how does addition/subtraction and multiplication/divison work in balances ternary? i cant find any videos related to that. addition and subcration i can figure out, but multiplication i struggle
I think this is one of only like 3 videos on all of RU-vid about this topic, and it is fantastic! Super interesting ideas, and very well-presented. Maybe a sequel which considers other bases? Would be a shoo-in for the next SoME
Honestly I love balanced ternary a lot (used it as the seed of a speculative world building project some years back), and it's a really cool topic to see.
4:44 this is true actually true in every base, even unbalanced ones the catch is that they become infinitely long in both directions and only one of them can be taken care of 57 = …999942.99999… = …999943 …999943 + 57 = …000000 = 0 with balanced bases though, the complement of 0 is still 0, which is Very neat
Most mathematicians will generally say that you can either work in ℝeal numbers with infinitely many digits to the right, or in the p-adics with infinitely many digits to the left, but not both at once. That said, that still doesn't stop every ℝeal integer multiple of each power of the base from having 2 expansions. (P-adics have no such weakness, with every p-adic number having 1 and only 1 possible expansion.)
Yeah, but no buddy. You can *hypothetically* imagine, that a number like …999 existent, but the usual way that ellipsis is defined is “if we keep adding that series of digits, what value will we approximate?”. So 0.999… gets closer and closer to 1, but …999 just gets… big. Infinitely so. As the person before me mentioned, p-Addis exist, but they usually get rid of infinite fraction expansions
A few things: a) There exist numbers in natural languages that use subtractive counting (e.g., 18 and 19 in Latin), so I don't think that it's *_that_* counterintuitive/contrary to natural thought b) 1/2 being .r1 or 1.rT is no different to decimal .r9=1 (as many others have pointed out) c) Using signed bases can quarter the multiplication table's size, making it easier to learn arithmetic (just remember that like signs multiply to a positive number and unlike signs multiply to a negative number) d) If you keep the size of the multiplication table the same, the base will roughly double, allowing for more compact representation of numbers e) The problem with odd bases is that the repetend grows exponentially with the number of factors of two in the denominator, not the initial repetend of 1/2
The fact that 1/2 in balanced ternary is either 0.1111... or 1.TTTT... isn't that much worse than 1 being either 1.0000... and 0.999... in our usual decimal notation.
I mean, even 1/2 is "ambiguous" in base 10: 0.5000... vs 0.4999... It just so happens that we don't write an infinite tail of 0's, so that representation _looks_ shorter.
When you were talking abou the deceased likelihood of a carry, I was thinking "yeah, shiw your work". When i looked back at the screen you were showing your work! Well done!
8:43 Pretty sure this whole process is equivalent to simply rounding the base-10 logarithm of a number. If you peek on the other side of the isomorphism, the whole "compare with the square root of 10" thing is the equivalent of "round up if the decimal part is at least .5". Another fun thing is that if you walk one way through the isomorphism (base-10 logarithm), perform a rounding to the nearest integer, then walk backward through that same isomorphism (base-10 exponential), you automatically get the power of 10 closest to your original number on a logarithmic scale.
So you're a new math youtuber, and your first video is about cool bases! That's cool, totally subscribed! But do you know something: there's not a single video on youtube on bijective numeration. Maybe you could do that next? I know the challenge is, what's interesting about another way to write numbers? But the beauty of it is that every string is a unique, different numbers. This means for instance that all string algorithms now work on numbers (do they do anything interesting on them? I don't know but maybe!). A part of a number becomes something you can work with that behaves well! Leading zeroes means regular numbers don't behave well if you cut them up and paste them together, flip them etc. but all that you can do in bijective number system. And it's much easier to do things like encoding a number in a number when you have this string-number bijection.
"It's fun to imagine what could be, if we stray from the norms that govern things we assume to be simple" This is an extremely philosophically profound idea, which I wish it was easier to communicate to the people who say "but why?" in response to learning to count in base 3, or constructing languages, or imagining new societies with novel hierarchies.
Very clean and professional. Great information and explanation. Interesting topic. Keep up the awesome work! You deserve a large following with content like this! Also do I sense some jan misali influence? o.0
There was obviously some. He showed the thumbnail of the better way to count video at the begging, while making examples for people advocating for other bases
For balanced time you could have -6 to +6 from noon and from midnight, giving a day range and a night range, or it could be from -6 to +6 with 0 at our current 6am/6pm (the bottom of the clock) which would correspond to AM/PM, and 0 could be “bottom of the morning” and “bottom of the evening”, or you could put 0 at the top. Both hands pointing up would be our 6:00. If you use a 24 hour analog clock you could go -12 to +12. At first I was thinking noon would be 0 but technically it maybe should be midnight. Not if you don’t also want to have balanced dates which is a whole other situation. We could have 13 months though so that would be cool, but that would be 28 days per month, great for a lunar calendar, but it’s an even number. You could drop months and use -182 to +182 for days. Well you could still have -6 to +6 months (13 months) you just would use days for the date and months would only refer to the whole month.
The most annoying thing with balanced bases is, For example, we want to map a range 0 through 1 represented with 8 bits in base 16, You can already see where that is going to fail... 00 -> 0.0 FF -> 0.99609375 100 -> 1.0 There's not enough information for the third digit because that's 9 bits. So just "smash it" and define FF as 1.0 and then what happens? You can't have 2^(n bits) subdivisions with a closed interval without an unfactored quantization. So I.e. there's no value representing 0.5: 00 -> 0.0 7F -> 0.4[9803921568627450]... periodic repeating infinite decimal 80 -> 0.5[0196078431372549]... periodic repeating infinite decimal FF -> 1.0 That's why FF usually represents a terminal 0.99609375 of a right-open interval.
There are solutions to the problem of storing exactly one half if you use a representation other than real numbers. In computing terms, floating-point values aren't meant to be semantically faithful anyway. A mature balanced ternary ecosystem might render 2.5 as 1T+1T^T or 1TT/1T, only expanding into 1T.1111111... when strictly necessary for arithmetic with irrational numbers or exact real values.
Interesting! Really, though, subtractive enumeration is not so strange. Roman numerals use it: IIII vs IV, for example. And Roman numerals make some sense when thinking about using a physical pan balance: IV means to place 5 units on the empty side and 1 unit on the side with the item being weighed, to end up with 4 units of something.
I don't think we would have _that hard_ of a time communicating balanced ternary numbers if our languages had an appropriate naming scheme for them. 0.11111… would indeed be a pain though!
The issue with that is, that you can’t immediately see, if the value is larger or smaller to other fractions, when written as a/b. If you see something like 1.1TTT1 and 1.1T1T11 you just have to look at the numbers to notice, that the second one is larger. It would be way harder if you can’t write it the the… (not-necessarily-)decimal-point
10:08 Well, in some languages, notably European ones. In East Asian languages the number system directly reflects the base-10 representation (e.g., 1024 = 1 thousand 2 ten 4, 1000000 = 1 hundred * (ten thousand) ), and reading the time is identical. The only case we use "X min to Y" is when X is very small (5 min or less).
Great video. I'm not at all a math guy but I've been playing with balanced ternary on my own quite a bit (since resources on it are scarce) and its been one of my favorite learning experiences. Love that you actually crunched the numbers on carry rates, something which I could never prove but just came to expect while using it.
This is a great video, but there's a conversation I wanna start about radix economy because as far as I can tell, there are competing definitions for it. In chapter 1 of the video "the best way to count" they talk about how binary actually has the best radix economy because the leading digit of any number in any base has one less possibility than the rest of the digits, since leading 0s dont convey any information but 0s between non 0 digits do convey information. (forgetting about sigfigs, that's not important rn) This means that in a binary string, the leading 1 can only be a 1, meaning it conveys 0 information, and can be encoded by implying its existence and only writing the rest of the number. The video probably explains this better than I did. I've been convinced by "the best way to count" that binary has the best radix economy, but I'm also probably not that knowledgeable on it and wanna hear what other's think. I'm gunna try to post the link for the video in a reply on this comment, but I think youtube blocks links now, so if you don't see it, that's why. The time stamp for chapter 1 in "the best way to count" is 11:41 Edit: typos
I think a balanced base actually could be fairly intuitive in spoken language, due to perhaps its most important property: you can accurately approximate a number using its most significant digit. This is helpful for quick mental math. It won't be so bad to keep track of adding and subtracting, because we can express the negative digits in terms of positive digits. Converting ternary 1T11 to decimal 64, but understanding and using t1T11 intuitively is not! "This device costs t1T11 dollars? That's about t1000 (about 81), not bad!" In a balanced positional system, you should be able to do any basic arithmatic with just the first digit or two and ignore the rest and get more or less the right result. It's not like this is wholly unrealistic; roman numerals are a prototype of a balanced system (just not a positional one.) IV, IX anyone?
I will have to agree with you. Although it might be hard to relearn from base-10 as an adult, I think learning balanced bases to children from an early age could be just as easy as learning base-10. This is because magnitude representation in the brain is not bound to any base (but rather works on ratios when comparing that are more analog in nature)(and can also be found in animals!). Therefore balanced bases are just another way of converting symbols into these magnitude representations, and humans are incredible at processing symbols (just think of the complexities of language). I would love to see research on this!
The other place where a balanced numbering system would be useful is that a balanced base-255 system can be implemented on a modern computer in a way that allows the efficient storage of integers of arbitrary size (well, limited by system memory). Using the zero byte value as the terminator (just as is done with C-style strings), and the byte values from 1 to 255 map to the range -127 to +127 (the stored value is 128 greater than the value represented) makes storage and parsing easy. The calculations, and conversion to decimal for display, will be a PITA, though.
we already have signed integers (the 128-valued bit is on when the number is negative), so storing them this way will fit better (as opposed to a blanket x+128)
Re how to teach such number systems Might be interesting to talk about numbers as being made/crafted, like one would a physical object. Think carving away parts of a blank when making a sculpture. Start with some raw material (over counting), transform it by removing/adding parts (plus and minus) to get your final sculpture (the actual number) Makes numbers feel like they should be played with and manipulated right from the get-go, as opposed to just relating to quantities You get the quantity tracking aspect by applying a certain manipulation To me, it makes working with numbers the focus and not the numbers themselves like it is now
FWIW, I wonder if one possibly neat and aesthetic way to write numbers in this format would be to use color coding, like how it is done on banks today. You use a black "1" to represent positive, and a red "1" to represent negative, while "0" is always black. I also thought of this because negative numbers were first created in China, and the Chinese used black and red rods to represent positive and negative, though actually it was the other way around (i.e. red as _positive,_ not negative).
Actually binary is better in radix economy than trinary if you count the information density properly (the naive way to do it forgets the fact that the first digit of a number has one less posibility than the rest of the digits as it can never be zero, because while 102 is different from 12, 012 is the same as 12 is the same as 0012. So in decimal every digit has 10 posibilities but the first has only 9. This is not taken into account inthe naive way which is how it is done on wikipedia which is how that myth propagated). If you look up the video "the best way to count" it shows the derivation of this and other information about binary.
That final with multiple representations does happen in unbalanced bases tbh. In actual mathematics they often pick the non termination representation purely because that's the property that you can arrange all numbers to have.
imo negative bases are also somewhat elegant; in base -10 you can write every number you can in base 10, except you never need to ever negate, it's a built-in feature, just like in balanced bases. except in base -10, every other digit is negative so, 0-9 are written 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 as normal. 10 is then -10, because it's 1 * (-10)^1 + 0 * (-10)^0, so how do we make positive 10? we need to overshoot and subtract, the next place is the 100s, (not negative 100s, positive 100s) so +10 is written 190, because 100 - 90 = 10. +20 is 180 because 100 - 80 = 20, and so on until you hit 100 + 99 is 119, 100 - 10 + 9, 100 is just 100, and +899 is written 919. the 1000s are negative again, so to write +1000 we need to overshoot again, to write 19000, because 10k - 9k = 1k. interestingly, this also means to get 999, you'd need to write it as 19019, because 10k - 9k + 0 - 10 + 9 is 999, so the radix economy is a little all over the place.
A possible issue is that, when you see really long numbers, you’ll always first spend time counting the digits, to figure out if it’s positive of negative. Imagine your company looses money, because an intern typed an extra zero and your business partner just… didn’t see the problem
Please don’t. Base e needs infinitely many digits, if you want to represent all integers… Except for engineers. I guess they wouldn’t see the difference between base e and ternary
As a fast way to turn from unsigned to signed just subtract 10 from each digit above half of the base the same way you have 30 minutes til the next hour, and just like clocks have different bases for each different digit, the first being 12/24 and then 60,60, to convert it to seconds you need to do all of the math, it is just as intuitive as in: you have to get used to it. So 6 becomes 1(-4) and 25648->3(-4)(-4)5(-2) in signed decimal. Alternatively you can use the already there digits and just write 36658. Someone proof check for errors.
This is an awesome video subscribed. I have one issue though. You talk to fast and sometimes your words don't have gaps in them. This is perfectly normal when you are having a conversation with someone but for an educational video it add a lot of overhead. I had to turn on the subtitles and pause a few times to understand what you were saying.
I recently discovered balanced bases, which I'd never really thought about before and there's much good to be said about them, though being somewhat new to the subject, I still have certain fundamental question, and it is not as intuitive as I'd like it to be. You just mentioned that .111... and .TTT..., are equal, which stumbled me for a moment, though just a moment, because this of course also happens in even bases, just differently. For example the resulting value of 9 * 1/9 is perfectly obvious, whereas 9 * .111... is more tricky. Is it 1 or .999...? Well, the general consensus is that they are equal, but still, they're not. I suppose my question in this regard is if there are any, let's call them "complete" number systems, in which you can represent any value, where this cannot occur, where any possible value can be represented, in one unique way? I have a hard time seeing how. Perhaps prime bases, maybe even balanced, some combination of offsets and factors? As I said, I've can really see how. Next up, I miss a clear procedure to construct and deconstruct a balanced ternary number. 2*3²+1*3^1+2*3⁰ is perfectly sensible and there really is only one way and if the resulting value is to big, then one of the digits is position too much to the left, or the digit is too high and as long as the leftmost digit is just before which would, the position of that digit at least is correct, while the digit itself might still be wrong, and so on. Complicated though this may sound, the procedure really is quite clear, but not so with balanced bases, as far as I can figure. Take the decimal value 28 or 1*3^3+1*3⁰, easy, perhaps even intuitive, and the same goes for 26 1*3^3-1*3⁰, but how about say 13 or 14? they should be similar, being close and all, in binary 1101 and 1110, but they are not, 0111 and 1TTT, or are they? In the first case the leftmost digit is lower than the resulting value, and in the second case much higher. Is it just a matter of getting used to it? In any case, I have still to find a clear an concise procedure or algorithm Lastly, and this is where my greatest problems are to be found. Efficient binary coded balances ternary. I cannot for the life of me figure out a way that does not involve expensive operations, such as division end modulus, well, they are the same really and a lot of branching, or alternatively some 25% wasted potential, in term of what might have been represented in x number of bits. Certainly there will always be some waste, as the two bases does not quite add up, but not necessarily a lot. In 8 bits just 13 out of 256 potential numbers, or about 5%, is wasted, and on top of the, 8 is a number that computer like to work with, usually. Also, unless 27, 48 or 65 bit computers become all the rage, that's really the best you can do. however, the question remains, how on earth do you pack 5 balanced ternary digits into the space of 8 binary digits? Remember, there is actually plenty of room to stretch, though not enough that each ternary digit can use 2 binary digits, which would make things rather easy. Apparently there are ways of doing just that, only no explanations as to how, at least that I have been able to find. At an y rate, thanks for the effort :D
"How you communicate this intuitively": Don't you know the way number above 20 or 60 are spoken in French ( 70 is soixante-dix meaning 60+10, but 80 is quatre-vingts meaning 4*20, 99 is quatre-vingt-dix-neuf meaning 4*20+19 a weird mixture of groups of 60 and 20) or how time near he half hour is spoken in German (hint 3:34 is something like 4 past half to 4) So being intuitive is not really a condition in language.
No, that’s not how you say 3:34 in German. You only use it for rough values, 4 is rather specific. However, x:35 would be “fünf nach Halb”, to some people, which translates to “five past half”. In general, it’s quite common to say something within the form “5n before/after hour/half”. But, to pretty much anyone I know, 3:34 would essentially be the same as 3:35 (actually, Germans aren’t *that* strict about timekeeping) and would just say “fünf nach Halb”
@@Nihil2407 Probably not in current use but is vernacular. I heard fünf vor halb drei or fünf nach halb drei or similar several times. And when minutes are important I heard this form for other numbers smaller than 5.
you didn't finish the thought. balanced bases reveal a deep issue with conventional philosophy of mathematics, which is namely the notion that + and - are operations. we can see from simply carefully examining conventional notation that + and - are never operations, for instance: 4 +3 = 3 +4; this is referred to as the 'commutative property of addition', and is why addition is said to be abelian 4 -3 = -3 +4; this is taken as proof that subtraction is non-commutative, or non-abelian, but... look closer the - is stuck to the 3, and when the 4 moved out of initial position it's suddenly prefixed with a +. and if you've ever done any amount of algebra you know that this is the generic behavior of terms. the + and - are absolutely never operators, but always the signs of the term they prefix. which means that expression-initial positive terms are not being marked as positive because of a grammar rule in the standard orthography of mathematics. that is, if an expression-initial term is positive then its sign doesn't need to be indicated. thus we get: 4 +3 = +4 +3 = +3 +4 = 3 +4; now the commutativity is not a property of addition, but of listing terms generally 4 -3 = +4 -3 = -3 +4; and that's confirmed here and this has further implications. for one, it means that addition of unsigned numbers doesn't even make sense. so the successor function which underpins the Dedekind-Peano Axioms is definitely just nonsense, and so is the proof by Whitehead and Russell that 1+1=2. and everyone should have noticed this in the first place, since the majority of mathematics exists specifically because 1+1 has no general solution. for example: 1 dog +1 dog = 2 dogs; this is the only example Whitehead and Russell considered in Principia Mathematica 1 dog +1 quail = 2 wings; so 1+1=2 because 0+2=2... that's definitely different 1 dog +1 quail = 6 legs; so 1+1=6 1 foot +1 yard = 48 inches; so 1+1=48 because unit conversion 1 half +1 third = 5 sixths; so 1+1=5 because fraction addition 1 frog +1 pond = 1 pond; so 1+1=1 because frogs live in ponds 1 stone +1 mountain = 1 mountain; so 1+1=1 because mountains are made of stones 1 C water+1 C dirt = 1 some mud; so 1+1 is between 1 and 2 because fluids can fill the spaces between grains of dirt 1x +1y cannot be simplified; 1+1=1+1 because of like terms and if you want to object on the grounds that Logicism came back later to address the problems of units via type theory, or some such nonsense, no... notice that this means that 1+1 only reliably yields 2 as PM predicts if the units on all terms are identical. which means that it's not even possible to add the number 1 to the number 1, let alone get the number 2 out that. the only time addition can occur is over vectors, the combination of a unit and a number. this then poses problems for notions like Naturals vs Integers, because the Naturals are supposed to be unsigned bare numbers, but they're also identical to the positive Integers... despite not having any sign by definition... you see the problem here? and the source of this confusion is failing to notice the grammar rule about expression-initial positive terms. which is ironic since a number of the founders of Logicism, including Giuseppe Peano, considered themselves expert linguists.
Natural numbers are called as such because they represent quantities. -4 pieces of gold makes not sense outside the context of debts. Now, since debts already have a negative connotation A debt of 4 pieces of gold and a capital of 3 pieces of gold mean the same as -4+3 pieces of gold. In that sense addition and subtraction exists as verbs. Put in another way, they exit as how 5 year old are taught. As for the successor function, Dedekind only needed to define the positive one since the negative one essentially follows the same logical construction. The example equation you gave is a matter of units. Equations with different units only make sense if you can convert them in some sense, i.e. a function that converts different units into a common one. 1 dog + 1 quail = 2 wings or 4 legs only make sense with the contexts you gave. At that point, that’s not the fault of the ZFC axioms, but rather yours, for not explicitly saying what’s being counted. With that in mind, the following are obvious: * 1 + 1 = 2 iff the terms in the left hand side are the same. * 1 + 1 = whatever iff the terms in the left hand side are the NOT same That’s what everyone who’s been doing arithmetic have been doing.
I'll still never understand why people say nonsense like "quarter to five". Just say "four forty five" like a normal person. I'm asking what the time is *now* , not what it will be or was in some fractional hour. It isn't even any easier to say (note: both sides of the example I used have four syllables, thus take equal time to say). It comes from the same kind of people that will say "half dozen" instead of "six". I get it, you know what a fraction is. I'm not impressed.
This is great, I was trying to think about how (1000 - 1) * (1000 - 1) should evaluate faster by hand than 999 * 999 by the property that even though there are more digits in the first expression, it is simpler to multiply, add, and subtract 1's and 0's than it is for 9's.
2:39 does anyone have good resources on how trits are actually encoded in hardware? With binary it's an on/off but I'm having trouble understanding the trit equivalent described on the Wikipedia page. I'd be interested to know about trinary logic gates and whatnot that are used to construct higher level computing.
Either you risk ambiguity by using 3 distinct voltage levels, or you waste space by using 2 or 3 wires. Note that this is related to why some people claim the standard way to calculate radix economy is wrong and that it should actually be strictly decreasing over the positive numbers, making binary the most efficient usable base rather than 3. You could argue that many hardware circuits are actually a form of ternary with 3 distinct signals on each wire, but rather than representing a ternary digit, the third value is more often used as "no signal" (more formally: "high impedance") and it has very particular physical properties that make it useful for this, but less useful as an actual data signal.
Ternary typically adds a low-low state to the binary high-low and low-high states of a signal pair (with low-high now representing T instead of 0). Paper tape typically used 2 hole positions per trit, with both positions being punched either disallowed, read as an additional 0 state, or used for out-of-band signalling. This format also led to the binary-coded ternary still used by some CNC machine tools.
@@adiaphoros6842 that would be amazing, imagine if 0 is the "no value" state, then 1 + (-1) would physically be an electron and positron annihilation and you would be left with nothing, a literal "no value"
