Can any Number be a Base? 

16:27 Small error here. You say "21 + 2i", but it is written "21i + 2".

and before this i didn’t think my number universe could get any bigger…. thanks!

I can see base pi being useful for trig. imagine cos(10)=-1 and sin(10/2)=1, etc. Also sum of reciprocal squares would be 100/(whatever 6 would be)

Interestingly, tally marks or even just counting with your fingers are an example of base 1, and probably the oldest number system we have. Roman numerals were also derived from tally marks, and they could be considered an example of a system with multiple bases, where the auxillary bases are 5 and 10

That's "based"

my favorite type of number system that wasn't brought up here is factoradic, where instead of having one radix that you keep squaring, you take each digit as the next factorial, so each position can range from 0 up to the position number. to give you a feel for the system, here's some numbers counting from 0 to 24: 0, 10, 100, 110, 200, 210, 1000, 1010, 1100, 1110, 1200, 1210, 2000, 2010, 2100, 2110, 2200, 2210, 3000, 3010, 3100, 3110, 3200, 3210, 10000, and so on to go back and forth it's very similar to a normal base; for example to render 5835241010(!) into base 10 you would do: 5*9! + 8*8! + 3*7! + 5*6! + 2*5! + 4*4! + 1*3! + 0*2! + 1*1! + 0*0! =1814400 + 322560 + 15120 + 3600 + 240 + 96 + 6 + 0 + 1 = 2156023

For a number base a/b, you use digits 0..a-1, but that allows numbers to be written many different ways. In fact, you only need 0..floor(a/b). Let's say for your example of 265, converted to base 7/3. You write it as 64366. But you could also write it as 1110020.001... Similar for other non-integer bases.

12:55 But this problem should also happen for some Algebraic numbers. There are Algebraic numbers that you can't write in terms of radicals, for example a solution to some general quintic equation.

There's also base φ, with digits 0 and 1, no two 1s in a row. 2 is represented as 10.01 in this base. You can use base 2-i with the digits 0, 1, i, -i, and -1. Similarly, you can use base 2.5-√-0.75 with digits 0, 1, -.5+√.75, .5+√.75, -.5-√.75, .5-√.75, and -1.

Funnily enough, base 1 has a fun application where you can represent a string of numbers by having the “length” of the number represent a number in some other base, like base 9 for example, using 9 as a separator. This allows you to write any number of numbers in a string in base 1. Fun thought experiment.

This man put so much work effort to show us the beauty of math. I’m highly appreciating your videos dude. I hope u get a good job and good life mate

Mind = blown Respect for explaining such far out concepts in a way that is so easy to follow

I came up with a numbering system that was "like" base-phi in an esoteric programing system. You could represent integers with strings of two commands: '+' to add one and '@' to redo part of the substring. It was like base phi, because it took about n log_phi commands to represent a particular integer n, similar to how it takes n log_b digits in normal base b.

You make videos about topics I really wanted to know, but you can't really find them on the internet. Thank you sooo so much! ^w^

Very well expressed and executed video. I never thought of this before. Thank you.

base 12 is superior we must change to it. It can be divided in half, thirds, fourths. Vs base 10 which can only be divided in half cleanly.

I like base prime. 1 = 1 10 = 2 100 = 3 200 = 6 120 = 210 = 60 = 12 In this base integers can have multiple representations based on the factorization chosen. Also the next prime is easy to find 😂

what a great video, I enjoyed the insights and the production quality - thank you very much

I remember reading several research papers in the 70's about unusual number bases. It was a long time ago, so my memory has faded, but I do remember being intrigued by negative number bases. Their main attraction is that no sign is needed to handle negative numbers. Nowadays, of course, two's complement arithmetic is so entrenched in all computers that nobody ever uses anything else.

this video is absolutely brilliant. i have been trying to figure out the implications of non-natural bases but i have never been able to figure it out myself. This video is exactly what i have been looking for for years, subscribed!

Great content. I never thought bases could be something else than integers, but it actually makes sense. I just spot a very little mistake at 16:25 it shows 21i+2 (which is absolutely correct) but the voice says "21+2i".

this is a beautiful video. the topic is so absurd but you explained it in the most understandable way possible

For 10:00 you could just say for bases a/b, We use the digits 0, 1, …, max(a, b) - 1

Very good video. I kinda always wondered this. Good to see. I would like to see a tetration video of different group of numbers, that is a very difficult operation can be made by hand.

I really liked this video! You explained in very well and the animations were fitting and eass to understand. I am really looking forward to watch more videos! Good job!

This is so mind blowing and really well explained. I barely can believe what I see

really cool video but i dont think you covered about the golden ratio base? whats interesting about this is that if the base is the golden ratio, you get an interesting phenomenon. (btw base golden ratio only needs 2 digits, 0 and 1) let the golden ratio = phi we know that phi = 1 + 1/phi multiply both sides by phi. we get phi^2 = phi + 1, (a(b+c) = ab + ac) rewrite this as phi^x because we are in base phi phi^2 = phi^1 + phi^0. (x^0 = 1) remember that we can always multiply both sides by phi to increment all of the exponents. its really cool cause we get 100 = 11 in base golden ratio. just something to note. if you found this comment interesting, consider checking the combo class, another channel covering this topic and is the source of all these equations.

i’m not gonna watch this video but, fractional bases… mind blown. haven’t thought about something that crazy in a while. thanks. ❤

3:00 it is, however, possible to use infinitesimals as a basis. They aren't gonna be good for covering R (you could still do it if you allow infinite ordinals) but they can have neat properties such as a ε³ < b ε² for any real a, b. This can actually be useful. I used it to calculate a sequence dependent on low probability events in the limit where the probability is 0. (This is one particular way to get the Thue-Morse sequence, and using this "infinitesimal basis" number system you can extend that to more than 2 separate states)

Mind blown. What a great explanation!

This is getting really deep into number theory, had a hard time keeping up with the transformations, and still don't understand the utility if imaginary numbers or imaginary base number systems. Also why I don't think I'll cut it as a mathematician, that being said great video, thanks for the breakdown.

Yes, I've thought about this question from time to time. And here is answer. Thank you!

These feels like changing the clef in music theory.

Your video and animation are incredible! I hope you will continue to post other video.

great vid, and coolest end transition i've ever seen

Sres. Digital Genius, reciban un cordial saludo, gracias por ampliar los conocimientos en este tema apasionante. Éxitos.

This is a great overview!

Wow, it took a lot of pause-replay and pencil work, but I sorta-kinda-got-it. What fun! TY.

I just started the video and already hear about transcendental number. I don't think I have attained such a realm of understanding yet 😅

You're way of explaining is so great

What a remarkably beautiful system

If we used the base 1 you suggested (basically just tally marks) we can only write natural numbers, and there would be no (functional) decimal point. I certainly wouldn’t call that a counting base, it seems much easier to just put it with 0 as “bases you can’t count in”.

please talk about non-constant bases as well, where digits can scale by different multiples so e.g. factorial base, where n-th digit can be from 0 to n-1 and has value of n!

Already knew about the topic but the visuals are great

One of the craziest videos I've ever watched in my life, period. I knew how to calculate base 2 and stuff, but never even cared to think about other numbers as base. I'm absolutely mind blown, you deserve all the subs and views in the world.

Very interesting video. In positional numbering systems each base has particular characteristics to them, as for example the divisibility criteria vary from one base to another. In the decimal system, for example an integer is divisible by 5 if it ends in zero or five. In a base n, n ∈ ℕ, numbers ending in zero are multiples of the base. In base π, the sine function reaches zeros in integer positions of that base: .... -2, -1, 0, 1, 2, ... The study of mathematics today has a bias to base 10. There are many things related to this particular base. Developing mathematics using other numerical bases as a center could lead to interesting discoveries within mathematics and beyond. Thank you very much for the video.

Wait wait wait! I want to point out a lot of things: First we want to state what a "generalized base b" should be. L'll start with just real numbers. I think we should ask that, with a finite set of digits (natural numbers), we want to be able to write any real number as a sequence like: ±an[...]a0.b1[...]bn[...] (where we have a finite number of "a" digits on the left of the dot and an unlimited sequence of "b" digits on the right). This is what the "traditional" base b allows ud to do. We trivially notice that base 1 des not work, since we are not alloed to represent 1/2 in any way (just like any other fraction). Moreover base -1 does not work even to represent just integers with just one kind of digit... Using a negative integer just like you said causes no problem and could be done just fine (except for -1, of course). Up to now we should point out that each number can be written in base b in just one possible way (with the exception of "b-1" periodic, where for exmple, in base 10 we can write 3.000000... and 2.999999... and they are the same number). With non interger numebers we have to renounce to this property, but we'll be ok with that. Now think how to write 1/3 in base 2, it should be 0,0101010101... (and that's the only way to write it). Now how can we write 3 in base 1/2? It should be the reversed of the previous writing, namely: ...0101010.000... Here we have an unlimited sequence of digits in the left of the dot, and this cotraddicts our defiinition (see above). We could stretch that definition to include unliited sequence of digits on the left. Quite strange but ok, let's do it. We could now prove that, using any real number b (besides 0, 1 and -1) the amount of digit we need will be the maximum between b and 1/b, rounded up. Witch is much better than your proposal, since fof 3/7 we will just need 3 digit instead of 7. Talking now about complex numebers: it's not clear which digits are you allowed to use in case of complex numbers like 2+2i. Since it's a fourth root of -64 I suppose you will want to allow us to use all the integer digits between 0 and 63. If so 1 can obviously be written as 1, while i will be -0.08 (you can easily check it). Since you can wrote any number in base -64 using those digits and you can represent those numbers in base 2+2i by using just fourth powers (like 3000500020003.00040007... instead of 3523.47... in base -64), you can write any complex number like a+ib doing the operations: x-0.08y. I'm not sure about which complex numbers cannot be used as a base b, but I'm pretty sure that the n-th roots of 1 cannot be use, regardless of the amount of digits we will allow us to use. I don't know if other numbers with modulus 1 can be used (and, if I have to guess, I think they could work, with the proper, finite, amount of digits) nor I can extimate the amount of necessay digits for generic complex numbers, like e+iπ. I hope I was understandable (I'm sorry but I'm not a native English speaker) and please answer me noticing my possible mistakes.

You make me like the bases Thank you !

Between delight and eye opener 😊 thnks. A meaningfullless universe lues ahead.

The most interesting video I have seen in a long time! Now I want to know what every number is in every base! 😂

To represent all real numbers, the largest digit must at least be b-1. Hence, the digits 0,1,2 are insufficient for base π. For instance, 3 has the representation "3" in base π. Note that d/b + d/b² + d/b³ + ... = d/(b - 1) Is the largest number with digits at most d and zeroes before the decimal point, but 1 is the smallest number with nonzero digits before the decimal point. If d < b - 1 is the largest digit, the numbers between d/(b-1) and 1 have no representation, in fact all numbers x where dbⁿ/(b-1) < x < bⁿ have no representation (we just shift the argument by n digits). In particular, the number 3 has no representation in base π with digits 0,1,2, as 2π/(π-1) = 3 - (π - 3)/(π - 1) < 3 is the largest such number with 1 digit before the decimal point and π > 3 is the smallest such number with at least two digits before the decimal point.

My favorites are base Fibonacci and base factorial, but I’m not good enough to succinctly explain those here. Look them up if you’re interested! Micheal Penn did a good video on it

Fascinating, thanks! Would you be willing to cover the p-adic numbers sometime?

This video would be more interesting if some applications for the various bases could be shown. Most humans use base 10 (decimal). Almost all computers use base 2 (binary). Base 16 (hexadecimal) is useful for displaying binary numbers in a form that is easier to remember and read. The same holds for 8 (octal). If somebody use base 5, that would make sense since we have 5 fingers per hand. For clocks we use base 12, base 24 and base 60. For weeks we use sort of base 7. But what is the use for base -10, 3/7, pi, or 2i?

woah, cool video!

There's a reason why there are "big groups" of powers with the same sign on the form of z=|x| + |y|i (a similar point could be made for z=|x|+|y|i) If we write z on the polar form, we have z = |z| cis(theta).... where theta is between 0 and pi/4(or 90º)... since we are on the first quadrant in the complex plane. [cis(x) = cos(x) + i * sin(x) = exp(ix)] And z^n in the polar form is z^n=|z|^n cis(n theta) Note that the smaller the theta, the higher the n you need to change from quadrants(which happens when you change the signs of |x| or of |y|)... which means the higher the sequences of powers with the same sign.

Fascinating subject, great explanations. My one nitpick is that you consistently say "nominator" instead of "numerator".

Well that's a new argument for which base to use. We should be using base 10 because we have 10 fingers. We should be using base 12 because it has lots of divisors. We should be using base e because it has the best Radix Economy score.

thanks for this. please less ads, this is knowledge sharing and not entertainment

Base Grahams Number?

If there was a way to represent the ideas of percentages with one set of symbols , it'd be possible to represent each digit as a percentage of the digit, with none being 0 and full being equivalent to (1/A) in the next digit for base A The logical conclusion, however, is that you'd encode all the information in the leftmost digit, which means you only need that one digit's fill-percentage plus whatever power it's raised to... so we've effectively rediscovered scientific notation

part of me hopes we keep finding more types of numbers that branch outside of complex numbers so I can see what their base number system looks like

Great video! Super interesting. Where would come across this or apply it in mathematics?

thank you for this xD ive asked this question before but couldnt find an answer

Keep it up!

Great content to think about the way we write numbers. But I see some problems which limits the use of such bases: It includes counting and number comparison. In a Natural Number base you can count easily and any carry will go to the next column on the left. With a base of √5 a carry will skip one column and comparison 4*√5 > 5 while in natural bases a non zero digit more left is always of greater value. Base 1 does not make sense because all digits have the same value (always 1) and each digit can take only Zero. Incrementing to 1 would result in a 0 with a carry to the next digit on the left which is then incremented and leads to an infinite number of digits affected.

Upto this point I found, You are the second person on this planet who is seriously working on base system representation. Well, In my work, I'm trying to extend this for polynomials to represent polynomials with base of other polynomials. Very nice vedio. All the best for your reasurch and future. 😊

2:33 No, a tally system uses multiple instances of a single digit to represent numbers; you simply count the number of digits to get your value since they all have equal weight.

Base 1 is a very old counting system. Scoring.. though scoring eventually evolved into grouping per 5...

this is the first time I've seen someone actually talk about base 1, I always wondered about it since everyone I know doesn't understand how bases work so they just think, "Base 1, 1 digit, boom, tally marks."

LOL I've been making jokes about "base π" for few years and I never knew that base π was some serious stuff! Thanks for your good work❤

2:09 For clarity's sake, you should've used your previously presented indexation method to show in which base were those numbers written, as it was a bit harder to realize that we weren't exponentiating the number *fourteen,* but _nine expressed in base 5_

Your videos are useful!

There's also *multi-base* positional numeral systems, which remind me of HashCat's "Mask Processor". They open up a literally infinite multiverse of possibilities, by using *arbitrary sequences* of numbers instead of the typical powers of N. The Fibonacci numeral system uses the Fib sequence, which is essentially just rounded powers of the Golden Ratio. So, in theory, Fib System should have the same radix economy as base φ. There's also one for Primes. And if your sequence is finite, you can simply repeat it using powers! Imagine a *"Collatz3"* num-sys, it would be: _×3 + _×10 + _×5 + _×16 + _×8 + _×4 + _×2 + _×1... (then repeat using all squares, then cubes, etc...)

Hmm, this method for fractional numbering systems is different from I know. The other method is based on logarithms: you take logarithm of the number, write integer part of logarithm, take remainder, again take logarithm, etc. As a result, the representation of a number in base 7/3 is very similar to binary representation, but with occasional rare digit 2 appearing.

Very much enjoyed this :]

This reminds me of that age-old question that has vexed many an erudite academic: how many angels can fit on the head of a pin? Not saying this of you, but this is a good example of the quasi-mystical nature of cultic mathematics. Numeromancy and Pythagoreanism are very much still alive. I did enjoy the video, though. Cheers!

One of the things that fascinates me about negative bases is that the negative sign is useless, because that would create two representations of each number (eg. 1011 and -101 in base -2)

Nice synopsis; good explanations. My feedback: IMHO, "base 1" is strictly the digit 0, and no other. Allowing any other digit breaks the rule for other positive integer bases, and is "cheating." Speaking of breaking that rule, though, here's a neat trick I came up with some years ago, but which I strongly suspect isn't original. In base 3, instead of (0,1,2), use digits (-1,0,1), for which let's use the marks (\,0,/). Because then, all numbers, + and -, can be written without using algebraic signs. So e.g., +1 is just /; -1 is just \; the integers (..., -5, ..., 0, ..., 5, ...) are (..., \//, \\, \0, \/, \, 0, /, /\, /0, //, /\\, ...). Fractions can be converted from base 3 in the obvious way. The negative of any given number is represented by vertically flipping the number's representation. Other odd positive integer bases, b, would be analogous, using ½(b-1) new (non-mirror-symmetric) symbols for the +ve digits and their mirror-images for the -ve digits. Another crazy idea of mine - "factorial base." Your discussion here was all about fixed bases, but what if each successive digit is in a different base? Specifically, make the least significant digit binary, the next ternary, then bases 4, 5, 6, ... Similarly, after the "point," or radix, the digits are in bases 2, 3, 4, ... The non-ve integers would be 0, 1, 10, 11, 20, 21, 100, ... I call it "factorial base" because "1" followed by (n-1) zeros, n ≥ 1, represents (n!) in this scheme. Carrying out additions and subtractions in factorial base isn't too hard, but multiplication is totally impractical. A couple nice features are that every rational number terminates; and that e = 10.11111111... forever. A drawback is that there have to be an unlimited number of digit symbols. Fred

it`s best video for learn english, thanks

jesus christ, the way you explain is beautiful

I remember there was a proof of the Fundamental Theorem of Symmetric Polynomials that uses base ω (where ω is the first trans finite ordinal)

Gotta admit that your videos are so well-made! I've managed to understand each one of them so far! Wondering what your next video will be about, but your content is very interesting! and it can make anyone learn something new! ^^ Keep up the great content Digital Genius! ;)

you make unequal last step in the irrational bases. It should e like you difine pi, or whatever on some equal intervals. Basically, you may redefine what 1,2, are.

Can you write -2*Zeta(3)-Gamma'''(1) in base (EulerGamma + Pi/Sqrt(6))?

Really like the look of those negative integer bases. Those look rather compact and don't need a sign. Kewl.

There is of course nothing to say you have to use just positive digits. For me balanced ternary is the best base (uses -1,0 and 1 as digits). 3 is the best integer base balancing number of digits and length of number and it makes negative numbers better.

the (square) root ones feels like a copout, just treating it as its square by using more digits than should be necessary and treating the spare digits as a bonus for the niche situation a number has multiples of the base. The numbers leading from 0 to the floor of sqrt(n) should be sufficient. By the way, in the case of Pi I think you also need 3 as a digit. Otherwise there's going to be gaps. For instance how would you write 9.8? it's less than pi², but greater than 2pi+3 which is greater than all the remaining digits being 2.

Wonderful! Could you give us some bibliography? It would be so helpful!

Nice summary in the video 😃👍

I got the idea that there could be complex digits. That would allow writing complex numbers with integer bases. So complex binary has digits 0, 1, i, 1+i. I will write them 0, 1, I, C. Then, 2+3i is CI, 2+4i is I10, and so on. Technically, that is having a Cartesian product of real digits (0, 1) and imaginary digits (0, i), and choosing the appropriate real part and the appropriate imaginary part. These two parts could be in different bases, which would work but be confusing because the value of the real part would diverge from the value of the imaginary part at higher digits. So base real 3 and imaginary 2 has digits with values (1, 1), (3, 2), (9, 4), (27, 8), (81, 16)…, where the first part is the value is multiplied by the real part of the digit, and second part is multiplied by the imaginary part of the digit, and these are added together.

Also, in base n, you have to use the digits within the value of n. So in base √10, since it is equal to 3.16227766..., we use the digits 0, 1, 2 and 3, although 3 will be rarely used. The value of 4 in base √10 is approximately 10.220012.

I love the fact that e has the highest radix efficiency

What is called base 1 here number here is Peano numbers and they are used in proof assistants(such as coq/idris/agda) as they are easy to implement and for their version of addition/multiplication they can derive lots of rules like associativity.

Since we can't use base e, the best integer base to use would be 3 because it's the most radix economic. Or, we could use some rational approximation of e !

Hi. I got some interest on the topic. Do you know about any bibliography on the topic?

Another example using algegraic irrational numbers is the golden ratio phi as a base. It's called "phinary base"

Underrated!

You should make a video about non-positional numbering systems.

On the imaginary numbers, my mind just blowed up