Can any Number be a Base?

[@brunnomenxa]

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

[@brunnomenxa]

*16:26

[@MRBnessGamerz]

similar issue at 18:28 where he calls -8i "negative real"

[@lukatolstov5598]

Agree.

[@lukatolstov5598]

@@wham_sandwitch !?!?!?

[@brunnomenxa]

​@@wham_sandwitch, My objective here is to point out a "minor" error that appeared in the video, with the aim of potentially correcting it to avoid confusion, especially when the content involves mathematics. I'm doing this in a constructive manner. So, stop taking offense on behalf of others.

[@ryan20202]

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

[@DimkaTsv]

You tecnically can count to 1024 on fingers because it is possible to interpret finger position as binary. And if you assume intermediate states, then even tertiary is possible which allows to count up to 59'049

[@SgtSupaman]

@@DimkaTsv , yeah, the problem with that is that it is too much to be able to realistically keep track of and definitely too much to be able to recognize. Even with your fingers moving up and down to help, trying to keep track of what you are counting while concentrating on intricate finger movements will be virtually impossible as you continue for several hundred or even thousands. Even making tally marks, which is a far easier task, can make you lose track at such high numbers. But even worse would be trying to recognize what number is being represented. Say you asked me how many people I counted coming into the stadium for an event. I hold up my hands with my left pinky halfway up, my left index and thumb fully extended, my right index and middle halfway up, my right ring fully extended, and my right pinky, due to how my hands work, is potentially halfway up or trying to stay down. What number would that be? Before you even start to work it out, you have to ask how I was counting. Did I start from the right so it looked left-to-right readable for me, or did I start from the left so it would be left-to-right readable to other people? And, because this is positional, how were my hands in relation to each other? Did I have my hands facing away from me (to start and end with pinkies), facing towards me (to start and end with thumbs), or one facing toward while the other faced away (to make the smallest on each hand consistent with either pinky or thumb)? All that to say, if you really need to count *that* high, there are far better methods than using fingers. --addendum: Now that I think about it, you could use those states of your hands to encode even more numbers (using those four possibilities I listed as a leading 0, 1, 2, and 10 to get all the way up to 236,196), but, seriously, why would anyone want to?

[@DimkaTsv]

@@SgtSupaman that is why i said "technically". It doesn't mean that counting in such way is efficient or practical at all.

[@nielskorpel8860]

think of the hour:minute:second format, where every digit space has a different base.

[@nielskorpel8860]

well,other than the minute and second marker, but the hour and millisecond markers have different bases.

[@TheArtOfBeingANerd]

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)

[@KingOf_B]

I mean we do basically use base pi for trig already. We just do it in a way where we can still use base 10 but also make it obvious we are counting in increments of pi. Ie sin(n pi).

[@LucasFerreira-gx9yh]

base tau (2pi) could be better

[@CUSELİSFAN]

measurement in radians comes close to what you are saying imho.

[@s14011]

​@@CUSELİSFANand better. Because the pi oftentimes cancels out during calculations

[@astronemir]

Radians

[@RandyKing314]

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

[@unowong3084]

look up "apeirology" and "googology", thank me later

[@matsv201]

Well. Have you heard of j and k numbers?

[@RandyKing314]

if you mean quaternions, i remember having a similar experience!

[@aweebthatlovesmath4220]

Actually it haven't gotten bigger it's just a new way to write old things...

[@blerghhhhhh]

have u heard of p-adic numbers

[@landsgevaer]

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.

[@supernt7852]

According to this logic, 1 can also be written as 0.999999999… in base 10

[@supernt7852]

(which is correct as they have been proven to be the same number)

[@lox7182]

Even that can create problems with, for example, having more than one representaion for 1.5 in base 1.5.

[@landsgevaer]

@@lox7182 Yes, I am aware that there are always some numbers that can be written in many ways. But I don't see a reason why one should use MORE different digits than required, right? The video is like using decimal, but also allowing B to write eleven. No use for that.

[@Faroshkas]

​@@supernt7852You can write every non-repeating rational number in three ways. For example: 1, 0.999..., 1.000...

[@yunogasai7283]

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

[@pierreabbat6157]

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.

[@Sasha123-d1q]

Also base gamma(≈0.57721)

[@Theforgotten-b7g]

Unusualy a golden radio equals 1 divided by (square root of 2 divided by 5)

[@matroqueta6825]

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

[@DoctorIknowWho]

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.

[@GustvandeWal]

I have a hard time following this explanation. Care to give an example?

[@DoctorIknowWho]

@@GustvandeWal for sure! I made a terrible job of explaining but here we go with a “real world example”: Imagine you have a typewriter with number keys and a spacebar and are tasked with writing down a string of numbers given to you. The string of numbers can be of any length and the numbers themselves belong anywhere in the set of natural numbers. If you were to find that, one day, the type writer was modified so that you no longer had a space bar, you would still be able to write down strings of numbers by converting those given to you to base 9, and using 9 as the separator. To further extend this, if you found that your typewriter now only had one key remaining, by using our base 9 rule established earlier, we can write any string of numbers as a string of numbers in base 9 using 9 as a separator, and using THAT number to represent the list using tally marks. Example: 1, 10, 18, 27 Can be written as so in base 9 using 9 to indicate separation: 1911920930 And this number above is itself an integer that we can represent in base 1 with tally marks. That way, we can decode the original string of numbers!

[@GustvandeWal]

@@sobhansyed4482 This just seems like the explanation of unary counting (tallying). Where is the "base 9; use 9 as a separator" part of the thought experiment?

[@kronostitananthem]

@@GustvandeWal maybe he means using unary tallies as digits. 23 in base 4 = 11 in base 10 You could write 23 in base 4 as "||4|||". If you represented the base 4 in this system in a recursive way "||(||||)|||" it would be writing-system-independent. Although technically the real representation of that would be "||(|(|(I(I(...)))))|||" to infinity as the "(||||)" should also ideally be represented in base 4 which would be "|(||||)" which then the (||||) needs to be represented again...

[@GustvandeWal]

@@kronostitananthem Tysm! Ends up making a bit of sense, but not lots. Numbers with lots of digits would need lots of separators. I just wish this @DoctorIknowWho replied...

[@leave-a-comment-at-the-door]

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

[@alexandertownsend5079]

Is there a number system where you represent numbers as a sum of sqyare numbers?

[@WK-5775]

The final 0 of each number seems to be redundant. Btw: What about fractions in this system? I.e. what would be the meaning of digits to the right of the "decimal" point?

[@leave-a-comment-at-the-door]

​@@WK-5775 yes, the first digit is in 'unary' so it can only take one value, 0. the most logical way to do fractions would be to count the other way, but factorial is undefined for negative numbers so mathematicians came up with a smarter way. if you have each digit after the decimal be 1/n! then you can represent any fraction with a number of digits equal to the denominator+1 or less. (on this end, 1/0! and 1/1! both evaluate to 1 and so neither can be anything but 0. like that ending 0 you mentioned before, they are sometimes just omitted but I will include them for completeness' sake). examples: 1/2 = 0.001 1/3 = 0.0002 1/4 = 0.00012 1/5 = 0.000104 1/6 = 0.0001 1/7 = 0.00003206 1/8 = 0.00003 1/9 = 0.0000232 1/10=0.000022 1/11=0.00002053140a 1/12=0.00002 any multiples are multiples of those just like any other system. to get an idea of what's happening here in your head: each number 1/n starts at the 1/n! position, and the number that goes at that position is (n-1)!. so a third starts at the 1/6s place and 1/3 is 2/6; or a fourth starts at the 1/24s place and 1/4 is 6/24(since 6 > 3, the biggest digit at this place value, you carry over to the next place and subtract 4, like how in addition if you add 8+8 you carry a 1 to the 10s place and put 16-10 in the ones place). the numbers work the same way going the other way; that is the 3rd digit can be 0 or 1, the 4th digit can be 0, 1, or 2, the 5th digit can be 0, 1, 2, or 3, etc.

[@leave-a-comment-at-the-door]

@@WK-5775 also if you use this system for fractions a handful of transcendental numbers have fun decimal expanions: e = 10.0011111111111111111... sin(1) = 0.00120056009A00DE00HI00... (each group is +4) cos(1)= 0.0010045008900CD00GH00... sinh(1) = 1.0001010101010101010101... cosh(1)= 1.0010101010101010101010...

[@thetinkerist]

it is called factorialadicpoint man 😂

[@yanntal954]

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.

[@mzg147]

Although it was not explained in the video, the non-radicals work beautifully. If you have a quintic equation and some root γ, then by the fact that it is a root of a quintic polynomial p(γ)=0 you can move the 5th term to the other side and obtain γ⁵ = q(γ) where q has a lower (4th) degree. In other words, γ⁵ can be written as number with digits being the coefficients of q. So to write any number in this base, you will need maximum of coefficients of q digits.

[@yanntal954]

​@@mzg147 Doesn't this assume that all coefficients are integers though?

[@mzg147]

@@yanntal954 Yeah, the integer case is easier. I still think it works in the general case too, but then you have those pesky digits reversals just like with the rational bases in the video.

[@yanntal954]

@@mzg147 I am not fully convinced yet 🥺

[@chrisengland5523]

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.

[@DoxxTheMathGeek]

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^

[@legygax]

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".

[@juibumgeilheit]

i saw that too!

[@nbspWhitespaceJS]

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.

[@vampire_catgirl]

Oh yes combo class, the annoying guy who's constantly dropping shit and yelling, great

[@almscurium]

@@vampire_catgirlhow old are you?

[@mcrow312166]

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

[@claiiyn]

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.

[@WardR3]

HolyCow!! This channel is extremely underrated! It should have at least 2M subscribers ❤ Where are you ITE guys 🤨

[@videogameplayss]

if americans made the number system:

[@JimmyMatis-h9y]

Dogmatic. It's not like Brits dont still use it, including even weirder units like people's weight being expressed in stones.

[@Definitiv_Nichtkurz]

May I remind you who got people on the moon?

[@limehello1797]

NASA

[@Naniblocks]

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

[@cmilkau]

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.

[@yurenchu]

Unless we allow digits that represent values below 0 . Such as in so-called _balanced_ representation systems. For example, the _balanced ternary_ system is basically a base 3 representation system, but instead of digits {0, 1, 2} it uses digits that represent the values {0, 1, -1} . There is no convention for which symbol to use that represents -1 , but suppose I'll use the letter "h" for that. So 0₃ = 0 1₃ = 1 1h₃ = 2 10₃ = 3 11₃ = 4 1hh₃ = 5 1h0₃ = 6 1h1₃ = 7 10h₃ = 8 100₃ = 9 101₃ = 10 11h₃ = 11 110₃ = 12 111₃ = 13 1hhh₃ = 14 1hh0₃ = 15 1hh1₃ = 16 1h0h₃ = 17 1h00₃ = 18 1h01₃ = 19 1h1h₃ = 20 1h10₃ = 21 1h11₃ = 22 10hh₃ = 23 10h0₃ = 24 10h1₃ = 25 100h₃ = 26 1000₃ = 27 1001₃ = 28 etcetera. The negative of a number is then obtained by simply swapping 1's with h's and _vice versa_ : h₃ = -1 h1₃ = -2 h0₃ = -3 hh₃ = -4 h11₃ = -5 h10₃ = -6 h1h₃ = -7 h01₃ = -8 h00₃ = -9 etcetera.

[@Kram1032]

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)

[@NoNameAtAll2]

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!

[@spaghettiking653]

That was an entrance exam question for the Oxford MAT. They called the factorial base "flexadecimal".

[@Henrix1998]

Wouldn't you need infinite amount of number symbols if the scaling gets bigger?

[@zlodevil426]

@@Henrix1998yes, but you can express integers up to n!-1 if you use n different digits in base factorial

[@mikerickson01]

Yes, any number can be expressed as SUM(i=-∞ to ∞) ai*r^i (a are the digits, r is the base) where r≠0 and 0≤ai

[@eylonshachmon6500]

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”.

[@rahevar3626]

Exactly what I was thinking If we look at any base to represent number adding 0 to the left of the number and to the right after decimal point shouldn't change the number but since 0 is the only number we can use this rule breaks here Also if we convert any base 1 number we are multiplying every number with 0 so the answer is just 0

[@lefishe5845]

In my opinion tallies being restricted to natural numbers is slightly better than having it being entirely unusable but it's not a big loss to lose one base.

[@Mr_Yeah]

12:27 You also need the digit 3. Otherwise, you can't represent 3 in base pi because 0.22222… = (2 π)/(π - 1) ≈ 2.9339

[@beamathematician2487]

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. 😊

[@JoshuaNichollsMusic]

You should look up Combo Class, they have a number very similar to this that looks at negative, square root and transcendental bases too. Fascinating stuff!

[@felipevasconcelos6736]

Second person? Then may I present you to imaginarybinary, an extremely underrated channel that created a very unique way of using 2i as a base.

[@archivethearchives]

Combo Class with Domotro is also a fun channel that often works with maths theory and number base systems

[@CheckmateSurvivor]

Please check out my latest video about the most difficult puzzle in the world.

[@Henriiyy]

​@@felipevasconcelos6736Base-2i was first proposed by the legendary Donald Knuth.

[@gustavojacobina9796]

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

[@RGAstrofotografia]

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

[@brightblackhole2442]

simple. this is really an extended modifiable diophantine equation (EMDE) which is simplifiable by the isomorphism M -> ą_0 \ {S_sum, 0, +} according to an addition-like operator in a field of 0 nondifferentiable manifolds, where there are actually _3_ nth riemann roots of unity, so the determinant can be approximated by the limit as x approaches [y : y(x) not in S_product *] and the rest of the solution has been reduced to a trivial kirimeta-vu 3-model partially integrated gödel system

[@lloydbotway5930]

As I worked my way through grad school as a teaching assistant, one of the full professors sat in on a class I was teaching, where I explained negative bases. He was shocked and amazed and thrilled, because he'd never considered the possibility. No minus sign required! Cool!

[@OBGynKenobi]

Base Grahams Number?

[@trueluscao]

imagine

[@Goobfromdandysworld-sv7e9x4]

imagine the possibility

[@janusaj1337]

all the possible symbols that has and ever will be written would still not be enough to fit the observable universe. Even if the symbols were plancks length in size.

[@benyseus6325]

You would require as many symbols for that - 1 and there aren’t enough letters in the alphabet or any of Earth’s logographic systems to satisfy that (if we’re going by convention of bases that are >10).

[@godelo000]

I think that we’d need only one digit to express all numbers we regularly use 😂

[@maxwellarregui814]

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

[@Doogsonai]

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.

[@Anonymous-df8it]

May you clarify what you mean by 'redo[ing] part of the substring'?

[@kronostitananthem]

hello fellow esolanger

[@DurianFruit]

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!

[@dominicellis1867]

Isn’t this just polynomials evaluated at the specified base? In essence, every number can be represented as a power series with coefficients a(n) representing the specific digits and x^n representing the place value in that base system. Then, you could perform term by term operations in any base system. Id be interested in whether base dx, 0, or infinity are possible.

[@angeldude101]

Yup. Every decimal number you've ever written has secretly been a polynomial.

[@yurenchu]

The difference being though that the coefficients of polynomials are generally unbounded , whereas in a base representation system they are limited to a particular finite set.

[@dominicellis1867]

@@yurenchu there is a restriction on polynomials: the coefficients are generally elements of the reals with a 0 imaginary part. The fundamental theorem of algebra breaks down when the coefficients can be complex. You end up with roots like x = j without the conjugate.

[@yurenchu]

@@dominicellis1867 I'm not familiar with the "fundamental theorem of algebra", but the Wikipedia page about it states _exactly the opposite_ of your claim.

[@dominicellis1867]

@@yurenchu solve the polynomial equation x - j = 0. Normally a polynomial with complex roots always includes the conjugate, but when you allow for complex coefficients you can generate any root combination. jx + j^3 = 0 has x = +-1 and x = j as the solutions. Notice how x = -j is not a solution

[@blim8777]

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.

[@Definitiv_Nichtkurz]

I ain't reading all that.

[@DSN.001]

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.

[@MuSic-ok7dh]

Aaaaaaaaah! I was wondering about base Pi. Then you go "base imaginary numbers". Mindblown.

[@xminty77]

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

[@SirKenchalot]

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.

[@j4mster]

im high af and have absolutely no business watching this but for some reason im here anyway lmao

[@the_agent_z]

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

[@FebruaryHas30Days]

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.

[@IAteAnAK47]

what about base 3.5

[@ensiehsafary7633]

No you can use 0 to 9 since it's exactly like in base 10

[@IAteAnAK47]

@@ensiehsafary7633 bro, he probably had to watch like 7 whole 30 minute long videos about that just to be proven wrong.

[@FebruaryHas30Days]

@@IAteAnAK47 For base 3.5, you'll use 0, 1, 2 and 3 with 3 being rarely used

[@FebruaryHas30Days]

@@ensiehsafary7633 We're using powers of 3.16, so the highest value a digit can represent is 3.16

[@patrickmaline4258]

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

[@ffggddss]

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

[@Trineal23]

Yes, others have thought of it en.wikipedia.org/wiki/Balanced_ternary But it speaks well for you that you came up with it on your own as well :)

[@GameBoy-ep7de]

You should look into ternary computer systems then. I wondered about there being base 3 computers, and after some very shallow research that confused me, they do exist. I remember (-1,0,1), (0,1/2,1), and (0,1,2) being some possible representations. You might understand it better than did.

[@JoshuaNichollsMusic]

Yep this is called balanced ternary and it’s used by some computers :)

[@larsokkenhaug148]

The factorial base is sometimes called flexadecimal

[@ffggddss]

@@larsokkenhaug148 Interesting. But not the best name for it, as it doesn't have anything to do with the decimal system. Maybe they should have named it, "flexanary"? ;-) Not as catchy...

[@dinoeebastian]

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."

[@BKNeifert]

Just like with English, Base 10 is probably all I'm ever going to understand. I'm not a dummy... but I can't even conceive of a number system in base 2 or base 7. Let alone in i or complex numbers. I'm just too fluent in that language, and understand the world through it. Although, seeing you explain it in 7:04 it gives me some subtle impression of how the number system I already use works. I can kind of grasp this, but it's way above me. Good video. One of your videos helped me conceptualize how Calculus was discovered. So, thanks for that. But, this... woo... this is hard.

[@SilviuBurceaDev]

Funny, you're actually using base 60 successfully. Ever thought about the clock? :)

[@BKNeifert]

@@jijijijijajajajajajajiYour English is fine... better than most people's. You just need to learn how to use paragraphs. I understand math through base ten, and its geometry. I think you made a few accusations. First, that I don't understand Base 10... I actually do. It's just I can't conceive of using a different Base, as I just don't understand its geometry enough. I see the world through the geometry of Base 10; like, addition and subtraction, or even factions I understand through Base 10. And there's nothing wrong with that. I don't plan on learning Spanish or German any time soon, or any other language.

[@BKNeifert]

@@SilviuBurceaDev Yeah, that's true. But, that's actually more fitted to the geometry of a day. That's why I understand it. There has to be something real on which to base the system on. Which 24 hour days, and 60 minute and 60 second intervals fit perfectly to the geometry and cycle of the moon and sun.

[@Ryan_Thompson]

​@@BKNeifertThe arithmetic and conversion is straightforward. The number 134 in base 10 is 1x100 + 3x10 + 4x1 = 134. Every digit's value goes up by a factor of 10 as you go from right to left. Binary (base 2) is the same, except every digit's value goes up by a multiple of 2. So, 134 in binary = 1x128 + 0x64 + 0x32 + 0x16 + 0x8 + 1x4 + 1x2 + 0x1 = 10000110b. This much, I think you already know, right? If not there are some great, simple guides out there. Are you just more thrown off by the intuition of how big certain non-base 10 numbers are? For example, how tall are you in binary centimeters? I have no clue either, and I'm a computer scientist who works with binary daily. I don't see the world as 1s and 0s, although some numbers, for me, are more intuitive in binary (or hexadecimal, or octal). Like SilviuBurcea1 alluded to, you arguably use a variable base all the time: time! 12 values for the hour, 60 for minutes, 60 for seconds, and decimals for fractions of a second. We also use base 10 symbols for the hours, minutes, and seconds (or sometimes Roman numerals!!), so now it's even more complicated, yet we're used to it. But I bet you'd have no intuitive idea what time to have lunch if we simply used base 10 seconds. (43200 seconds would be 12:00:00 = 12x(60x00) + 0x60 + 0x1.) Anyway, you're not alone in being so accustomed to base 10 that it's difficult to imagine most numbers in other bases. If you have to work with another base for any length of time, you get used to it. Learning how different bases work is orders of magnitudes easier than learning a different language, at least!

[@BKNeifert]

@@Ryan_Thompson No, I think it's the Arabic Numerals, that they're based in Base Ten, and the notation or logic doesn't make any sense when using them for other bases. Like, I understand Roman Numerals, though they're hard to do math with. I think I prefer the geometry of Base Ten, though. It's more intuitive, and easier to comprehend. Like, from what I understand, that's why it's basically the universal standard, is because it does work, and is more efficient than anything else. Binary is a close second, but that's for computer programing.

[@user-jt1to1ry6q]

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)

[@modolief]

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

[@aventurileluipetre]

jesus christ, the way you explain is beautiful

[@japalaciosh]

Interesante artículo, nunca pensé que un número cualquiera se puede expresar en términos de otros números en infinitas bases.

[@becomepostal]

What?

[@MohandeepSingh007]

Beautiful storytelling of a somewhat sophisticated Math topic! 👍 Loved it, but a few bits were very interesting and few comments/questions: 1. @8:20 Love the sound effects denoting arithmetical operations 💨💨 2. @12:49 The revenge of Irrationals through transcendentals (pi, e, etc.) against rationals. 😄😄 3. @20:30 This is how you encode complex numbers in computers?? 4. @20:43 Beautiful Ending. 5. Can state your References and Further Reading? I am trying to learn these nowadays and a good reading material can really help me.

[@blacklight683]

That's "based"

[@artkim2334]

Base 10. 37568 Base 6/4. 4200020200002002004402

[@liquidcashews]

badoosh

[@Tartarus4567]

Based on what?

[@kropotkinvore316]

thank you based god

[@apogee792]

Damnit i was gonna say that 😂​@@Tartarus4567

[@7lllll]

11:05 i don't think so, i think you have to use only numbers smaller than sqrt(5), so that you have some crazy decimals in each digit, but still maintain the property that each digit represents the powers of sqrt(5)

[@thomaskember3412]

When I was 13 I came to realise that numbers did not have to be expressed in base 10 and could be expressed in base 8. In a notebook I wrote out the method for converting from base 10 to base 8 and back to base 10. I also wrote the multiplication table in base 8. I never showed this notebook to anyone, not my teacher or my fellow pupils. I wish I had now; I might have been hailed as a child genius or at least one who could think for himself.

[@orion6able]

Cool! I used to use base 8! Mainly because I used a kind of binary roman numerals for math and base 8 was way more compact. Using those are how I passed math

[@liteseve]

Seeking validation from others is toxic for your ego. Youll be more content in life when you dont care what other people think of you. Just do what makes you happy (including math)

[@jaskarvinmakal9174]

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.

[@Rayleigheffects]

Hi I’m the first guy to see this and like this

[@CharlesShorts]

at 13:15 I swear I heard "we need to understand greece's economy" LMAO

[@DanielJoyce]

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 😂

[@oliviercomte7624]

There also integer bases with supplemental digits to represent -1 or -2 for instance that are used internally in CPU to avoid carries in order to parallelize digits computation. They are sometimes refered as fractional base.

[@Oli1974]

All your base are belong to us

[@Adventurerrallyzombie]

PvZ references

[@josueantoniofloresmedellin651]

WHAT YOU SAY?!

[@Nemo_Anom]

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!

[@sylv512]

0:16 We use base 16, not base 6.

[@zoladkow]

must be an honest mistake, as in some circles it's called "seximal"... heart wants what heart wants 😉😁

[@Salkauski]

he said we can use them, not that we do

[@MURDERPILLOW.]

Nono he said base 2 is used in computers

[@deleted_handle]

he said IS used not ARE used so only talking about base 2.

[@TheZerothbase]

I've had the screen name "zerothbase" for 25+ years now....precisely for the reason mentioned in this video at 2:55

[@mike1024.]

I have never heard of using number bases outside of natural numbers >= 2. Having never studied a field or read any papers using such ideas, perhaps this fits in the area of mathematics for which we haven't found an application yet haha. Nevertheless, as a mathematician, I found this to be a very interesting watch. My only complaint: This might depend on your country of origin, but I'm accustomed to hearing numerator and denominator rather than nominator and denominator. It might just be my American education.

[@momowo1509]

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!

[@DonkeyScourge]

My head hurts

[@wolfelkan8183]

9:26, that "point" is not a decimal point. It would technically be a "trivotseptimal point"

[@MasterGhostf]

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.

[@dan_2247]

then why not base 60?

[@yurenchu]

I used to think like you when I was a kid. I preferred even numbers, and multiples of 6 , 12 and 60 because they are easily divided by many of the smaller natural numbers. However, as I grew older, I realized that there is actually more beauty in "irregularity" . Whereas "regularity" is monotonous and boring, "irregularity" creates character and is more exciting/surprising. I think it was also in my teenage years that I encountered the phrase "Pefection lies in imperfection" (or something along those lines). By the way, I think that number properties that really matter mathematically, are properties that are not dependent of the used representation system. (For example, {three squared} plus {four squared} equals {five squared} , regardless of whether we write this in decimal, binary, octal, hexadecimal, ternary, or whatever representation.)

[@zix2421]

Это интересно, это великолепно, надеюсь, это понадобится в моей жизни)

[@AK-vx4dy]

@18:00 Why we can't use (2+2i) to power (-1) in this case ?

[@matesafranka6110]

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

[@AK-vx4dy]

Do you have examples where such bases are used ?

[@Brant92M]

Now I'm really wondering if computers can be gently coaxed into operating in base -1-i. Or more generally, if the type of binary arithmetic our machines do applies only to base 2, or if it also works for any number system using binary.

[@lj823]

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

[@perplexedon9834]

4:38 tiny correction, the top part of the fraction is the "numerator", not the "nominator". Probably just a mispeak though

[@nicholasleclerc1583]

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_

[@denysfisher2316]

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

[@ibperson7765]

~ 12:45 Great video but now Im wondering if using the digits 0,1,2 can express every number. If so, could it be just 0,1 used? How to know?

[@Kohlmam]

What a remarkably beautiful system

[@BigDBrian]

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.

[@jsilverwater]

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❤

[@Bob94390]

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?

[@0q9s2]

17:00 Could i be represented in base 2+2i with the digits 1, 0, 0, 1, 61, and then a 4 after the decimal point? This would, in base 10, become -64 + 2+2i + 61 + 1 - i, equaling i, right?

[@MateMagoHacker]

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.

[@rogerc4748]

Another thought. Is it possible to make the exponent of place value (ie 10*1 10*2 10*3) a non integer, or perhaps a member of some set

[@saumitrachakravarty]

4:23 Subtitle is wrong here! The placement of *odd* and *even* is switched. Please correct it. 6:10 and 8:51 Instead of *nominator* if should be *numerator*

[@sirnate9065]

Super interesting. I'm a little confused about the selection of digits for a rational base. Do you always use the larger of the numerator and denominator?

[@nanamacapagal8342]

Yes

[@spieleflo8543]

"Can any number be a base?" Me at 3:00: "Nope, because there's base 0. So why explain all that stuff when the answer is no?"

[@askemervigbahnson333]

I think the number 97 in base (sqrt(5)) at 11:23 should be written as 111020.0201001102000010100102 instead of 30402. I would argue you can’t use digits larger than 2, as having 3 of something is more than having (sqrt(5)) of it, so it would carry onto the next place. I would say 97 can’t be written as a whole number in base (sqrt(5)). Breakdown of that long string of digits: (cumulative) 1 x (25 x sqrt(5) ) = 55,9 1 x (25) = 80,9 1 x (5 x sqrt(5) ) = 92,08 0 x (5) 2 x (sqrt(5)) = 96,55 0 x (1) “Decimals” places: (“squarerootoffiveimals” places?) 0 x 1/sqrt(5) 2 x 1/5 = 96,95 0 x 1/(5 x sqrt(5) ) 1 x 1/25 = 96,994 I don’t think I have to include more “decimals” to show my point, but I got overly enthusiastic and rounded to 22 “decimals”, at which point the number in base sqrt(5) equals 97.000000001 in base 10. Otherwise amazing video tho, I never even thought it was possible to go beyond integer bases!

[@edwardblair4096]

This assumes that we are limiting "digits" to natural numbers. Is there any advantage to allowing rational, irrational, or complex "digits"?

[@MrGatlin98]

These feels like changing the clef in music theory.

[@Balfoneus]

Math is so fucking cool. Only as an adult I’ve come to really appreciate this science and just in awe of its ability to be complex and simple at the same time.

[@josir1994]

With just 0,1,2, how to write 3 base 10 in base pi? Another thing is that when you use up to a-1 for digits in base a/b (a>b) or √a etc, you actually create the problem of multiple expression for the same number, which is not ideal for a number system.

[@ryotoiii]

Wait, so how would base 1 work? How would you go about representing fractions?

[@cielprofondinfo]

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

[@Maker0824]

3:04 I have never heard 0^0 be undefined. I’ve heard it be called 0, and also called 1, and both depending on what works better, but never undefined

[@kresovk5]

My thoughts exactly, although it is just 1. Can't remember the exact place where I've seen it, it was another math channel (asian dude with glasses :)), but he explained it well. Also, some parts of physics and math rely on it being defined value.

[@eviem5658]

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.

[@peter-d9f3l]

Base 2+2i can certainly express of multiples of i, you just need a lot of symbols. For example, i itself is expressed as 0.00000(512) where you need to use the 512th symbol. This is because (2+2i)^-6=i/512.

[@ewengoisot808]

0:47 Real numbers ℝ and imaginary numbers 𝕀 aren't completely disjoint : 0 (∈ℝ) =0i (∈𝕀), so, it shouldn't be disjoint circles. But… actually, it's probably harder to draw like this.