How to Count in Fractional and Irrational Bases

Combo Class links:

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

[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.

[howdy]

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!

[atanvardecunambiel]

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

[laz]

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

[jaypaans]

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!"

[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.

[lexinwonderland]

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!!

[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.

[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.

[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

[nbboxhead]

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.

[steelegagnon]

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

[thepeanuts]

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

[science_gang]

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

[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 :)

[BillyBob-whsq]

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.

[minamcvinnie]

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

[imcwaszec]

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<=a<b. Because every polynomial of the form f(X)=g(X)*w(X) is ZERO in Z[X]/w(X), so we can assign "number" [1ab] to zero (-0), and [100]=aB+b, where B is one of the solutions w(X)=0; number of digits is b, and we got nonstandard numeral positional system (I call it C-a-b). There is arithmetics in this system - addition, myltiplication, negation, factorization, integers, rationals, hyperelevewns (repunits)... Addition and multiplication is similar to positional system with negative integer base: carry is representation for integer number b, negation we get using "mask frpm zeroes". For example w(X)=x^2+3X+7 is septenary system for base B=(-3-V(-19))/2 with ZERO having two forms 0 and 137, but 7 is "big digit" and we use it only in arithmetics to mask and reductions. Mu notation is [0/137] for 0, 1=[1/136], 2=[2/135], ... and -1=[136\1] etc. Notation [A/B] means that representation for A is shorter than for B. Carry looks here like +[1240], double carry +[1110], borrow +[136], , "-" is shortcut for [126\1]. Lets do something. Addition OK we will count infinite number of carries like in negative base, so time for "reduction of zero" [...137...]=0 and finally we get [1]. Formally Multiplication works too (leads to a series of additions) but we must do more carries and zeroes reductions (beware of infinite calculation). Negation works like this: [3460] = 35, so -35=[13700+1370-3460], where 13700 is mask for 34 and 1370 mask for 6, then (not formally, with "big digits") Factorization [11610\3460] = because 5 and 7 are not primes here. Check this 😉. This is numeral positional system for a ring in the field Q(V(-19)). One of infinitely many others. Now if we substitute X:=X+n in w(X), n integer, we get systems with 5 digits for n=-1, 11 digits for n=1, 17 digits for n=2, ... etc. Arithmetics in these systems looks different, buf factorization in the same. For every 0<=a<b we have such "tower" of numeral systems, although most of the rings are not UFD. For C-3-7 ring is UFD but not euclidean. For C-3-4 (rings with V(-7)) or C-4-9 (rings with V(-5)) isn't, because in the recent [6/143] = [2/147]*[3/146] = -[11/138]*[13/136] = 6. This is just alternative notation for coming from Dedekind one and copied bklankly from coursebook to coursebook😝Btw every such "tower" of positional system for every ring is like "irdinary tower" for ring Z integers, that is for normal and negative bases. Negative base is also "natural" in such sense that repunits with even index have factorizations R(B;2n)=R(B;2)*R(B;n)*R(B;-n) if n is odd, and use recurence if n is even. For negative base B the same. Of course for nonstardard bases with square root (see above) this is also true.
More general and by analogy we get arithmetics for Galois fields using two reductions (coefficients mod 2 plus some prime ideal) or "double" reduction Z[X, Y]/(2, w(X)), because (2, w(X)) when w(X) is irreducible with coefficient mod 2 is prime ideal in this ring. Much more general we can use "numeral systems" for Z[Y][X], where Z[Y] with degree less than some n are our "digits" and last "digit" is with degree equal n, especially for some well known algebraic curves (elliptic or hyperelliptic curves).

[ag]

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 ❤️