How to Count in Fractional and Irrational Bases

Let me show you the rare abilities of a bunch of fractional and irrational number bases! Also, I just started a Patreon like some of you requested. Check it out to help support these videos and get cool rewards: / comboclass
Combo Class links:
Bonus Channel with more videos: / @domotro
Discord: / discord
Reddit: / comboclass
Patreon: / comboclass
Note: these irregular bases still follow the rule that they are named after what number the places in a number represent powers of, but they sometimes take more characters for digits than the typical "having a character for everything from 0 to 1 less than the base number" that integer bases require. In each case here, I was using the minimum amount of characters that allows each base to work functionally (and be able to express any integer in a finite amount of digits).
Some bases mentioned in this video (all are types of "positional numeral systems") include: base 10, base 2, base pi, base e, base 1/2, base 3/2, base square-root-2, base square-root-10, base cube-root-10, and base phi (golden ratio).
Disclaimer: Do not copy any actions you see in this video, apart from the mathematical knowledge. This is for educational purposes.

[@ComboClass]

Okay, I just started a Patreon like some of you requested. Please consider checking it out to help support these videos and get cool rewards: www.patreon.com/comboclass

[@user-pr6ed3ri2k]

41st liker

[@chanthachalmers8798]

Can you talk about base 1,-1 and 0?

[@CudaWudaShuda365]

Hey how could I make an algorithm to do this in python?

[@ComboClass]

Note: since some people are commenting "...but a base b should always use b symbols", nope you just learned that from integer-numbered bases. For these non-integer bases, that won't always be enough, so the goal here is to use the minimum amount of symbols that still lets you represent any rational number in the base, which is the amount of symbols I used for each of these bases.

[@michelcarlini]

But is there a general rule for that?

[@ComboClass]

​@@michelcarlini Some types have their own specific rules (although by "rules" it's more like suggestions for which the ideal set of symbols is that has all the properties you want without having unnecessary symbols). Like for an integer n, base square-root(n) will need n digits (0 through n-1). Other types have their own "rules", which may come up in future episodes. Base 2i (which will be in an episode soon) needs 4 symbols to fully function

[@deltalima6703]

Thank you for reading the comments sir. Great videos.

[@michaelwoodhams7866]

Base 3/2 using digits just 0 and 1 is doable. However, it means that integers (other than 0 or 1) end up needing infinite digits to express, e.g. in base 3/2 with digits 0 and 1, the number 2 is slightly more than = 10.0100000100 (base 3/2). So mathematically possible, but too painful in practice.

[@supereyes82]

I already tried seeing what base 3/2 would look like long before this video did, and I used 2 digits. I got these results: 1/4 in base 1.5 would be 0.0001 repeating, 1/2 in base 1.5 would be 0.01 repeating, and 3/4 would be 0.100000 repeating. 2 would be 1.01 repeating, 3 would be 10.100000, etc. Also the 2 in base 1.5 is TECHNICALLY unnecessary, as you don't need to use a 2 to represent the number we call 2, but it would take a repeating fraction, and would be a tad more complex. The same number in fractional bases can have 2 different representations, and numbers that look less than other numbers can be greater. For example, 10/9 can be represented in base 1.5 as 0.11, and 10/9 is greater than 1. This says the base 1.5 number 0.11 is greater than 1, which to our integer-base-trained brains, looks off. However, in these bases inbetween 1 and 2, this is normal. I kind of did what this video shows, but I didn't use the digit 2, which gave me repeating decimal fraction results. 1/3 in base 1.5 is also a repeating fraction. The dice analogy for my version would have large foam dice and would work like this: Put all the dice on the one's place. Then, for each 1.5 dice, slice one dice in 2, toss one half, leave the other half, and move the other one. Repeat this until you have less than 1.5 foam dice in each place. If you have any fractions, then try to do the opposite to the fraction.(i.e. do this to to all of 0.5, but to only to 0.25 of 1.25.) Take the fraction, then add half of that faction onto it and move it back. Repeat this until there's enough digits that are only one or zero for you, or you run into a pattern that allows you to make a repeating fraction.

[@howdy832]

Taking a hint from this video, you should look at base sqrt(2I). Knuth uses it to encode the folds of the dragon curve, as they're intimately related. Good stuff!

[@ComboClass]

The dragon curve will definitely be mentioned in the imaginary/complex bases episode which will be a few episodes from now.

[@cheeseburgermonkey7104]

​@@ComboClass Can't wait to see all the other math videos you put out! Keep it up!

[@petevenuti7355]

I've only recently heard of using a matrix as a power. I can't wrap my head around that. Is that going to be in that vid? Or a third?

[@nanamacapagal8342]

Strange that 2i in base (1+i) can be written 2 ways: 100 or 1002.

[@nanamacapagal8342]

@@petevenuti7355 it's best to think of that as not multiplication. e^t is the base exponential with the defining property that it's derivative is itself. the key is to let t increase over time. so if i was at 1 i would be moving at a speed of 1 to the right. if i was at -2 i would be moving at a speed of 2 to the left. e^At is a variant of this, it's derivative is e^At * A, which means you're moving A times your speed. if I picked A = 2 and I was at 1, I would be moving 2 to the right. if I picked A = i and I was at 1, I would be moving 1 upward, and because physics this means I wind up moving in a circle (this is why e^i*pi = -1, btw). if I let A be a matrix (and it needs to be a square matrix), my position needs to be recorded as a vector instead of a regular number. 1 is now [[1], [0]]. i is now [[0], [1]]. the matrix will deform the graph in ways complex numbers can't: doing e^[[2, 0], [0, 1/2]]t will squash points vertically closer to 0 but spread them out horizontally. if I was dealing with 1-dimensional vectors the matrix A only needs to be a 1x1 matrix, which is essentially a real number. if I was dealing with 2-dimensional vectors, A needs to be a 2x2 matrix so it can properly map 2d points to 2d points. 3-dimensional vectors need 3x3 matrices and so on.

[@laz001]

My word - this is the *clearest* way i have ever seen this concept presented. Well done

[@kittyn5222]

Correction my 14 words

[@DavidSartor0]

@@kittyn5222 Haha.

[@jaypaans3471]

You are golden. the overflow property of phi (11=100) is incredible. In "normal" bases that would never happen They'd be like "thats cheating!"

[@Anonymous-df8it]

I wonder if you can go the other way for compactness

[@Jay1024]

Binary technically has 2=10; just like how phinary can't have a 11 as it can be replaced with a 100, binary can't have a 2 as it can be replaced with a 10. It isn't a perfect comparison, though they are similar in that sense.

[@lexinwonderland5741]

oh man, like i've said on previous videos your delivery is fantastic, your enthusiasm is infectious, your style is what i've specifically aspired to for years in my own teaching, i gotta say just A++ and I am so glad you're here in our community sharing your content! Even if it's things I "already knew", you deliver it in such an entertaining way that I am happy to "re-learn" from your videos. Please please keep it up!!

[@thangnguyenvan2788]

school in ohio

[@atanvardecunambiel8917]

1:15 pinary 2:59 vötbinary 6:46 trivötbinary 10:05 dynabinary (cube root bases could be tryna-, fourth root bases tetryna-, etc. \-sna- at the end of prefixes for prime numbers greater than DEC17 becomes \-syna- for roots) 12:15 dynadecimal 13:37 phinary my beloved

[@fgvcosmic6752]

Are these named by Jan Misalis conventions

[@iantaakalla8180]

Yes but dyna- and \-syna- are not in his system

[@atanvardecunambiel8917]

@@iantaakalla8180 The Misalian system doesn’t have a root for, well, root bases, so I invented one. Should I ask the jan themself on Tumblr about that as well as imaginary bases?

[@Anonymous-df8it]

@@atanvardecunambiel8917 He won't be able to reply as he's @ut!$t!c

[@asheep7797]

Is the vöt from "Vötgil"?

[@MCGeorgeMallory]

My favorite base is phi^2, all integers are properly represented as palindromes, with a phiphicimal point just after the zeros place. Like how there are no 11s in base phi, 21.2 = 100.01, 211.12 = 1000.001, and so on. 1, 2, 10.1, 11.1, 12.1, 20.2, 100.01, 101.01 are the integers 1 through 8 as written in base phiphi.

[@santerisatama5409]

Wow! Thanks! Hmm. Some connection with the 3-periodic palindromicity of the continued fraction of e seems apparent: 2; 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8, 1 etc.

[@rkirk]

Perfect combination of a well-structured lecture with just the right sort of chaotic energy and tons of enthusiasm. This channel is fantastic and I just love the vibe. I'm 33 and watching this takes me back to being a little kid watching Bill Nye and Beakman's World.

[@steelegagnon5273]

i don’t know what demotro is studying right now, but i love how it always feels like we’re learning with them

[@JerusalemStrayCat]

I just watched a cool video about phinary yesterday! Apparently it's useful for encoding Fibonacci-related things. Also, it never occurred to me that one could write in base 3/2 using digits 0, 1, and 2 - I thought it was necessary to have only symbols for numbers less than the base. It certainly makes writing integers in base 3/2 a lot easier than if one just used 0 and 1.

[@ComboClass]

There would be integers you couldn’t make with just 0 and 1. In these weird bases you sometimes need more characters than the typical “rules” so I was using the minimum amount of characters necessary in each base for it to be fully functional

[@doggfite]

@@ComboClass you know, I just made a spreadsheet that calculates base 10 numbers into other bases and I messed around with base 3/2 also, but I only used digits less than the base in all cases. I could be wrong, but it appears to me you can represent all integers, just not rationally. Like 2 and 5 just have infinitely many decimal places, wouldn't they? I'm genuinely curious about this because it felt "obvious" that every fractional/irrational base should use the rounded down number of digits to represent, but I also spent too long trying to wrap my head around a single digit that would represent (in the case of 3/2) 1.5 lol

[@LouisOnAir]

Yeah, I tried writing 2 in base 3/2 using only 0 and 1 once and it did not work out

[@michaelwoodhams7866]

@@LouisOnAir It can be done, just not with a finite representation (like 1/3 has no finite representation in base 10.) 26027/13122 (base 10) = 10.0100000100 (base 3/2) < 2 (base 10) < 10.0100000101 (base 3/2) = 236291/118098 (base 10), so 2 in base 3/2 starts 10.0100000100... I found that by trial and error, I haven't figured out the rule for which digits are 0 or 1. (The 'no finite representation for integers' problem is the same as base pi or base e, but for nice fractional bases like 3/2 I expect there to be a simple rule for what the infinite representation is.)

[@Anonymous-df8it]

@@ComboClass You can though. It's just an infinite recurring expansion

[@DerMathematicker]

10:00 Base a/b: move a, throw a-b out.

[@nbboxhead3866]

You really need to see my equation for digits in bases if you haven't yet, Combo class, it's pretty good! Turns out, for any X, Y, and Z, digit X of Y in base Z is equal to floor((y mod z^x) / z^x-1). It's a lengthy(ish) formula, but could be very useful for future videos. I'm ready to explain it, and I really want you to see this.

[@ComboClass]

Sounds cool, feel free to email me at the contact link on my “about page”

[@witherschat]

based on this, how many digits does base Pi have?

[@АндрейФейгельман-ш4т]

@@witherschat Well, this formula cannot be greater than Z for any X and Y, so 4 (0,1,2,3)

[@АндрейФейгельман-ш4т]

Does it work for non-integer Z? If i plug in x=-5, y=3, z=√(2) then it returns 1

[@sv_an]

@@АндрейФейгельман-ш4т русскоговорящий😳

[@BryndanMeyerholtTheRealDeal]

Did you know: 1) Base e is technically the most efficient number base. 2) Base 0 is nonsense as 0⁰ is undefined. 3) Complex number bases can produce mathematically beautiful patterns.

[@Anonymous-df8it]

Base e is the optimum size. We've been using a number base that's way too big!

[@marcushendriksen8415]

Oh my god dude, your style is just too perfect! Really sells the "crazy mathematician steps outside his ivory tower for a breath of fresh air and attempts to enlighten an unsuspecting villager", lmao

[@servvo]

i genuinely had my mouth stuck upen for like 2 minutes when suddenly you wrote two using the golden ratio and then 3???? and then it kept going??? your videos are always so incredible, keep em coming pleeeaassee

[@baconheadhair6938]

yyyyyttttttttt you lil guy

[@thepeanuts55]

we need to use threeven more in our daily lives, such a funny but weirdly useful word

[@veronicawilliamson4526]

What’s threeven again?

[@duccline]

​@@veronicawilliamson4526divisible by 3

[@peridotisthegoat96]

​@@veronicawilliamson4526even but with threes instead of two

[@troyDM9330]

That demonstration of base 3/2 with the dice... idk how to say this but I felt that in my throat. I wasn't ready so so quickly understand something that i did not understand 10 seconds ago and so elegantly too. Thanks my guy

[@science_gang]

Your growth in subscribers is truly impressive! Keep up the good work!

[@BillyBob-wh4sq]

Wow you blew my mind when you explained how to make integers in base phi! I'm both grateful and impressed at the many perfectly understandable yet really neat math concepts that you present in your videos.

[@RigoVids]

I’m amazed at how good the content on this channel is. He comes off kinda like a klutz on camera but his knowledge and explainations are top notch for these esoteric and generally useless questions.

[@firaswaffle]

HE need to comedown he is clearly going insane

[@servvo]

HE need to keep going up he is clearly goin goofy goofy wacka mode

[@lexinwonderland5741]

He needs to go deeper. Idk if that means more adderall or less adderall but the man is a prophet and has beauty to share with us

[@Gunbudder]

did you read my mind??? i've been thinking about counting in irrational bases and what that might mean. how is this channel so good?

[@lexinwonderland5741]

Bonus points question: how do you write sqrt(2) in base cbrt(2)...? what about writing sqrt(2) in base sqrt(3) or vice versa?

[@pahulpreet-singh]

is base sqrt(2) we can write cbrt(2) as cbrt(100)_# I think. Similarly sqrt(1000)_#, or 10*sqrt(10)_# to write sqrt(2) in base cbrt(2). The representations are irrational since 2,3 are co prime. (the numbers marked with # are representations in special bases and others are regular base representations)

[@a2g555]

As always a w combo class video from the two legends domotro and carlo, i really enjoyed this one so i wanted to thank y'all. I'm really excited to see what topics will be delved in the next couple of videos(I am totally not biased and or looking forward to the 4 dimension video) ((I totally am)). Love y'all ❤️

[@mpalin11]

Damn. I'm impressed by how you found that "take three, throw one"-game. Cool!

[@minamcvinnie4629]

I've been waiting for you to discuss different bases more in depth. Thank you D0m0tr0, today's class is great. 😁

[@tankfire20]

I love this class so much. It actually makes me happy when he uploads.

[@winteringgoose]

You madlad, you absolute chad, you threw 0.999...=1 into this and I don't even see anyone arguing in the comments. You've done it!!

[@Randomizer903]

The glare on the lens at 16:00 made it seem like you were enlightening us with transcendental knowledge.

[@Vearru]

I know it doesn’t make the most sense to write numbers in base pi or e but I am curious what the most reasonable selections of symbols would be, or if perhaps it would be better to use noninteger powers of the numbers for the digits to change what it means for it to be a base.

[@offbeat4772]

Base e has a surprising amount of value in computer science, as it is the most "efficient" base for storage size. Usually, in order to actually be represented physically in experimental processors it is rounded to 3, which is still close enough to gain an advantage over just using binary. Still an interesting tidbit though, I think.

[@Lugmillord]

Interesting. I had a section in my master's thesis (actually computer science and not math) where I also talked about real-valued bases (>= 2). But that was in the context of a completely different number representation, where I used intervals on the number line instead of digits. It was fun, but probably pretty useless.

[@Anonymous-df8it]

???

[@anthonym2499]

i would say that is what he is doing here. sqrt(2) is not a base, it is a standardized unit of length. And we already have those; if I want to represent 1 yard by using feet; then a write 1 as 3. If I want to write 1 foot as inches, then i can write 1 as 12. The metric system is just better at defining less arbitrary smaller units. I could build my house using handspans as my standard unit of measurement .... but then someone can come along and "convert it" into footballs. An arbitrary unit metric is arbitrary. The only nonarbitrary unit metric is in a discrete metric space; 1. But apparently thats called hyperelevens.

[@Anonymous-df8it]

@@anthonym2499 ???

[@Lugmillord]

@@anthonym2499 that's not quite what I meant. You still used discrete, whole numbers in your representation. I meant that both the base and the "digits" can be real-valued. Written out, a number could be something like 2.57 45.2 0.8 - that is supposed to be one number with three real-valued "digits". Of course it looks awful in written form. Three bars of different lengths stuck together are a bit more readable. This also requires a canonical representation or else each number has an infinite amount of ways to write it down. It was a weird topic. Kinda fun to mess round with, especially since I haven't seen anyone do that so far.

[@anthonym2499]

@@Lugmillord In the current video; I cant help but to think hes "invented" the polynomial. c0 x^0 + c1 x^1 + c2 x^2 + ... = A if cn is a real number, instead of restricting it to an integer; what advantage or goal would that be used for in computer science? Does it speed up a computational process?

[@the17thvoyager89]

I think at about 2:30 was when my mind started folding in on itself, and I love it

[@CZghost]

- How complicated counting system do you want? - Yes.

[@matthewhafner962]

Threeven is not a word I knew I needed, but now I need such a word for *every* number.

[@santtusavola]

The golden ratio base was epic! Great video.

[@SatisfyingWhirlpools]

9:00 WOW! I'd played that "game" with integer number bases before, and I hadn't thought to use it for 3/2 base. It was a beautiful aha moment

[@imcwaszec937]

Salut! OK one more step 😄. Do you know why base V(2) has 2 digits and base 2i four? V(.) stands for "square root". This is norm of algebraic number. Look at this formula Z[X]/w(X), where w(X)=x^2+ax+b. Here we have ring of polynomials with integer coefficients divided by a principal ideal generated by w(X). Some of polynomials have special form with 0

[@bennekin]

I DIDNT EVEN KNOW THAT WAS A THING BUT IVE BEEN OBSESSED WITH NUMBER BASES LATELY OMG THANK YOUUU I just assumed it was impossible until I looked at a base conversion formula and seeing that having a natural number was never specified it got me thinking… now I’m here :)

[@419]

I really love your content. I wish schools would inspire people to be more creative with math because I believe creativity and convention are both equally important, but they never teach the beautiful creative side of math. There's an infinite amount of fun puzzles to solve if you just know how to look for them.

[@TheKingOfToast]

Something extremely cool; similar to how base 1/2 is just a "flipped" representation of binary, base phi is a kinda flipped(but weird) representation of base Fibbonaci in that all numbers can be represented in a binary representation without ever having an 11.

[@radostin04wastaken]

this man is an international treasure

[@Scrambleking]

I love how these videos are the equivalent of a scientist who continuesly becomes more unhinged as they experiment

[@kioarthurdane]

You earned my sub by promising to cover base imaginary numbers! Glad to have found you!

[@dovesr0478]

Your channel blowing up is imminent. I noticed your sub count has almost doubled in just the past week. Just a matter of time until you're at 100K. Great content man, and well deserved success.

[@EvilSandwich]

I'm not going to lie. Base Root 10 actually sounds hella useful for logarithmic scales and anything involving exponentials. You can even stretch it out more with 4th root letting you get evenly inbetween factors of root 10s. Or Root 8, 16... all the way to any Root (2 to the n) base. Depending on how stretched out you need it. I know the golden ratio is your personal favorite. But in terms of sheer real life applications and practical utility, root 10, root 2 and root 16 (or just base 4) bases show huge potential. Hell, if we want to approximate Pi in Base (2 to the n) root 10, we can just say that Pi roughly equals 10 in Base Root 10. Or 1000 in Base 4th Root 10. 10 in Base Root 10 is a LOT closer to the true value of Pi than 3 in Base 10.

[@GameJam230]

What gets REALLY interesting is multi-base systems, such as how we measure time. One more interesting example I've worked with before is what I called "Base-Primorial", where each digit has a place value of consecutive primes. So the rightmost digit is in base 2, then going to the left we have base 3, 5, 7, 11, etc. What ends up happening is that all primorials land on a 1 followed by an amount of 0s. And, much like how all primes are 1 above or below a multiple of 6, (or end in the digits 01 or 21 in this base), it allows you to analyze further divisions of primes for larger primorials than 6. In a way, it's a numeric visualization of Dirichlet's Theorem of Arithmetic Progressions, but I learned about that LONG after making this system.

[@packered]

This was great, would love to see a video exploring how these bases do other kinds of mathematical operations. Addition, multiplication, etc.

[@daniielah.7569]

This is truly the Explosions&Fire of math

[@enpeacemusic192]

While I already know most of the subjects you cover in your videos, you always teach it at a way that is refreshing, Big up!

[@jellyfish0311]

Came here through recommends. This is delightful, your explanations are very clear. Thank you so much for this video.

[@jimmygarza8896]

The golden ratio base segment was amazing, but personally, I still prefer to think of the representation as a sum of Fibonacci numbers as base phi.

[@djadj_]

base phi sounds like it's the least bijective number system, i love it.

[@Anonymous-df8it]

Introducing bijective base phi. Digits are (phi-1, phi)

[@ra1nman_mashups]

This video is awesome. Now I’m intrigued what imaginary and complex bases would be like

[@ComboClass]

Those will show up in an episode pretty soon. Next time I dedicate an episode to weird numerical bases it's going to be about base 2i and others like that

[@gheffz]

Love it... and love where you imagination runs!!!! Someone who definitely thinks outside the square... I'd even say lives outside the box!!! What about various complex bases? Oh, you touched on it at the end!!! What about Base 0? Base 1? I am tired so probably they don't exist!

[@VeteranVandal]

Been a while since someone taught me a thing I never heard about. This was a very deserved like.

[@ericgolightly8450]

That's pretty dope. I've always wondered how these would work.

[@WhattheHectogon]

What an awesome video! I’m currently editing my first video in quite a while and it’s about a “hypereleven” fact that you didn’t mention in your video! Also, I’m curious, what mic and camera do you use for your videos? Are happy with those products, or are you looking to upgrade?

[@ComboClass]

Cool! The mic/camera can vary but often I just use an iPhone camera and mic which isn’t necessarily my ideal situation and I’m looking to upgrade soon when I have the funds for it

[@WhattheHectogon]

@@ComboClass Thanks for the quick reply! Keep up the wonderful, wacky content :)

[@Anonymous-df8it]

@@ComboClass What do the base-phi hyperelevens look like when simplified?

[@poorman-trending]

Dope as shoit! Can you get a 2nd hand washing machine though? You could use it as a 3D whiteboard while you’re washing that lab coat. That would be ultimate DOPE AS SHOIT!

[@orterves]

This is a fantastic series, keep it up :)

[@ChongFrisbee]

I paused the episode in the middle to imagine what base 2i would look like. Glad I didn't decide to comment on it until the end of the video, haha

[@morgangraley1049]

8:42 That looks just like a 2

[@bongo50_]

Brilliant video. One interesting I noticed with your table showing the base phi representation of the integers was that, for prime numbers, the number of 0s (excluding the infinite leading and trailing 0s) in the representation was equal to the number. For example, 3 in base phi is 100.01 which has 3 0s. It'd be interesting to see if this works for all primes. Edit: doesn’t seem to work for 11 (10101.0101). Also, carrying doesn’t work as normal when performing addition.

[@Anonymous-df8it]

How does carrying work?

[@bongo50_]

@@Anonymous-df8it I didn’t look any further into how carrying works other than that it doesn’t work like normal. I specifically said in my original comment that I’m excluding leading and trailing 0s. You could use them to make any number have any amount of 0s.

[@trueriver1950]

@@Anonymous-df8it he said he's ignoring leading and trailing zeroes

[@NickiRusin]

Now that's what I call based.

[@thegamesninja3119]

Goldbach's Conjecture, the insane part of me in hubris thinks I can prove, sees value in this.

[@princessLilytea]

Quality maths, never thought about non-integer base systems. surpisingly interesting

[@SatisfyingWhirlpools]

Base Phi is my new favorite thing

[@myownlittlworld9427]

He’s like Bill Nye, the math-clock guy.

[@anthonywestbrook2155]

This answers the question I've been wondering about for years, but whenever I looked them up the answers felt completely unapproachable! Thank You! Is there another video describing ways in which these bases end up being useful? I'm sure they are, I just trust that Combo Class is where I'll grok it fastest.

[@cancerguy5435]

I cannot stop myself from giggling over the constant "ratio" in the base-ф segment. Internet has taken its toll on me.

[@flanjunk]

Interesting. As a beginner I would appreciate some explanation as to how many digits you use for the different bases.

[@TalsBadKidney]

Nice work, Domo-kun

[@Nachthorn2ft]

Thank you for the video. By the the way, in germany the name from the late mid age about the number 1.5 survived since today. Its the single number name today that name a fraction. We called the number 3/2 or 1.5 between 1 and 2:" "anderthalb" (which comes from "(the) other half").

[@Qreator06]

A Blessing From The Lord!

[@lexinwonderland5741]

(sorry for spam, commenting while i watch) HOLY WOW, YOU TURNED FRACTIONAL BASES INTO A BOARD GAME??!!! that's even better than the normal animal population example!!!!! Absolutely brilliant example, that'd be a great mechanic for an actual board game too! Fantastic!

[@antemajetic2394]

The whole 11 -> 100 thing in the base golden ratio makes perfect sense since that is exactly how the Fibonacci numbers work.

[@iamdigory]

You could generalize the root bases to a continuous place value system, a system where you can write numbers in between the places

[@clonefighter1996]

5:33 So in theory, writing an infinite number of 1s in base 1/2 will represent 2-epsilon. -(That's "2 minus epsilon", if I were multiplying, I'd write 2epsilon)-

[@santerisatama5409]

Love the production. Very crisp math presented in highly informative and entertaining manner in the dystopian environment of backyard trash and bloody white coat of a psycho scientist.

[@snoowbrigade]

This is like if a homeless mathematician is teaching me math at the junkhard

[@yurisich]

In terms of the Fibonacci sequence, two terms in sequence are equivalent to the next. I wasn't surprised to see 11 yielding 100 when considering this, but it could be a coincidence.

[@karlwaugh30]

So because phi^2=phi+1 or more generally phi^(n+1) = phi^n + phi^(n-1) is exactly the property that leads to the link with the Fibonacci sequence. Eg. Because the Fibonacci sequence has its property then the limit of the ratio of terms would have to have the above property , which thus defines phi as well (by converting to a quadratic and solving).

[@Anonymous-df8it]

Can you have a Fibonacci base (one's place, two's place, three's place, five's place, eight's place...)?

[@karlwaugh30]

@@Anonymous-df8it precisely this. You only need to use 1 digit in each place, and if you add the restriction that no two 1s can be adjacent (because they can be shoved into the new position) then it becomes a unique representation of all integers

[@lexinwonderland5741]

Well shit, this is crazy and entrancing and entertaining and genuinely educational! Fantastic work m8! Keep it up!!

[@blacksniperbeats.]

I like your magic words funny man

[@jucom756]

You can represent each base b>1 with only less than b+1 symbols, however that does result in a lot more infinite decimal places needed

[@luckydancz4960]

10:00 to do that in base x/y you just do the same process, you take x dice and keep y amount of dice

[@Frownlandia]

An interesting companion to base phi is Zeckendorf representation, where each position is the next fibonacci number. It also has the property that 11=100, but it's easier to count: 0, 1, 10, 100, 101, 1000, 1001, 1010...

[@AliceYobby]

The thing that made me drop my jaw was learning that Phi - 1/Phi^2 = 2. In a nominally unrelated video. What!!!

[@SimpleBach]

The way he scribbles to mark the numbers on the number line at the beginning ❤️

[@contentishogi]

Oh boy the base 3/2 system should be interesting for thinking about ratios of musical notes

[@KapaMi72]

You are the craziest math guy on KZbin! Keep up the good work👍

[@Redjd2098]

Reciprocals just flip the number line. Neat!

[@karlwaugh30]

Lovely stuff. The phi stuff reminds me of (what I know as) Fibonacci base - where the first place is the number of 1s, the second is 2s, then 3s, 5s, 8s etc. You can count through the integers uniquely with the same "no elevens" property you talked of. Eg: 1,10,100,101,1000,1001,1010,10000,10001, 10010, 10100, 10101, 100000 being the integers 1--13 And this form has applications in coding theory and compression, is relatively easy to do some arithmetic in. Similar "sequence bases" can be used but this is by far the most elegant I've encountered.

[@michaelwoodhams7866]

I invented this once, and called it "fibinary". Then I though, "I bet I wasn't the first", did a web search, and indeed I wasn't. The first interesting thing about fibinary is that number representations are not unique. Anywhere you see "011" you can replace it by "100" (or vice versa) and the result is the same number. E.g. 1001 = 1x5+0x3+0x2+1x1=6 and 0111 = 0x5+1x3+1x2+1x1=6. Although most numbers have multiple representations, there is a unique representation for which there are no adjacent '1's, so we can call this the canonical representation for that number. It isn't too hard to figure out an algorithm for adding in fibinary. Basically we write the two numbers one above the other. We are going to do manipulations which don't affect the sum, but which drive the top number to zero. Once it reaches zero, the bottom number is the sum. Manipulations we can do are: Rule 1: If at a given position top number is 1, bottom number is 0, we can swap this 1 and 0. Rule 2: We can do transformations 011 100 to give us opportunities to apply rule 1 Rule 3: although normally we have only one digit in the '1's column (i.e. from right to left, columns are 1, 2, 3, 5, 8, etc) we may need to add an extra '1s' column (i.e. from right to left, columns are 1,1, 2, 3, 5, 8, etc.) This allows us to sidestep a nasty special case where our strings are alternating 1 and 0, for which no rule 2 modifications are available. (E.g. to add 10101+10101). With a bit of work on how we apply rule 2 (left as an exercise for the student) at each step, we can guarantee that every time we apply rule 2 it lets us next apply rule 1 at least once. Because rule 1 always decreases the value of the top number, and the top number is bounded, we can prove that the addition algorithm will finish in finite time. I haven't figured out how to represent non-integers, nor a multiplication algorithm other than 'translate into a sensible numeric notation, multiply, then translate back'.

[@plav6067]

goated channel

[@crepls]

could you mention the factorial base in one of your next videos? i think it's an interesting enough topic

[@3Cr15w311]

Around 30 years ago I played with factorial base and there was a magazine that had code for it, which I entered and experimented with. Maybe Dr. Dobb's Journal in the early 90s.

[@PBalint817]

Evaluating the factorial, it becomes the same as any other integer base

[@AjCohn]

this video is so cool, you're a fantastic educator

[@ethanolcott3960]

12:13 There is a way to represent this notation without using bases it is like Z[x] Where Z is the integers and x is whatever number you want to represent sqrt(2), 1/2, etc. It doesn't work exactly like bases though as it is just refering to the collection of integers with some x added ex:(1+x) or (584+x).

[@SuperYoonHo]

Thanks sir👌👌👌Also explained at the Great Courses The Power Of Mahematical Visualization

[@yiannchrst]

I love playing with these bases!

[@cstarr3240]

I actually looked into base-phi a couple dozen years ago. There was a website that referred to base-phi digits as "phigets." I thought that was a pretty witty way to describe the base system. And there is a tendency for larger integers to have more ways of being expressed in phigets than smaller integers.

[@kitkat47chrysalis95]

This guy is legit the best