Base 1010 and the Realm of Number Representations

[@ComboClass]

Thanks for watching! Leave a comment if you enjoyed the video, and check the description for cool links and stuff. And if you want to send anything to the new mailbox I mentioned in this video, watch this short video on my Domotro channel for guidelines and the address: kzbin.info/www/bejne/lV7KhKmYq7RrjZY

[@mesplin3]

You might like this sequence: en.m.wikipedia.org/wiki/Euler%27s_totient_function

[@445supermag]

You came very close to something I've been thinking about, which is multiplication rules for other bases. Just like for base 10 where the sum of digits are divisible by 3 or 9 when the number itself is divisible, this works for all base-1. for instance in base 8, the digits of a number divisible by 7 add to 7 (fourteen in base 8 is 16). There are also parallels with the other rules we are familiar with, like a number divisible by the base ends in 0, half the base will end in itself or 0 (like 5 in base 10).

[@ComboClass]

@@mesplin3 I already have episodes planned for Grade -2 involving this sequence. It also was involved (although not explicitly mentioned) in the math behind my earlier episode about "spiky shapes" on clocks :)

[@ComboClass]

@@445supermag Yeah some of this has come up in my earlier episode about the magic powers of the number 9 and my episode about why the word threeven should be in the dictionary. It also will come up more in future episodes :)

[@mesplin3]

@@ComboClass ah. I must have missed that one.

[@seedmole]

I've always loved the fact that "Base 10" is either redundant or ambiguous, depending on perspective.

[@KeoniPhoenix]

Spoken like a true Math Jedi.

[@NeatNit]

I've had an idea *opposite* of the one offered at 6:30 in the video: a base should be written using exactly one digit, using any base higher that itself. So base ten would be written base A, dozenal is base C, hexadecimal is base G, and so on. No room for ambiguity!

[@lugyd1xdone195]

​@@NeatNit Ive had the same idea but how do you want to write base 3718 then? Do we invent some new symbols? Keep adding apostrophes? That wont be readable at all.

[@SgtSupaman]

@@lugyd1xdone195 , just stop at the most easily identifiable sequence of symbols. 0-9 + A-Z + a-z will let you name everything in base sixty two. So, base three thousand seven hundred and eighteen would be written as base-xy. Numbers go on forever, so no system can maintain a single digit forever, but any base people would actually use would be single digit. If anyone goes beyond that (like trying to use a base of multiple thousands for some reason), it still reduces the number of digits used pretty well without any additional strain on memory since it only uses what people already know. However, if people were open to learning something new, there is a great set of symbols that exist out there that only require you to learn nine parts, but allows you to write in base ten thousand (or base-2bI, according to my aforementioned sixty two suggestion). So I would say all base names should be written via that system, but I can't type those symbols, so that won't work at the moment.

[@lugyd1xdone195]

@@SgtSupaman while that's cool you never have infinite symbols. Or alternatively from a system accounting for this you will have a very detailed symbol that you can easily screw up and takes a lot of space. The whole number writing system is also an convention, where in arabic numbers you count any repeating pattern as a separate symbol. You could write with the roman numbers if you want and in some cases it would be beneficial. You could make a system where a fractal pattern expands for higher numbers. You could denote it as a geometrical shape. It'd just take a lot of paper and patience. And nobody wants to learn ten thousand symbols. Not for a thing you can do fairly well with ten. You will always have to establish the base of your conventional number system or it will be the wild west with higher number bases. There's no point in making the pool larger if it only confuses people and doesn't solve the issue (denoting every base with one definite symbol). I dont know what I would want to use base three thousand seven hundred and eighteen for, but if I do I'd want to denote it clearly so that someone not well versed with the system can understand what I mean.

[@pythx]

Five seconds in a clock already falls. Purely impressive.

[@stevenlubick2689]

Yes!!

[@mathguy37]

classic combo class

[@MichaelKah4712]

and a squirrel after 25 seconds

[@RedLuigi235]

jan misali has an incredible video on the "base 10" question called 'a base-neutral system for naming numbering systems,' highly recommended.

[@janKepi]

was also gonna comment this. that video is great, i've watched it at least 3 times

[@MrAlRats]

I think we should just use a non-positional numeral system to represent numbers as a subscript to denote which base we are referring to. For example, the number ten can be represented as 1111111111 and this can be used as a subscript to indicate that a number is being represented in base-ten positional numeral system. So twelve would be written 12₁₁₁₁₁₁₁₁₁₁ in base-ten, which is 1100₁₁ in base-two.

[@NoOffenseAnimation]

proud to have been part of this journey, excited to graduate from grade -1

[@aashsyed1277]

Actually his grade -1 is actually equivalent to normal number theory.

[@NoOffenseAnimation]

@@aashsyed1277 ... So? You know I don't mean actually graduating right?

[@neiladrian]

That lab coat will soon be standing on its own. 🤣 Nice content as always, Domotro!

[@justinwatson1510]

Your realization about primes ending in 0 only in their own base was quite clever.

[@aelamf]

This person is the pure embodiment of a great mad scientist

[@Amipotsophspond]

20 is the same as knight's movement in chess. I wonder if you could put coordinates to this matrix of numbers and put functions to those coordinates, so like a knight's movement would be like (xa+1,ya-2)=(xb,yb) or y-2=x+1 I don't remember how to do it. and like maybe the lines/functions are useful in some way or something. like maybe you can describe math operations like cubes and squares by the function they make on this matrix of numbers.

[@ComboClass]

I actually filmed a scene that's somewhat similar to the concept you're describing (graph-like "slopes" that the appearances of certain digit strings have) but I ended up cutting it from the episode and saving it for a future episode where I'll be able to show a clearer visualization of it

[@SpencerTwiddy]

@@ComboClass Sequence A237048 if the link disappears

[@whistling.citrus]

Thank you for the mailbox I can't wait to send you something and see all the wonders that appear! Also fascinating discussion, I bet you are naming all kinds of new concepts!

[@Qreator06]

Only 3 more classes... Time flies by so fast when you're having fun

[@ComboClass]

Grade -1 will have "100" (written in Base Six) episodes. Then Grade -2 will come shortly after

[@plushloler]

I love making as-close-to-a-square representations of numbers, this is the first time seeing someone talk about it, other than myself

[@guyedwards22]

This video made me realize a neat fact about perfect powers. When wondering about when a number can be written in the form a^b with a,b>1, those are exactly the numbers whose powers on their prime factors do not have any coprime pairs. If there are no coprime pairs, then there must exist some number n>1 which is a factor of every power. You can factor n out and represent the number as (p1^x1*p2^x2*p3^x3......pm^xm)^n, which suits the a^b format

[@kmatlockii]

Super clear explanation of how different bases work and relate to one another!

[@bakerrboyywoo]

That squirrel is trying to ace his next math paper, and Im 100% sure he's in the right place Thanks for the video your love for the squirrel made it

[@vanevarnel7131]

Nice birds sounds good choice Domotro.

[@morganjonasson2947]

I dont know if anyone else did this, but i used base 5 at my work in norway to count invoices and making sure no was missing. I counted it in norwegian as usual up to five: en(1), to(2),tre(3),fire(4),fem(5). After that the base 5 bullshit started: femoen,femoto,femotre,femofire,tofem,tofemoen,tofemoto,tofemotre,...,...firefemofire,TJUFEM (100 base 5 or 25 base 10). Totjufem (50), tretjufem(75), firetjufem (100), and then, we have the HUNNERTJUFEM (125, or 1000 base 5). Why base 5?at first, i grouped the invoices in tenths, but i noticed that it was hard to get groups containing exactly 10. Sometimes i got 11, sometimes 9, causing lots of recounting and confusion. However, five was way easier to make exactly correct groups as i can place 4 letters between each finger on one hand and the fifth one would make a group. Also, there was this great pattern where 4 in any position would make great numbers. Ex: firefem (40b5) makes 20, firetjufem (400b5) makes 100, and firehunnertjufem(400) makes 500. The best part, was how i could place it on the table. Lets say you have 86 invoices. That becomes tretjufemotofemoen(321b5) and would make 3 groups of 25, 2 groups of 5 and one group of one. Place the three groups of 25 in one row, to groups of 5 in the 2nd row and the group of 1 in the 3rd column. You now have a representation in base 5 of the number 86 fitting a narrow space of 3x3 invoices. If you did the same in base 10 you would just get 1 llong line of 8 groups of 10 and a remainder of 6. The question is then if the table is long enough to fit 8 invoices in a row. Iit sounds complicated, i know.

[@ohmisterjeff]

I see Combo Class in my feed. I click. It really is that simple.

[@codatheseus5060]

I've recently been thinking about the utility of introducing numbers as matrices from the beginning, which would make some stuff easier later on for everyone This video title made me think of this again It would make imaginary numbers easier, higher dimensional maths easier, and things like dual numbers easier too At least initially

[@1234567zeek]

RIGHT ON MAN! Please keep it up because your awesome!

[@Apophlegmatis]

I love the quirk of this channel. Sets whiteboard on folding chair, so we can enjoy plenty of headless instructor time

[@chickenbonelives]

You look like my crazy friend, but you make a lot more sense than he does! So when are these maths gonna make wormholes? Love the channel keep em coming.

[@CallenderMan]

Technically, isn't the leftmost column base 1? Since every number is represented with one single object, which could be seen as a digit. Like, if you used 0's instead of dots then you'd have 1 digit, making it base 1.

[@BryanLu0]

Yes, but it isn't a basic base 1. You actually can't represent any numbers except 0 in a basic base 1

[@Debg91]

Right, although not a regular positional base, but what is known as "bijective base 1" which is basically tally marks or dots in this case

[@AlphaFX-kv4ud]

​@Bryan Lu actually the first digit doesn't need to be zero, if you make it 1 then through standard positional notation you can write any number except 0, which is fine, we went a long time without needing zero anyway

[@landsgevaer]

@@AlphaFX-kv4ud It is kind of different though. If I write 3 in 'unary' as 111, and I ask what goes in the "4s-position". Normally there are leading zeros that we just do not need to write. So 3 is ..00003 (in any base > 3). In your system, you are sort of 'cheating' as it were because those leading zeros are still there implicitly, so you are using two digits under the hood, one of them being the empty character. If you really have only one possible digit x in each position, then there is only the number ..xxxxx you can write. Convention is to choose x=0 so you can only express the number zero. Of course you can claim there isn't any 4s-position for the number 3, but then you are deviating from the base-N system and use a completely different notational convention (like Roman numerals use a completely different convention, for example). That would not be a regular 'base-1'.

[@Tletna]

Thanks for the fun video. I liked how you used the dots in the column where base one would have gone. That was cool. I wonder if there is a pattern where if you add certain numbers but leave them in different bases if the result would be correct in a 3rd base. EG: Two base two is 10, four base three is 11, 10 + 11 = 21, 21 is seven in base three. It seems like just a coincidence, but I wonder if there are patterns to that as well.

[@arrendor]

2+3=5

[@elliotcelliot]

10+11 is only 21 in base ten

[@SgtSupaman]

You can't add two numbers from different bases. Besides, you said that 2 (base-2) plus 4 (base-3) is equal to 7 (base-3) and that is somehow correct to you? 2 + 4 =/= 7 In base 2: 10 + 11 = 101 (which translates in base-10 to 2 + 3 = 5) In base 3: 10 + 11 = 21 (which translates in base-10 to 3 + 4 = 7) There is no coincidence here, just basic math.

[@cheeseburgermonkey7104]

Early to a Combo Class video! Can't wait to see the new and cool thing Dimotro is going to teach us...

[@ericgolightly8450]

I've found these things called Argam numerals, which can be used in large bases. There are 479 of them so far.

[@scali5948]

WOO WOO NEW COMBO CLASS EPISODEEEE

[@areadenial2343]

We could avoid ambiguity in names of bases by using base 36 to name them. So base 9 is followed by base A (decimal), base B, base C (dozenal), base D, base E, base F, base G (hexadecimal), et cetera, until base Z which is followed by base 10 again. So for most commonly-encountered bases, this naming scheme would be ideal.

[@landsgevaer]

If we need to agree on a base to express the base, then just agree to make that ten. Or perhaps binary if it must be "simplest". 36 would be such an artefact: why stop at Z, why not 64 like in the Base64 encoding scheme, or why not use the entire unicode system as digits?

[@lugyd1xdone195]

If we substitude 1s for the dots in the chart isnt that basically basic base 1 representation?

[@arcycatten]

would there be an infinite number of representations of a given number, if you considered non-basic bases?

[@ComboClass]

Theoretically, you could "invent" any type of base / numeral system that you want, so yeah. You'd always need to make some sort of restrictions about which types of bases you are considering/including when you try to make a "realm" of bases like this

[@arcycatten]

indeed, you could theoretically have as many bases as numbers (real or not!)

[@goop9086]

Would it be possible to have a base that is the amount of numbers ending 0s in each base. Like the clolum on you chart that is on the right.

[@guslackner9270]

The smallest base is actually ones only, not binary (maybe we should call it 'monary'). So base ten 1, 2, 3 would be monary 1, 11, 111. This is why numbers in set theory are defined as (+ 1), (+ 1 1), (+ 1 1 1) which is, interestingly, the same as the list data structure in the list processing (LISP) programming language.

[@matthewmorrison2071]

Tbh, I really like the lack of music. More relaxing :)

[@marcelinomoreno4506]

I enjoy Vido was easy to follow the lesson.

[@B3Band]

Domotro on Numberphile when?

[@Hamza-zz3zc]

Very very very interesting 🤔🤔🤔

[@Gamesaucer]

At least up to base 35 you can easily write it with a single character going 0-9 and then A-Z. I don't think there's a widely agreed upon convention of what happens after we run out of letters, though. How would you write base 36 in an unambiguous way without resorting to writing the word for that number? Or any base after that? Just running through normalised Unicode code points, foregoing the ones that are invisible? ...Binary definitely sounds easier, hahah.

[@orsonzedd]

I have a bunch of fire damaged d7s, I should send one

[@Voiding210]

D7s???

[@Jkauppa]

what reps

[@Jkauppa]

rep who, for what, nope

[@Jkauppa]

squirrel is a rat

[@hkayakh]

Squirrel 🐿

[@stevenlubick2689]

More Squirrels 🐿️🐿️🐿️🐿️🐿️🐿️ may live there.🙃🙃

[@litigioussociety4249]

Mice like clocks according to a children's song, and squirrels are rodents, so I guess all the clocks are attracting them.

[@hkayakh]

@@litigioussociety4249 “my clocks bring all the squirrels to the yard”

[@memeteam2016]

Do prime numbers in other bases

[@stealthmaster0]

Matrixes! Oh wow I love bases but like, can you explain me the computer equivalent of base64? Cuz, does it use the alphabet?

[@Ivankarongrafema]

You can also use base 1, where TEN is 1111111111, even less ambiguous

[@OhhCrapGuy]

I'm a little surprised that you suggested using binary to represent the base number, when there's an completely clear and ambiguous way to represent every base canonically: Just already use a base higher than the one you're trying to describe. Base 2 can be represented in base 3 and higher without ambiguity while we still don't know which base I'm writing "2" in, except that we know it's just not in base 2. It's in a higher base, but there's no way to know which one. Sure, we still have to change how we write base A and base C, but we were going to change it to write them as "base 1010" and "base 1100" anyway. Plus this way we can communicate clearly while still leaving the method of communication to be perfectly ambiguous, and that appeals to me. What appeals to me most of all is that we're equivalently defining our system of base description to use an undefined base-infinity, while leaving the symbols for that base to be an implementation detail.

[@ComboClass]

I like the idea, but that would be very dependent on notation, like to describe Base 60 (which some cultures have used in history) we would need to invent a whole new set of symbols and memorize their order, as opposed to always just using 0’s and 1’s. Your way would minimize length of saying “base ___” but would take way more rules to set up and describe (and really wouldn’t be very possible to describe arbitrarily large bases without a ridiculous amount of new symbols). EDIT: and I know that if you used base 60 for example you would need that many symbols anyways, but it’s far more common to talk about base names than to actually need to use those bases. Thanks for sharing the thoughts though!

[@stickfiftyfive]

Cool. Just going to have to watch that one no less than 1100 times(in Base 10 aka Base Two)), or 12 times (in Base Ten). Not confusing at all. 😂

[@BleachWizz]

7:04 - base ten ten. >:) now tbh base A sounds better you write it as the digit the number is in a base higher than itself.

[@love2o9]

I feel like you're not even phased by the clocks falling anymore

[@stevenlubick2689]

No 😂😂🙃🙃

[@JimmyVermeer]

Arguably 16 is a perfect power in 3 ways because it is 2^4 and 4^2 and also 16^1.

[@ComboClass]

When the exponent is just 1, it doesn’t count as a “perfect power”, otherwise every integer would be classified as that

[@aashsyed1277]

You did add a little bit of background music .

[@ComboClass]

I actually didn't (in this episode). There's only the natural background noises like birds and train noises and wind and stuff

[@Dr._Geno]

how has it taken me this long to realise that you've been slowly growing a beard throughout these videos, I should have noticed much sooner but I didn't at all.

[@ComboClass]

A few months ago, I decided to keep growing out my hair and beard until Grade -2…. (Some grades I may get a haircut at the beginning and then let it grow all grade haha)

[@YoutubeBorkedMyOldHandle_why]

The ancients kind of stuck us with base10. It boggles the mind to imagine how much compute power has been spent over the years, doing nothing more than converting between base10 and base16. Too late to change I suppose, but it would have been nice had they got this right. I've long since figured that our primary number system should be one where the base can be represented by 'the base' raised to a 'power of the base.' e.g. base2 which is 2^1 where 1=2^0, base4 which is 2^2 where 2=2^1, base16 which is 2^4 where 4=2^2, or base256 which is 2^8 where 8=2^3. The beauty of this is that 8 binary digits can be represented by exactly 4 base4 digits, 2 base16 digits, or 1 base256 digit. Conversion becomes trivial. (Octal is completely ridiculous of course. It breaks this rule so it's just about as hard to convert as base10.) I like to think that extended ASCII is 'sort of' like base256. This is especially true when we use the characters like numbers for things like passwords etc. Taking this up a notch, we could do the same with base3, base27, base19683, but of course these numbers get big really fast.

[@vincehomoki1612]

If you didn't understand, he meant bases which can be represented as 2^2^x. (in the bottom that's done with 3 instead of 2)

[@dddono]

For some reason everything comes back to Pascal’s triangle. Maybe I’ll make a video of a pattern I found

[@kingkiller1451]

17:45 I like that they can have it an arbitrary number of times. 256 = 16², 4⁴, and 2⁸ then you can go again and say 65536 = 256², 16⁴, 4⁸, and 2¹⁶

[@AloysiusScrimshaw]

Indeed… 🦈👁️

[@calebclark9114]

That thumbnail though 😂😂😂😂😂

[@guilhermepanarellirangel6663]

Combo x2 🤯

[@krazykilper]

Sorry I was late to class!

[@DanDart]

In bases > twelve, we would need to alter English again, because thirteen or thirty prefer ten as the base of the word. Base score is probably the exception. It's also insane that "hundred" used to mean what we now mean as hundred-and-twenty. Long hundred? Bah!

[@defenestrated23]

The left column is base 1

[@stefandempf818]

The only thing i dont get is why not more people join the livestreams

[@gnikdroy]

livestreams?

[@MipCat]

Combo "Class"

[@BenOrBreak]

birds so calming SQUIRREL

[@vivalameme6341]

how in the world did you make learning fun

[@piepiequals]

I had an Idea about that if you take base ten blocks and change them to some thing like base 7 blocks or something like that. another idea is that base ten blocks normally start as a 0d dot called a unit and then turn into a 1d line called a rod then a 2D square and so on, but what if you go to 4d and so on?

[@islandwoodfired]

ben, is that you?

[@ToanPham-wr7xe]

😮

[@theBATgoesUPoh]

Funnily enough the dot representation of numbers is base 1

[@mircoi.3256]

you should have named the first column "Base 1" and made zeroes instead of balls... then the number in the last column would contain the exact number of divisors, not one less. It is perfect.

[@ComboClass]

A series of 0’s doesn’t accurately describe the numbers though. Like “three” doesn’t make sense to be “000” because that would mean 0+0+0 by the rules any of these other bases follow

[@Voiding210]

25000 now!

[@jonbezeau3124]

Base A

[@louconover767]

There is a base 1 system: Roman numbers

[@xtieburn]

Ahh yes, base ten. Which of course in decimal is base 26543...

[@taru4635]

For most practical applications, base 10 is objectively better than base 10 and you can't prove me wrong.

[@smbs47]

But base 10 is objectively worse than base 10, and try proving me wrong.

[@HomieSeal]

Wait… would 16 be 10,000 in binary? Holy shit it would be

[@CMDRunematti]

somehow i feel youtube is starting to take over scientific publications' place...

[@cubefromblender]

The best base is base-10

[@nayutaito9421]

Funny way to exclude 1 from prime numbers

[@davidrogers8030]

Spo minority opinion, find trashing stuff annoying. Otherwise, great show!

[@J_i_m_]

There are 10 kinds of people: those who understands binary, and those who doesn't.

[@WindsorMason]

And those who realized this is a base 3 joke, and those who realized this is a base 4 joke, and those who realized this is a...

[@Skippery_Mapping]

Math class in ohio