or which I can think of in a completely different way like music or art or whatever) but I do enjoy noticing patterns and understanding proofs even if I'm not very good at making the proofs myself.

[ker]

Damn man, I really want to spend more time offline just thinking about math and stuff and you are a serious inspiration for that❤

[solvexdubstep]

Cool video! I figured it out pretty quickly once I saw that it was related to different bases. If given any number of /2's and at most one consecutive +1, I imagine it's possible to represent any nonnegative real number less than 2.

Also, I wonder how this idea might be connected to the Collatz conjecture? The 3x+1 thing would of course require two different numberations (*3 followed immediately by +1), but it does make me wonder if one could come up with some sort of Collatz conjecture numbering system.

[GamingKing-jopy]

there is a visualization for categories in category theory where you draw "objects" and "arrows" just like in your visualizations. now lets take the category of integers as objects and the arrows corresponding to the numberations +1, -1 and ×3 for each object, and also use the functors P: x → x+1, N: x → x-1, and T: x → 3×x. these functors are just like the numberations we're doing. if we follow 0 after applying these functors, we can see what number it will make.
now here's the real power
take the category of strings containing, lets say '0', '1' and 'T' with the arrows and functors corresponding to concatenating '0', '1' or 'T' instead of the numberations, and we find that they're actually the same.
this would be like finally seeing the same diagram you made, but in balanced ternary.