How to Count in Fractional and Irrational Bases 

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

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

But is there a general rule for that?

​@@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

Thank you for reading the comments sir. Great videos.

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.

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.

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

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

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?

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

@@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.

Correction my 14 words

@@kittyn5222 Haha.

Are these named by Jan Misalis conventions

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

@@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?

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

Is the vöt from "Vötgil"?

I wonder if you can go the other way for compactness

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.

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.

school in ohio

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

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

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

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

@@АндрейФейгельман-ш4т русскоговорящий😳

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

@@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

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

@@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.)

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

yyyyyttttttttt you lil guy

What’s threeven again?

​@@veronicawilliamson4526divisible by 3

​@@veronicawilliamson4526even but with threes instead of two

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)

???

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.

@@anthonym2499 ???

@@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.

@@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?

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

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

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.

How does carrying work?

@@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.

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

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

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

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

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

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

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

Complex numbers of the form (N + M.i) where N and M are integers are known as Gaussian integers and effectively are an integer lattice in the complex plane. See my separate post about base i

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.

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

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).

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

@@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

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'.