really well put together video! these are some very compelling arguments for binary. you're right that I dismissed it for superficial reasons without second thought in my videos; the immense advantages it has for arithmetic and information density shouldn't be overlooked, and there definitely is a case to be made that these may matter more than the things I focused on. in the few contexts where the notion of an "objectively best base" actually makes mathematical sense as a thing to care about, binary is a clear winner. I don't think I'm completely convinced that binary is necessarily the absolute best choice for a human-scale base (the "coincidental" advantages seximal has for working with small primes are just too good) but I am convinced that it can work as a human-scale base to begin with, which I hadn't even properly considered before. it definitely deserves a seat at the infinite table with the other SHCN bases.
@@wilh3lmmusic thats basically what the video said, you can combine 2 or 3 digits of binary to make it shorter like base 8, but with all the advantages of base 2
@@leggyjorington3960 Not exactly. The grouping system in the video is distinct from hex or octal in a few ways. First of all, if you just want to use hex, then the notation suggested is very inefficient. I'm not a big fan of how we usually write hex (0,1,2,3,4,5,6,7,8,9,a,b,c,d,e,f) but it's much more efficient than iiii iiil iili iill... etc. The same is basically true for octal. Second of all, the advantage of his system is that it isn't hex or octal, it's both (and any other 2^x base). You can very easily re-analyse a number as any of those bases to get its benefits. Sevens aren't great in hex, but they are easy in octal, so just group by three bits. Octal sucks at fives, so just group bits by fours and treat it like hex. Wilh3lm is, imo, correct that the benefits of the more flexible system are outweighed by the practical issues with actually writing these numbers. I spent several months using almost exactly the notation suggested in the video and it simply isn't practical.
I frequently see children using this system to express the number 4 to me when a school bus drives past. It's truly amazing to see something adopted with such enthusiasm at a young age and gives me a lot of hope for future generations.
Another fun fact, since microprocessors out number humans, and most of them do arithmatic significantly more often than the average person does, technically binary is the most often used base currently.
I think I found my favorite genre of videos: the hour long math rabbit hole. Videos such as this one, "HACKENBUSH: a window to a new world of math" by Owen Maitzen, or " The Continuity of Splines" by Freya Holmer. Really loved the use of music and sound in this one.
I love both of the videos you mentioned so much, if you have any other similar ones to share I'd love to know! I can also go through my favourites and find similar ones if you want me to tomorrow ^^
@@codenamelambda The recent 3b1b explanations of light slowing down when travelling through matter are definitely a must-watch, if you haven't seen them already. Oh, and also the "Why can't you multiply vectors?" by Freya Holmer
Hi, not sure if anyone’s mentioned this yet but your use of pitches following the harmonic series to accentuate numbers you’re talking about is absolutely incredible and I didn’t want it to go unnoticed.
This point is addressed in the video. Using just three fingers on each hand still lets you go further than seximal, and binary is the only base where you can use any amount of fingers for finger counting
The amount of times I got caught by the “‘well actually’ -You“ moments scared me. Every time I felt like I had a valid argument to make, there was a direct response to it.
James Grime convinced me of 12, almost 12 years ago Misali of 6, almost 6 years ago You just convinced me of 2 Who said there was no progress in history?
in 2 years some will convince you of base 1 and then 1 year later someone else will tell you base 0 is better, on which there will be a imediate response claiming base -1 is optimal, while then you realized that you actually had memories of someone claiming 1 year ago that base -2 was better and so on
The bit about the square root algorithm made me literally get up out of my chair, scream "what??" at my phone multiple times, and roam around my apartment for several minutes rethinking life
honestly, this video has now converted me into a true binary supporter also i did not expect all of the arithmetic stuff with binary to be SO simple and easy to do
re: chapter six. i'm not sure your naming scheme really does a great job here. it definitely feels like "ninety one" is less information to parse than "four one hex, two four, three", and in general, standard names tend to feel less cumbersome. perhaps there's a less mathematically elegant, but more linguistically practical, way of handling binary number naming.
"four one hex, two four, three" is more analogous to "nine ten one" like some languages do. For me it's more natural with -s on the things that should be plural, so four one hexes two fours three. (like "nine tens one") I think something that could help solve the cumbersomeness is basically doing base 16. So make all numbers less than hex compound words, so you can treat each as a single unit. Like ninety one would be four-one hexes two-fours-three. Compared to base 10 English, the digits are more syllables, but that happens in other languages. Like in greek the numbers up to 10 are all 2 or 3 syllables. With this system the numbers up to 16 are all 1, 2 or 3 syllables. The fact that it's base 16 means it's actually less cumbersome than base 10 for large numbers. It also keeps all the elegance, since it's just a conceptual reframing to help interpretability. There's still a slight problem that, for example, "four-one hexes two-fours-three bytes hex one" could be read as "four-one hexes" + "two-fours-three bytes hex one", you'd need to say "(four-one hexes two-fours-three) bytes hex one" somehow. I think this is just a problem that comes with the b^(2^n) system, you might have to switch to b^(kn), maybe 2^(4n) so powers of hex. Tho you could have the power names be like hex, byte, hex-byte, short, hex-short, byte-short, hex-byte-short, int and so on, so it'd be "four-one hex-bytes, two-fours-three bytes, hex, one".
@@d.l.7416 in written form, using the 2^(2^n) system, your latter example isn't technically ambiguous. but when spoken it's unclear what order of magnitude is being referred to at any point until the number ends, because at any point while telling you the number, someone could just say "short" and suddenly the number you were thinking of is 16 orders of (base 2) magnitude too small, etc.
my proposal for naming numbers is to simply read off the bits, using short syllables that can be flexibly strung together and said quickly. If 1 is pronounced like "wun" or "nun", and 0 like "oh", "wo", or "no", then a sequence like 1101 1001 becomes "wununowun wunowowun"; and you could optionally insert the magnitude words like hex between gaps. This mirrors the written form and retains its advantage of being able to group digits into whatever sized chunks are most convinient.
Hearing "nine ten one" in English is a little hard to parse, but in my native Hokkien where numbers work exactly like this, it's perfectly natural (九十一 káu-tsa̍p it). I think this just goes to show that it's all a matter of getting used to it. The "four one hex two four three" system is perfectly fine.
@@jfb- I think that would become very confusing very quickly. People already mishear fifteen (wunununun) and fifty (wununo nowuno) in decimal numbers, and you're expecting them to parse wununowun from wunonunun in fast speech (not to mention that both of those numbers are a mouthful). At the very least I think you should pick sounds with greater contrast. Maybe something like ko and mi, where the sounds of each differ in as many features as possible. You'd probably still have people getting mimikomi and mikomimi mixed up if you rattled off a couple of mikokoko-bit numbers, but it would be an improvement.
In my humble opinion this video goes into the youtube's mathematical hall of fame. A deep and new point of view shedding light on a topic that everyone can relate to yet few thought consciously before. I have no other words than to thank you for your work.
Sadly, any way of speaking base-2 numbers will be non-portable. Some ancient cultures will say that int is 2^16, while modern ones say 2^32. And then there's the oft-forgotten cultures where byte (or char, as they'd say) is e.g. 2^31. Instead the names for the double powers of two should follow a simple and memorable system, e.g. 2^16 could be "int underscore least sixteen underscore tee". Since that's a mouthful you should also introduce "intmax underscore tee" as shorthand for "the biggest power of two I feel like thinking of right now".
just use the x86 names: byte, word, dword, qword / quad, and dqword / double quad. i can't remember what the next two were but i could've sworn it was ddqword and qdqword. super convenient and not confusing at all
@@rubixtheslime after qword is either oword (octa) or xmmword (128 bit simd register) after oword is xword (hexadeca) or ymmword (256 bit simd register)
I'm incredibly surprised this video didn't mention that the GREATEST advantage binary has, is that it's a system we could ACTUALLY switch to without nearly as much hassle as any other system. I'll be honest, I went into this video thinking "huh, interesting. Id like to see a new point of view", got to the twist reveal that it's about binary and went "ok, this is either a joke or I'm in for an interesting if unconvincing response", but now you've really convinced me. Holy cow, I had no idea what I was in for. The counting is fun too since I don't have to remember so much, and somehow these numbers are easier to understand with my dislexia too. They could also be made easier to understand for dyslexic people with a few simple tweeks so that's comforting. And on top of all that, binary numerals would be so fun to make fonts for, as you can pretty much make the symbols whatever you want as long as one is "less" in some way than the other. Hollow/full circle, down/up arrow, Mario/Luigi, literally infinate options lol. Anyways awesome video, please make more! I'd absolutely love more specific video lessons on how to use binary with your numeral and naming systems!!
I really don't like the way that binary is proposed to be written in this video. The bottom connection thing is actually really nice, but the whole "short ticks for 0, long ticks for 1, and downward short ticks for the radix point" thing seems like it'd be really prone to accidental slip-ups and unnecessary ambiguity, especially without some sort of guide on the paper.
Agreed. A good way to distinguish them could be to write 0 like the lowercase Greek gamma and 1 like the cursive lowercase L. It would be fast, less prone to errors and easy to read.
@@polymloth I just sat down to try and find a system based on this comment, and this was what I came up with lol, connecting adjacent characters like cursive. This is a pleasing analog of chiral topology, something like over- and under-crossings in a knot diagram. You can use word breaks to indicate digit groups, and a slash to indicate fractions as usual (I don't actually think having a positional notation for fractions gets you much idk).
the thing is, if natural languages have no issue making this distinction (take the word יוון in Hebrew), this shouldn't be an issue with numbers. we can also always take inspiration from Hebrew crossword solvers and turn the short tick to a short cross.
I think language is the part where I'm least impressed here. There are studies suggesting that languages that express numbers in fewer syllables lead to people doing faster mental arithmetic. A lot of languages have specific words not only for the singular digits 1-10, but also for some numbers into the teens. English has "Twelve", for example, instead of "Twoteen". French has a single digit words for eleven, twelve, thirteen, fifteen, and sixteen. And I think there's also some linguistic benefit to having words for numbers like twelve rather than "ten two" than just a historical linguistic artifact--small integers are just the numbers we will use the most, being able to express them quickly and unambiguously is inherently valuable, even if it makes the language harder to learn than Toki Pona. It's also worth noting that languages tend to have specific words for 20, 30, 40, 50, etc. In English: twenty, not "two ten"--now there's no syllable advantage here (although in some languages the word for 20 is one syllable) but there may still be a linguistic advantage to having a separate word--even if hearing is an issue, you'll never mistake "twenty" for "ten" or "two". Rather than express number words in base 4, I think it would probably be advisable to have language work in...probably hexadecimal honestly? Either that or Octal, but I would lean hexidecimal for two reasons. First because it's 2^4, and 4 is a power of 2, so it'd be easier to deconstruct if you were already thinking in binary. Second having number names for numbers higher than 10 is fairly reasonable, cause a lot of languages already have that (1-12 in English and German, 1-16 in French and Spanish). But honestly, this is all hypothetical, really. People are never going to switch number bases unless governments force it by changing money to a different number base. As long as people have 10s, 20s, and 50s in their wallet, they're going to think in decimal.
Finaly managed to crack the text at 8:20 on my own. The substitution is 啊 = A, 痹 = B, 雌 = C, 低 = D, 婀 = E, 付 = F, 佮 = G, 喝 = H, 乙 = I, 咳 = K, 刕 = L, 冪 = M, 妳 = N, 我 = O ,仳 = P, 儿 = R, 絲 = S, 偍 = T, 無 = U, 予 = V, 劸 = W, 牙 = Y. And the actual text is that one comment from jan Misali's Ido video: " YOU HAVE GOT TO BE ABOUT THE MOST SUPERFICIAL COMMENTATOR ON CON-LANGUES SINCE THE IDIOTIC B. GILSON. DID I MISS THE ONE WHERE YOU SAID WHICH CONLANG YOURE FLUENT IN AND READ AT LEAST THREE TIMES A WEEK AND CAN READ NEW BOOKS IN EVERY WEEK OF EVEN ONE YEAR OR LISTEN TO RADIO SHOWS IN EVERY WEEK? NEW RADIO SHOWS? "
Great video. Although the "speaking system" is quite flawed. So i made a new one. Firstly, the biggest flaw is recursiveness. It's new and neat, but when you want to say a number to other person, it's better to convey number's magnitude right away. For example in decimal you would say "world population in 1975 was 4 billion and dot dot dot". However in binary it would be "three four three hex two BYTE two four one hex four three SHORT dot dot dot", and only when you say SHORT the person can sense the magnitude. As a follow up, what if the person is writing the number down? For example when he hears "four int two..." how many zeroes should he put before writing down "two"? If the number is "four int two short" then 14 zeroes. If it's "four int two byte" then 22 zeroes. If it's "four int two byte short" then 6 zeroes. Secondly, phrases are a bit bigger. A small number in decimal (255 - "two hundred fifty five") would be "three four three hex three four three": 22 symbols versus 37 symbols (or 6 syllables versus 7 syllables). We sometimes omit hundreds, so it's minus 2 syllables. Thirdly, "speaking/writing system" interferes with the idea of grouping bits into groups of 2, 3 or 4 bits. System works with groups of 2 and 4, but does not with groups of 3. As it's said in video there is a learning curve, where person first learns arithmetic on group2 then group3 then (maybe) group4. But what if he considered group4 arithmetic too complex and stopped at group3 ? After he's done calculations on group3 he has no choice rather than regroup the whole thing and only then say the number out loud. For the first problem I'd kinda go traditional method (millions, billions, trillions etc.) For the second problem I'd compress numbers up to hexadecimal digits For the third problem I don't know. Either group3 people will have to regroup, or make another speaking system for group3 representations (which is quite bad). The digits Imho it would be worse to use digit name, that we already use, so I gave new names, trying to reflect "binariness". Also these numbers should be fast and easy to pronounce and phonetically distinct from each other , because they will be used a lot in speech. I'm not a conlanger, but I tried my worst wan du ti ro rówan ródu róti ko kówan kódu kóti kro krówan kródu króti hes Here ó is a stressed o. These words represent names for following hexadecimal numbers: 1 2 3 4 5 6 7 8 9 A B C D E F 10 I used "hes" instead of "hex" for sixteen, because I think it's faster to pronounce it this way. For 0 we could use "zero" I think "r" prefix for 4 and "k" prefix for 8 work quite good with binary. With this system we can count up to 255 (in a similar way we count up to 999 in decimal without involving power names, such as thousands, millions, billions). "hes" is used as a connecting word between quartets (similar to "hundred" in decimal). If the right half of number is zero, "hes" is omitted. ........ - zero .....|.| - rowan |||||||| - kroti-hes-kroti ...|.... - hes ..|..... - du-hes |..|.||. - kowan-hes-rodu Now for power names. Let's call collection of 8 bits as bunches (not bytes cuz we will use this keyword). Bunch in decimal would be 3 digits. Every bunch in decimal is followed by a power name (thousand, million, billion...) If we do the same in binary with suggested names (byte, short, int, long, overlong, byteplex), these "power names" (technically power phrases) will be like that: 256 ^ 1 = byte 256 ^ 2 = short 256 ^ 3 = short byte 256 ^ 4 = int 256 ^ 5 = int byte 256 ^ 6 = int short 256 ^ 7 = int short byte 256 ^ 8 = long (We suppose that most significant words come first) Here I would suggest other naming system (which in general will have more syllables). However each power name will be represented with one word, pronouncing these names I think will be easier, since "int" "short", "long" are not pronounced well together. Also every name will end on "-yte", indicating, that it is indeed a power name: 256 ^ 1 = byte -> byte 256 ^ 2 = short -> plyte 256 ^ 4 = int -> fryte 256 ^ 8 = long -> ksyte 256 ^ 16 = overlong -> znyte (I did not come up with an alternative for byteplex) Now instead of "overlong long short byte" we would get "znyte ksyte plyte byte". But we need to combine these words. The rules are: 1) Last word gains prefix "o-" 2) Other words turn into prefix form The prefix forms are: plyte -> pil fryte -> fer ksyte -> kas znyte -> zun So by these rules "znyte ksyte plyte byte" will convert into "zun-kas-pil-o-byte", or "zunkaspilobyte". Here are first 14 power names: byte plyte pilobyte fryte ferobyte feroplyte ferpilobyte ksyte kasobyte kasoplyte kaspilobyte kasofryte kasferobyte kasferoplyte And now one example with all of this: distance to the Sun in nanometers: |... ...|||.. ...|.||. .||..||. ||..||.. .|.|.|.. .|.||... ..|..||. .|.|.||. "ko ksyte wan-hes-kro ferpilobyte wan-hes-rodu feroplyte rodu-hes-rodu ferobyte kro-hes-kro fryte rowan-hes-ro pilobyte rowan-hes-ko plyte du-hes-rodu byte rowan-hes-rodu"
I'm pretty sure if we had developed binary & hexadecimal counting, we would not be translating numbers from base 10, or from metric, especially since considering the metric system was developed in a world where base 10 was well-estsblished. What if we had established a measuring system based on 100,000 [in hex) of the diameter of the earth through the poles? The "hex stick," if I may, would be about 16 cm in length. A little small, but still usable. As much as we like to make up words, I'm sure we'd still have million and billion, or an equivalent in hex language & a little different in magnitude. Mil & bil are kind of arbitrary labels, but if hex had an equivalent, you could again feel the magnitude. Let's just pretend for a bit that hun, thou, mil, & bil applied to the number in the 3rd, 5th, 7th, and 9th digit respectively. Distance to the Sun would be about 37 milhex, if I may. A population of 4 billion would be EE milhex, which is just under 1 bilhex. Magnitude problem solved. 🙂
@@livingpicture Well yeah, we solved magnitude problem in our ways. But thou, mil, bil are names for 3*n hex digits (or nibbles) which is kinda arbitrary. I tried to follow the idea in the video: to establish names only for 2^n digits. The traditional approach (3*n digits) is probably easier to remember, but idk, I didn't learn my system :)
I kinda prefer this (but what do you guys think about this?): 0 = zero 1 = one 10 = 2 = two 100 = 4 = four 1000 = 8 = eight | 1111 = 15 = fifteen (there should be alternative names for 10[dec] to 15[dec]) 1,0000 = 16 = *hex* 10,0000 = 32 = two hex | 1111,0001 = 241 = fifteen hex one 1,0000,0000 = 256 = *byte* 1111,0111,0101 = 3,957 = fifteen byte, seven hex five (fun fact: this is 3 syllables shorter than its decimal name) 1,0000,0000,0000 = 4096 = hexabyte (or hexbyte?) 1111,1110,0100,1001 = 65,097 = fifteen hexbyte, fourteen byte, four hex nine 1,0000,0000,0000,0000 = 65,536 = *"short"* (there's probably a better alternative name for this we can borrow from Computer Science) This gets absurd... 10,0000,0000,0000,0000 = "two short" 1,0000,0000,0000,0000,0000 = "hex short" 1,0000,0000,0000,0000,0000,0000 = "byte short" 1111,0000,0000,0000,0000,0000,0000,0000 = "fifteen hexbyte short" Maybe we should think of an alternative for "hexbyte"?
You present your arguments well, however: I am nonbinary. So I'll have to stick with Seximal, or maybe Balanced Ternary for the small number and negative advantages
Perhaps you would like base negative two, or negabinary? It uses alternating negative and positive powers of two so negatives can be written just like positives without the need of a minus sign.
I think the proposed method on how to say the binary numbers has a couple of major drawbacks: 1) The recursive, non-linear conversion between words and symbols makes it hard to dictate, and non-trivial to write a dictated number down. 2) The symbolically easy doubling becomes unintuitive, e.g. 3 4 1 doubled becomes H 2 4 2 3) It can bury the most significant part towards the end, for example 3 4 2 H 3 4 2 B. You have to listen to all of the spoken numbers to make sure you're even in the right order of magnitude.
i think 1 and 3 come from the fact that it's a 2^2^n system instead of a 2^kn system. you could instead make it a 2^4n system (powers of hex), but name the powers using the 2^2^n system. So hex, byte, hex-byte, short, hex-short and so on. like you'd say for example, three-fours-one hex-bytes three bytes hex two-fours, which is basically just how standard english base 10 words.
Take a cue from the world of programming, where our spoken numbers are base 16 but our math is base 2. If we used the numerals from this video it would make the conversion trivial.
Don't you love it when you have that one really strong opinion that no one else cares about, but then you stumble across an hour long video essay about it at 3 am
I always felt, instinctively, that binary "should" be the best base, but it just seemed too cumbersome to use in practice. Thanks for your excellent work.
I went into this with a strong preference for hex. I got pleasantly surprised by the suggestion of binary, and when you started grouping the bits it's essentially a multi base system around binary. Hex is really just groups of 4 bits, which is why it's good. I think the naming would sound more natural if it's done for groups of bits instead of what's suggested here
I feel like I've stumbled on a piece of forbidden knowledge. The funniest thing was trying to come up with some counter-arguments while noticing I had already used a binary representation of octal (tri-octal to be more precise) to convert an alphabet, and already figured on my own how easy it was to apply a vigenaire cypher mentally on it thanks to how trivial it is to do arithmetics on it
16:00 I thought I'd heard that somewhere; it's a "trick" used to store floating point numbers in the IEEE standard; the leading 1 in the mantissa is implied
This feels like an induction into a religion. Now I see how things should be, and my duty is to convert the non-biners to the true light of one and zero.
In the end, it's not about "which system is generally better", but "which system is better after I came to a conclusion which were my criteria and priorities".
i would love to see some examples of this binary notation used in some common settings. some examples >in a car for speed and distance. >prices and measurements in a grocery store. >numbers in a video game. >a deck of playing cards. that batch of 4 trick seems like it would be very readable. i would also like to see something like "babies first numbers video", that would really show how easy binary actually is. a webpage for "try math in a new base" could be a nifty demonstrator for it's usability. i think the symbols themselves need a direction notation, the underline may be enough but I'd like it more obvious.
I made a userscript that you can find on greasy fork called "convert to binary" that attempts to convert numbers on webpages to their binary representation. The only pitfall of it is that it doesn't accurately convert non-integer numbers (like it'll turn 10.32 into lılı.lııııı)
@@ARockRaider It doesn't look any better. The issue here is both conversion and size. It's pretty silly to suggest that using rather indistinguishable characters is somehow better here. There is *some* ability to parse a misread sign simply becuase of possible interpretation, but that's because we're operating within a presumed 'field' of possible speeds. It's also more important that I used a number like 22, which I can defend, and you instead focused on what the digits look like. Which tells me you either didn't take my critique seriously or you are too predisposed to the idea of it working without considering it's real-world applications.
@@jibbjabb43 I had assumed that you picked a number to look the most outlandish, i was pointing out that you wouldn't use actual 1s and 0s and that you would be using a notation that makes the numbers every bit as clear. that you picked 22 makes my assumption of an intentionally outlandish number very clear, on top of that the exact speed would be adapted to the number system 20 for example would be i .i.. and as a group with proper underlines would be very different from 40 or i. i... neither look anything like 30 or i iii. but i expect that you wouldn't be using multiples of 5 for speed-limits, rather you would pick a new batch that always stays neat and round.
My first reaction was "wait, binary??", especially as i watched the jan Misali's videos before, but then it turned into "oooh, thats how we can do it", and then "it's beautiful as heck"
Actually, I have a more general counterargument: Redundancy is good, actually. All of these arguments demonstrate that binary is the most efficient base in a similar way to how Ithkuil is the most efficient language. Sure, that might be true on a technical level, but if you miss more or less any information, it's harder to recover. For example, something weird happens with the HDMI on my TV, so the edges of the screen get cut off when I try playing a Switch game on it. But because all of the Hindu-Arabic numerals are fairly distinct, I was still able to follow the timer in the Star raids in pokémon, despite the top half of the number being cut off. Meanwhile, if the top half of your binary numbers got cut off, that'd be it. There would be no way to distinguish numbers anymore. Or similarly, consider seven-segment displays. They're actually *horribly* inefficient, since you can display 128 distinct figures with them, and we only use 10. But because we use so few, depending on which segment goes out, you can still distinguish most numbers. And even in a case like the middle segment going out and making 8 and 0 indistinguishable, context clues help. If you see 1-"thing that looks like it's supposed to be a 9" change to 10, you can infer that it's probably counting down and is probably supposed to be 18. But if it changes to "thing that's probably a 2"-0, you can infer that it's counting up and that it *is* supposed to be a 0. Or similarly, finger counting. Functionally, "binary" finger counting is actually duotrigesimal. No one's going to actually think of each finger as its own digit, so you're effectively expecting people to learn 32 different hand shapes. So if you're just using hand counting to show someone a number, like a kid going "I'm this many", it's going to be less efficient compared to just holding up a number of fingers and subitizing. Or similarly, if you're actually counting up or down, with both seximal and jisanbeop, you only ever need to change either 1 finger at a time or all 5 fingers. (And with all 5 fingers, it's also always either resetting the hand or changing 4 fingers to a thumb) Meanwhile, with binary finger counting, you might have to change any number or combination of fingers. And that's actually so well known of a problem that Frank Gray came up with the idea of reordering binary numbers so you only ever have to change 1 bit at once all the way back in 1947. (With the idea itself going back to at least the 1870s)
I actually agree with most of your points, but have a few nuanced points to make. 1) This one's simple, actually using binary doesn't mean blocking out half the digit would leave it unreadable. Binary is actually pretty good about that, you can *always* tell if you can see part of the digit. 2) I use decimal for readability purposes, but I find binary a better system for doing simple counting tasks. Remember that you could use different systems for different purposes, it's done in some cultures in other areas, such as speaking one language and writing another. (Looking at you, Switzerland...) 3) While your argument about finger counting has some merit, I actually think that's only a problem for children. Since we're actually using a *unary* system for that task, binary finger counting is strictly better but also more complex. (Seeing each finger as a digit with two states, the *number of digits* doesn't have anything to do with the counting system - that's just a physical limitation.) It's too complicated to teach to children without a whole course around counting systems, which doesn't make sense at that stage of life. For adults in the mathematics or comp sci sphere though, it's easy enough to adopt and therefore we probably should do so on an individual basis. Not sure what the video itself seeks to argue, as I'm not about to watch someone try to convince me of the merits of a counting system I already use, so don't take any of this as an argument toward the creator's points. I'm simply arguing for what I believe to be best in my own experience.
@@impishlyit9780 1. Eh, not necessarily. One of the video's weaker arguments has to do with number length. It basically makes a *font* based argument, where if you change binary numbers to tall (1) and short (0) lines, the horizontal width of numerals becomes comparable to other bases. And that really *does* run into that issue. Or more broadly, I forgot to include this bit for contrast, but imagine a binary display. If any one segment goes out, you cease to be able to distinguish anything in that place. At a minimum, you'd need to do something like pairs of dots, where only one can be on, for you to be able to lose a segment and still be able to distinguish numbers. Meanwhile, even though only the top left and bottom right segments lack minimal pairs, you can still generally make things out on a 7-segment display, even if one of the segments goes out. You'd need 2-3 to go out before it starts really impacting your ability to read things. 2. No counterargument here. 3. It's also more than just kids. For example, how often do you *actually* need to count on your hands? It's typically smaller things, like how you might silently count down from 5. And while I'll grant that people probably count down specifically from 5, in part because we have 5 fingers on each hand, unary feels a lot easier for that. In a way, it's sort of like sorting algorithms. Merge sort is more effective for really big lists, but it also requires a lot of overhead. So for shorter lists, the less asymptotically efficient insertion sort winds up running faster overall. The main time I can think of that it's potentially useful to display bigger numbers would be something like days, but there are also systems like the Medieval Arabic hand numbering system that do it more easily than finger binary. It's actually even more efficient than finger binary. It encodes one digit on your thumb and index finger and one digit on your other three fingers, so it can go up to 9999 on two hands, as opposed to 1024. But overall, my main criticism of the video really is that it successfully argues that binary is the least redundant numbering system possible, but not that redundancy is something we need to be avoiding
You should into your TV's settings and turn on either "overscan" or "just scan", or set the aspect ratio to "full" or "just scan". Turning on "game mode" might also fix this.
I really appreciated jan misali's introduction to the advantages of seximal, and have considered myself a convert for many years. You present a strong case here, and it's certainly convenient that your choice happens to be binary. One of the two systems that I have to know anyway. I'd like to see separate (short) videos on some of the topics here. How to write binary numbers. How to speak binary numbers. How to divide binary numbers. Etc.
Some notes as I go: - I see you don't want to be able to write the numbers 0 and 1, if the leading digit doesn't matter for information. - You seem to be neglecting the human tendency to ossify details. A binary system wouldn't stay a binary system for long. Within a few generations, eople would start writing shorthand octal or hex with a couple strokes rathet than four or five, and those would become the basic unit. That grouping trick is actually a negative when you look at language evolution. - "What's the Most Commonly Used Prime Which is Incompatible with This Particular Base?" got a laugh out of me, good work on that one! - Your repeating symbol and your grouping marks would be super easily confused in hasty handwriting. That's not an inherent binary problem though, just one with your notation. - 2⁴ should be called nybble or nyb. It's cuter and sounds better in practice. - ... I both love and hate you for pointing out to me that stack is a number name. It is though, saying something like "three stacks twelve" is perfectly normal in minecraft contexts. - I think seximal is still more aesthetically pleasing, to be honest. And that matters because the only context switching away from decimal will ever happen is art. ... though balanced ternary might win there, it's just the right amount of weird to be really fun.
For your first point: I do agree that they handled this poorly. But I think all is not lost. Instead, you can think of what they are talking about as a consequence of the fact that simply knowing the bit length of a number tells you the value of its most significant bit (which has to be 1). The “length” of the number does not give you so much information in other bases, this is definitely true. As for your second point, so long as people don’t forget where these ossified forms come from and they are visually similar to unossified forms and can be easily decomposed into them, then I don’t think this is actually an issue.
Since the measure of information per digit is considering all infinitely many numbers and the number of single digit numbers is finite, the amount of extra information carried by that first digit tends towards zero in the limit of considering all numbers since a finite portion of an infinite space is always 0% of the total space. I think this means that when considering information on the long scale this simplification is fine, since the context of the discussion was already the long term efficiency of a base and not its efficiency writing small numbers.
43:29 "Traditionally, to notate a recurring fraction, the entire recurring segment is marked, but that doesn't make a lot of sense: It's simpler to just mark where it starts." I think your opinion comes from the age of typing and digital text representations. In hand-writing, marking the recurring segment makes more sense for the following reasons: (1) Often, you will get this repeating decimal from just long division until you enter a cycle, and adding a line of the repeating part is something you can do after you've written the number, which doesn't require you to erase anything or rewrite the number, and also separates the repeating segment from any digits of the next repetition you may have written before realizing it was repeating. This doesn't apply if you're rewriting the number somewhere else, but if you're copying a hand-calculated quotient with a long repeating decimal, you'll want to use some mark like the over-bar to mark the repeating segment of the decimal anyway, so any other notation like the "r" you're using will just be another notation to learn IN ADDITION to something like the over-bar. Thus, I think this is the most important reason. (2) In hand writing, writing a line over several numbers is not significantly harder or more time consuming than writing an r before them. It's actually a slightly simpler shape than an "r", though the length and precision required mean I'll just say it's about as hard overall. This is different from typing or digital text, where putting bars over numbers requires special characters or text formatting, both things that are much more of a pain to deal with than just typing "r", both just in typing it, and in the sneaky problem that things like this sometimes won't render properly on other people's computers or on printers. (With typewriters, it usually requires some kind of tricks involving typing over the same text I suspect.) That being said, one downside of the over-bars specific to hand-writing is that, due to the imprecision of handwriting, it can often be unclear to readers exactly which numbers the over-bar is over. In typing, this would only really be a problem if your method of creating over-bars was bad, and using preceding "r" would get rid of this issue in handwriting (as would using parentheses or circling, although those aren't quite as convenient to add on to an already written sequence of digits as an over-bar is).
In my school they taught us to write repeating digits usung brackets: 1/7 = 0.(142857) This eliminates the precision problem as brackets are easier to write and mark exactly where the repetition begins and ends
Amazing. Genuinely, i was amazed multiple different times. I'm a computer science major myself, I had already discovered the bit about multiplication, but the section on factoring? I had never thought to do that, and it blew my mind. Division? So much better than base 10. Your notation is beautiful, I'm definitely adopting it for when I do binary work, and I'm geniunely strongly considering switching to binary in my everyday life.
As someone who played with numerals a lot as a teen, and who's done a lot of programming, I find it VERY telling that we programmers who work with it often will ALWAYS display it as hexadecimal. The 'best base' is really about what is the best base for humans, and minimizing how it pushes against our mental limits. I really liked your new binary-based symbols for hexadecimal numbers. I wonder if a society based on binary math and those symbols would work better and learn math easier. I think the binary naming in this video got really awkward and long to speak. Instead, use the groupings and name those. I think a society would need to use octal or hexadecimal for communication, while doing arithmetic in binary.
As a programmer myself, during my education (specifically when learning assembly), I always found working with the binary numbers far more intuitive / less cumbersome than working with the hex numbers. Like, at all points, I would think about it in the binary expansion, but then every time I had to actually write the number in the program I would have to group the binary by 4s and convert to hex, its just a mess. This is not to say hex is bad per se, but that it is necessarily just a layer of translation/abstraction over the actual numbers, which are binary. "x5555" is obscure, needs translation. "101 101 101 101" is the real number, the thing you actually need to work with. Anyways..
Well let’s see. The majority of programming is done staring at a text rendered with a monospace font. Why of course it would be quite a nightmate to juggle literals 32+ characters wide. There’s no real compressibility unless the programming language allows macros or something to use a custom number literal format (or the IDE used allows to work with proportional fonts in comfortably). I suspect there can be more mundane reasons other than this one.
@@05degrees Yeah, suppose we start with a system like the proposed. I think it wouldn't take much for people to start comming up with symbols and meanings that more readly represents the common quantities and groupings.
Me at the beginning: Worth watching, but unlikely. Me at the middle: Fantastic arithmetic, but I mostly say rather than play, and binary is long. Me at the end: I'm convinced.
I love the notation system you created and the grouping shorthand, it’s very elegant and also makes the fundamental patterns you discuss visually obvious at a glance, without having to translate into numbers, even for someone totally new to the notation without having built base specific intuition. I could easily learn to do a lot of that math with your binary notation without converting into or out of decimal, which makes an excellent argument for it’s naturalness and simplicity
Felt compelled to decipher the text at 8:20. It reads: "You have got to be about the most superficial commentator on con-langues since the idiotic B. Gilson. Did I miss the one where you said which conlang you're fluent in and read at least three times a week and can read new books in every week of even one year or listen to radio shows in every week? New radio shows?" Was a fun challenge, thanks!
Congratulations! You have been given the award for the most uses of "Fermat's Little Theorem" in a video that is not about Fermat's Little Theorem or about modular arithmetic!
I really loved the video, but I think, as many commenters have pointed out, that the legibiliy of your line system could be improved. Some might even compare it to a certain Minecraft optimised number notation system. My suggestion would be to change out the short line for a small circle. This not only de-clutters the lines by separating them more cleary, but also prestents a great opportunity for ligatrues utilising the existing latin alphabet and also reducucing stroke number. or for l. and and for .l maybe for l.l, and maybe for .l. Keep in mind the arcs would be proportionally smaller in written notation I'll flesh this out further and come back to this comment later, I'm expecting some grouping and priority conflicts, but there might be a nice way to deal with them.
I take some umbrage with the “efficient finger counting” argument, because finger counting is primarily a tool for teaching children. Binary finger counting (and even seximal finger counting) is more complex than simply counting the number of fingers raised. Bases are an arbitrary construct and humans don’t think in terms of them, so a child will never intuitively understand binary finger counting the way they understand “base 10” finger counting (though i would argue it’s more accurately described as base 1 finger counting)
Many cultures have counted in dozenal or hexadecimal on one hand. Many Asian cultures to this day count to ten on one hand and find the western way to count on fingers jarring. Your argument is eurocentric.
I’d argue the mentioned systems are still ultimately base-1 counting systems. They use 12 or 16 naturally ordered positions of the hand. Counting up to 12 with the dozenal thumb-thing is still just moving your thumb through a “number line” of hand positions. It still retains the simple linearity of western counting that’s lost with binary counting.
well actually maybe base infinity would be more accurate but my point is that neither 10 finger counting nor 12/16 one-hand counting have to actually deal with the things that make bases complicated (i.e. multiplying by powers of the base according to the position of the digit)
@@considerthehumbleworm During the Middle Ages doing arithmetics with your fingers was common enough that almost every book on mathematics had a chapter on it. For example in _'Liber Abaci'_ by _Leonardo Fibonacci_ finger counting is the first chapter. Systems to count up to 100.000 were around for quite some time. Conflating finger counting to what _we_ today in the west teach to toddlers is a mistake. Drawing conclusions from oneself about others rarely works out.
Finger counting is what we teach to toddlers tho. I think it’s a bad idea to offer binary finger counting as a replacement in those contexts. It’s useful to be able to count higher than only 10 in other contexts but it only muddies the water for learning basic arithmetic. Honestly, teaching toddlers systems that go up to 8 or 16 is probably preferable to 10 to ease the transition from “number line counting” to using actual positional binary (even binary finger counting, eh?). That transition’s a bit of a given with 10 finger counting and base 10, so it’s worth considering how that works with binary. Also we don’t have to hate each other because we disagree about numbers lol
The thing that fascinates me the most is that base 2 is every base 2ⁿ at the same time, so it is capable of using properties from all of them. Amazing video.
I think many of your ideas have basis. But i dont think that it works quite as well in practice. I am a fan of base 6, because of jan Misali's video, but as a programmer and computer scientist, i have to acknowledge the advantage of binary for raw computation strength. There is a good reason we use it in computers, and computation is a place where numbers shine. But unfortunately for us humans, we are not easily wired for base 2. It's hard to say how the real world efficiency of human calculations would turn out if we completely switched to binary, but i think that our brains are slightly better wired to understand larger bases. As someone who has poked around with basic arithmetic in both binary and seximal, i can say that binary is easier algorithmically, but if i had to do a lot of math, i would prefer base 6. Part of this is the practical problem of we need to write down the numbers we are doing math with and i feel like bigger numbers are better for cramps. I think it would be interesting to do legitimate experiments teaching people variou bases, and see who in the long run uses the bases most effectively. Seximal, for me, just seems to be a nice balance between small bases, which are easier for math, and big bases, which are easier for brains. Curious to see if jan Misali responds, but i still think i am on the seximal side for the best base for humans.
What a high quality video! I wonder if I’m really slow or most could keep up without extensive pausing and replaying. Congratulations on such a well made video regardless.
Programmers and hardware engineers have been doing this forever, using 8 or 16 as the "compressed" written format, but dropping back to bits for actual arithmetic, especially when making machines to do so. Anyone in those groups worth their salt can convert from 0-F their binary quartets and back intuitively, and when learning to do that, often use fingers in ways that resemble your grouped bits. The main problem with binary as a written or spoken system is that its hard to read at a distance, and the octal/hexidecimal are "error correcting" in that a smudge on a piece of paper or a dent in a sign makes simple tally marks unreadable, but leaves letters and hindu-arabic numerals mostly in tact.
This really is exceptional content. As a nerd I really appreciate the work that was put into this video. Great work. My only note is the pacing was too fast for me to grasp some of the details that were claimed to be “immediately obvious”.
@@matheuscabral9618 First, there's no need to make assumptions and attack people. Please refrain from it and maintain a civil discussion. Secondly, my point is that they said their video would've been longer, which is unexpected to me seeing how it responds to a 10 minute video; this video is already so much longer than the video it responds to!
"just use only three fingers on each hand" leads to communication issues. The great thing about addition only decimal finger counting is no matter how you count you always get the same result. It is unambiguous and inclusive to everyone who has any fingers. But yes it also means that you can only communicate very few numbers at one time. Basically decimal counting is TCP (sacrificing speed and efficiency for making sure it is as unambiguous as possible within the system) while binary counting is UDP (so prioritizing speed and data density while risking the wrong message being sent)
@@nukollodda For non-perfect squares, the algorithm continues past the decimal point to whatever degree of accuracy is required. As for how to adapt it to accept complex inputs and outputs, that's above my current level of understanding.
I want to point out that another word, while less common, for 2^4 is "nibble" if we're using "byte" for 2^8. Hex is fine, but I want more excuses to use "nibble", thanks.
@@minerscale personally I'd have to disagree here. Word only exists for its counterparts Dword and Qword. It would perhaps be better to either go with short, int, long or switch all of them to Word Dword Qword
18:14 What happens, is that you'd stop being able to count it easily by sight at a distance - counting "how many fingers are extended" doesn't care about positioning, binary counting does. And the issue with this, is that when using hand signals to convey numbers, you're probably doing that _because your voice can't reach the other_. Most situations where that happens are scenarios in which stopping for a couple seconds to reorient & count the exact numbers for yourself is an awkward use of time. As an example: if I am on a construction zone with operating machines, and I need the handyman to replanish the third team with beams, screaming "BRING BEAMS TO THREE!" will probably be only partially heard, and after hearing the expected "WHAT?", being able to just raise three fingers while hollering "BEAMS TO THREE!!!" adds a layer of important redundancy, as the handyman will be instantly able to note that one word is 3, and the other is the object to bring, allowing them to process it better across the other sounds. And yes, this is also true with binary, but with that system you'd be forced to use and read for the specific fingers which hold specific meanings, which can easily mean that the handyman needs to ask clarification _again_ because either they didn't catch if the fingers raised were for 2 and 4, or 1 and 2, or they caught that, but not what to bring. It's already hard enough with all the redundancy we use, and needing to clarify and reclarify everything is stressful and awkward for everyone involved. In this sense, using binary for finger counting is strictly worse. Seximal, or well, the other counting methods named came about because of that need to balance redundancy and information density. Sometimes, being able to alternate between asking for 2 of something and 60 of something is important, and between the thumb being quite ubiquitous and only needing to pay attention to "which hand are they raising", reading fingers at a distance using only 4 parts, is plausible. Using 10 parts, making order important for every finger, is simply too awkward to use. However, I will concede some other thing that's reasonable: if instead of an specific number you are asking for a general amount, then having each finger imply a different order of magnitude is useful, albeit there's already an order-independent system for that in which the amount of fingers raised is how many 0 are after a 1. So yeah, binary is really damned good for counting... In paper and computers. It isn't good for hand counting tho, but honestly it doesn't need to be - hand symbols can simply not change alongside written ones.
14:49 I can't speak for jan Misali, but I think he probably just meant that if you write binary with Hindu-Arabic numerals, the numbers get unwieldy in size, and base 4 in Hindu-Arabic numerals is a good way to compress it. 15:36 This fact isn't what makes binary the most efficient, it just makes it the most efficient by a long shot. Even if we didn't consider the fact that the leading digit can't be 0, by replacing b-1 with b in the formula, binary would still be more efficient than base 3. side note: I don't know how we're measuring which is better rigorously, even though it's clear intuitively. Because the graph for base 3 does sometimes dip below the graph for base 2, like for example between 8 and 9. Is it the one which is the lowest for the most numbers, which has the lowest average, or what? 16:27 I wouldn't be so hard on seximal. jan Misali admits that in seximal, numbers are written longer, and the justification isn't radix economy, it's that the square base is small enough to be used for compression, it's only too large to do basic arithmetic. In fact, I think it doesn't make all that much sense to use radix economy as an argument at all, our brains don't simply look at the representation of a number and take a specific amount of time to process each digit based on the log of what they expect the digit to be. Although it would be fun to see a study comparing each base in its own writing system and the speed of writing or reading numbers.
Personally as programer I chose binary with hexadecimal compression because it is the best of both worlds. Your number to long, just start compressing into hexadecimal, need to do math decompress it. It just allows better writing efficacy and in low level programming you just do base two operations on hex nums anyways.
I am also a programmer, I love the idea of having two different ways of writing numbers, I previously only thought about hexadecimal and even made my own writing system for it wich also uses a sub-base of 4, 2*2 = 4, and 4*4=16, I love the symmetry Binary for math operations and hexadecimal for showing numbers!
I feel like this system could be improved by using unique labels in spoken language for every permutation of 4 bits, effectively treating it like hexadecimal when talking. You already concede that spoken binary would create some very long number labels and offer quartery as a compromise, so you might aswell embrace the pseudo-hexadecimal nature of grouping 4 bits together by adding a horizontal line at the bottom by covering these groupings with a single label each.
Exactly! And since lower numbers don't have decimal based names (compare "eleven" to "twenty-one") we can start by borrowing those. In spanish we have unique names up to 15 (it's "quince" as opposed to "diecicinco") so we already have the firt 4 digits covered. From that point, I'd say the next 4 digits should have a prefix (so, I.II. would be something like hexasix), but we would need to find something that rolls of the tongue (I love stack for 64, but it sounds clunky as a prefix)
The music argument in the early chapter is interesting. Our notation is specified in fractions of powers of two - but they are fractions of four beats per measure. We also work just as often in three (triplets). However, we count in groups of 4 or 6, and often 8 (dance) and 12 (triplets over 4 beat measure). When dealing with irregular numbers like 5 or 7, the rhythm is usually done in syncopated emphasis of groups of three and two. For example, 5 is usuall 3+2 or doubled at 3+3+2+2; 7 is usually 3+2+2. Syncopated division of 8 is common with 3+3+2 (typical of latin and jazz) and 12 with 3+3+2+2+2 in flamenco compas. You further get the complication of one instrument playing a phrase in triplet over the rest of the band playing in duple. The final pattern is that phrases tend to also be groups of four measures, a prime example being the three sections in a twelve bar blues. This is super internalized, and my brain stops and pays attention if a phrase isnt that multiple.
4-bar phrases and 16-bar sections are common enough that when I'm counting measures of rest or something, it usually feels most natural to count in a base 4 system. On each hand I use my 4 non-thumb fingers and with 2 hands I can count up to 16. For example: 1 = ...| 2 = ..|| 3 = .||| 4 = |||| (1-4 had implicit |... right) 5 = ...| left, ||.. right 6 = ..|| left, ||.. right 7 = .||| left, ||.. right 8 = |||| left, ||.. right 9 = ...| left, |||. right etc. On both hands I start from my index finger, but with each one visualized palm-facing down, that results in the left hand counting right-to-left and the right hand counting left-to-right, but the point is that it feels right in the music.
The "best" base is the one which lies at the intersection between maximal efficiency (cramming in as much info as possible), maximal legibility (being able to get that info back out), and maximum versatility (being useful in multiple numerical arenas). While binary is objectively the most efficient way of storing data, it isn't the most legible or versatile, so that's just the starting point for finding our magic intersection. Legibility peaks around the concept of subitization, the ability for the brain to quickly count numbers, which begins to decrease after 4 and is gone for most people by 8. This means that any number longer than 4 digits is going to be harder to read, objectively, with separators as a bandaid patch. Thus we definitely don't want any common numbers to exceed 4 digits, so binary isn't necessarily the best choice on this front. The last point is versatility: can it express simple fractions simply, how easily does it convert to binary, can you count on your fingers with it simply, etc. At a guess based (heh) on all this, the intersection is probably around base-4, which means no one gets to be happy and DNA actually got the right answer somehow.
Phone numbers, IP addresses, credit card numbers, social security numbers, etc: these kinds of things usually try to group digits into groups of at most 4. If they tried to make groups longer than than that, they would be harder to remember. For decimal we tend to use groups of 3 so each group is about a thousand. But assuming we can handle groups of 4 just fine, and assuming we want a group to still get to around a thousand, a nice base would probably be around the 4th-root of a thousand? Base 6 looks pretty good for that. But a thousand was kind of arbitrary. With base 4 in groups of 4 you can get up to 256, and while that's not a thousand, it's still probably big enough for most people's everyday purposes. Like, it's larger than a Dunbar number so it's probably good enough, right? Part of the argument for binary radix economy was that if a digit represents less information, then you can get away with more of them in a grouping, but I'm not convinced that's true. I still think grouping by 4 is best. In music, we can deal with measures of 4/4, 4 bar phrases, etc. just fine, but when time signatures get more complicated than that we tend to group them into sizes 4 or smaller, like grouping 5 into 2+3, 6 into 3+3, 7 into 2+2+3, 8 into 4+4, 9 into 3+3+3, etc.
if you had groupings of groupings, though, what could that look like? Imagine if the regular groupings presented in this video are like music notation for 8th notes in groups of 4. Maybe groupings of groupings could be like 16th notes!
I think the whole argument that non-binary systems are "more wasteful" because they might have an extra leading zero is pretty enormously flawed and relies on the assumptions that A) pattern recognition and readability isn't a thing, and a series of two symbols that repeat over and over is at all functional. By this argument, and the whole ""efficiency"" radix argument, you could also make the same deduction, that all language should just be written in binary strings. After all, what is the English alphabet but a series of base 26 numbers?
Good point, except the alphabet isn’t for numbers, but for words and/or sounds. You’re not performing arithmetic with words. Also, no one is ignoring pattern recognition. The video simply argues that it can, in fact, apply to binary, especially with the right notation and proper parsing methods for the bits.
17:43 this is incorrect. While you are thinking in decimal, each finger only carries one numeral of information. It either "doesn't" exist, or it is a 1. You count the 1s to get the number. We have a name for that number system and it isn't decimal. It's unary. Tally marks. Base 1. Of course binary is going to be way more efficient, because you are doubling the amount of information that each finger can carry, you go from being able to count to 10 (or 20 if you're european) with 10 fingers (1:1) to 1023 (102:1) or 31 with one hand, because every single finger added doubles the number you can count to, whereas in unary, every added finger only increments that number by one. Surely if you actually gave each finger 10 different positions corresponding to the 10 digits of base 10 and somehow were able to reliably use, hold, and distinguish all of those positions simultaneously without the tendons in your fingers binding against each other like a shitty typewriter, you could count to 9,999,999,999, which is way more than anyone would ever need for everyday counting purposes (though so is 1023). Finger binary is super useful though. I use it all the time at the risk of people thinking I'm making rude gestures or gang signs (18 = heavy metal, 19 = I love you, 4 and 5 = fuck you, 3 and 7 = finger gun)
and I do happen to know a way to calculate square roots. So a square root of N is a number G such that G * G = N, we can formalize this: G^2 = N G^2 - N = 0 f(G) = G^2 - N f'(G) = 2G g(G) = (G^2 - N) / 2G g'(G) = G - g(G) Keep computing g'(G) on any initial guess number and it will eventually converge on the answer. It converges faster if the initial guess is closer to the true answer, as g(G) is essentially computing the relative "error" of the guess, and thus will oscillate past the true answer and slowly converge on it. If N = 9 and you start with G = 1, then you will get 1 -> 5 -> 3.4 -> ~3.1 and you can see how it converges to 3. But let's say you're trying to compute sqrt(2), you can take guess values based on the closest known perfect squares, which are 1 and 4, whose square roots are 1 and 2. So the answer must be between 1 and 2, so you start with G = 1.5 and it converges to 1.414 fairly quickly. Same with 10, you know the closest surrounding perfect squares are 9 and 16, whose square roots are 3 and 4, so the answer must be between 3 and 4. Choose 3.5 as your initial guess and it quickly converges to ~3.16228 within 3 generations. Here is that algorithm implemented in unix dc: [lGd*lN-lG2*/lGr-sG]sg to use, set the precision, G and N registers and then run the macro a few times. Here's what that might look like: [lGd*lN-lG2*/lGr-sG]sg 20k 10sN 3.5sG lgxlgxlgxlgxlgx lGp 3.16227766016837933200 People call me insane for using dc, but it's fucking awesome once you "get it". I even used it to implement a working base 64 decoder and encoder, which I will concede is insane, but it was fun. But yes, approximating square roots by hand is a pain in the ass. That sqrt(9) with a starting guess of 1? I did that by hand. Really pushed the limits of my mental math abilities when I had to divide 2.56 by 6.8 and make a multiplication table for 6.8 by adding 6.8 to itself over and over in my head. Look at this shit sqrt(n) = x such that x*x = n x^2 = n x^2 - n = 0 f(x) = x^2 - n f'(x) = 2x g(x) = f(x)/f'(x) g(x) = x^2 - n / 2x guess - g(guess) -> ans 9 -> g(x) = x^2 - 9 / 2x 1 -> g(1) = -8 / 2 -> -4 1 - (-4) = 5 5 -> g(5) = 25 - 9 / 10 -> 1.6 5 - 1.6 = 3.4 3.4 -> g(3.4) = 3.4^2 - 9 / 2(3.4) -> 11.56 - 9 / 6.8 -> 2.56 / 6.8 -> 0.3 3.4 - 0.3 = 3.1 3.1 -> 3.001 -> 3.00001 -> 3 (3 + .4) * (3 + .4) 9 + 2(.4*3) + .4*.4 11.56 0.3060706 6.8 | 2.56 25 20.4 4.66 46 40.8 5.26 52 47.6 4.46 44 40.8 3.26 1 6.8 2 13.6 3 20.4 4 27.2 5 34.0 6 40.8 7 47.6 8 54.4 9 61.2 10 68.0
Grouping "bits" by up to for 4 is also pretty practical in a human sense, since around 4 is the natural range of human "subitizing" i.e telling how many of a thing there are at just a glance
![Felix "Unfair" | [Stray Kids : SKZ-PLAYER]](http://i.ytimg.com/vi/Oswujxm2Ag0/mqdefault.jpg)
really well put together video! these are some very compelling arguments for binary. you're right that I dismissed it for superficial reasons without second thought in my videos; the immense advantages it has for arithmetic and information density shouldn't be overlooked, and there definitely is a case to be made that these may matter more than the things I focused on. in the few contexts where the notion of an "objectively best base" actually makes mathematical sense as a thing to care about, binary is a clear winner.
I don't think I'm completely convinced that binary is necessarily the absolute best choice for a human-scale base (the "coincidental" advantages seximal has for working with small primes are just too good) but I am convinced that it can work as a human-scale base to begin with, which I hadn't even properly considered before. it definitely deserves a seat at the infinite table with the other SHCN bases.
please do a responde video! not only. for exposing more of your points but also for giving this guy more attention here on youtube
I think the answer then would be some more usable power of two such as 8 or 16
+
@@wilh3lmmusic thats basically what the video said, you can combine 2 or 3 digits of binary to make it shorter like base 8, but with all the advantages of base 2
@@leggyjorington3960 Not exactly. The grouping system in the video is distinct from hex or octal in a few ways.
First of all, if you just want to use hex, then the notation suggested is very inefficient. I'm not a big fan of how we usually write hex (0,1,2,3,4,5,6,7,8,9,a,b,c,d,e,f) but it's much more efficient than iiii iiil iili iill... etc. The same is basically true for octal.
Second of all, the advantage of his system is that it isn't hex or octal, it's both (and any other 2^x base). You can very easily re-analyse a number as any of those bases to get its benefits. Sevens aren't great in hex, but they are easy in octal, so just group by three bits. Octal sucks at fives, so just group bits by fours and treat it like hex.
Wilh3lm is, imo, correct that the benefits of the more flexible system are outweighed by the practical issues with actually writing these numbers. I spent several months using almost exactly the notation suggested in the video and it simply isn't practical.
drama in the counting community
lol
When are the diss dropping?
Things are heating up in the number fandom
6 finally cancelling 7 cuz it 8 9
This is worse than the time there was drama in the circle community.
Edit: Relevant timestamp: 1:06:04
I frequently see children using this system to express the number 4 to me when a school bus drives past. It's truly amazing to see something adopted with such enthusiasm at a young age and gives me a lot of hope for future generations.
Wouldn't they also be expressing the numbers 128 and 132?
@@mehulpandya4761 The most avid mathematicians on the back seat usually flaunt their counting skills in this way.
New euphemism just dropped!
Finally, Something More Creative Than "But 4 Tho"
@@mronewheelerholy hell!
Another fun fact, since microprocessors out number humans, and most of them do arithmatic significantly more often than the average person does,
technically binary is the most often used base currently.
My phone did arithmetic to like ur comment
I mean the term “use” implies some sort of conscious input so no
We use computers which uses binary@@foggy8298
@@foggy8298 I mean, when we use a computer, they use binary. So...
I'm a bit excited
a
Because Vötgil has so many vowels
I am waiting for a formal debate
vötgil :3
Because this episode is a first in a few ways
59:03 Excuse me, but four bits is a *nibble* (half a byte!). I will not drop the cute name. I love it.
fr agree, how wouldnt you wanna say 8DEC as "a nibble"
"gimme a nibble burgers"
"i would like a nibble kilo of ice cream"
"give me nibble apples"
@@im-radio "Can I just have a nibble of your sandwiches?"
"NO!"
If you wanted to keep it monosyllabic you could shorten it to "nib" :P
@@im-radio36332 right?
"Seximal may win a sprint, but binary wins the marathon."
So that's why you made a 1-hour video to counter jan Misali's 18 min one.
i'm not sure you understand how a response works
youre missing the point but its still funny
Guys it's a joke, don't take it as him being serious
kekw
@@katie-ampersand
short vid = sprint
I think I found my favorite genre of videos: the hour long math rabbit hole.
Videos such as this one, "HACKENBUSH: a window to a new world of math" by Owen Maitzen, or " The Continuity of Splines" by Freya Holmer.
Really loved the use of music and sound in this one.
I love both of the videos you mentioned so much, if you have any other similar ones to share I'd love to know! I can also go through my favourites and find similar ones if you want me to tomorrow ^^
@@codenamelambda The recent 3b1b explanations of light slowing down when travelling through matter are definitely a must-watch, if you haven't seen them already. Oh, and also the "Why can't you multiply vectors?" by Freya Holmer
we're honored to be compared to videos like these! glad you enjoyed
RIP Owen maitzen
I just appreciate how unnecessarily hostile the video is. It's honestly hilarious.
Yes, and it makes the thesis so much more compelling
One cute fraud.
yooo wikipedia bisexual lighting skeleton
having beef with base four for over an hour
Hi, not sure if anyone’s mentioned this yet but your use of pitches following the harmonic series to accentuate numbers you’re talking about is absolutely incredible and I didn’t want it to go unnoticed.
Some fingers are significantly harder to extend individually than others. This applies both physiologically and culturally.
Ah yes, the four (l..) finger ;)
This point is addressed in the video. Using just three fingers on each hand still lets you go further than seximal, and binary is the only base where you can use any amount of fingers for finger counting
@@OMGYavanithree fingers per hand still means you can flip someone off, and arbitrarily skipping the middle finger would introduce ambiguity.
finger counting 19+113 💀💀
i can do it just fine, so... skill issue
The amount of times I got caught by the “‘well actually’ -You“ moments scared me. Every time I felt like I had a valid argument to make, there was a direct response to it.
James Grime convinced me of 12, almost 12 years ago
Misali of 6, almost 6 years ago
You just convinced me of 2
Who said there was no progress in history?
in 2 years some will convince you of base 1 and then 1 year later someone else will tell you base 0 is better, on which there will be a imediate response claiming base -1 is optimal, while then you realized that you actually had memories of someone claiming 1 year ago that base -2 was better and so on
2 months ago
@@LucasFerreira-gx9yh he may end it up studying Gaussian integers. Hahahahaha.
the pattern shows that in 6 years youll be learning base .5
@@smoceany9478 12/(t/6)!?
+1 for playing overtones behind fractions
i loved that
There are 10 types of people, those who understand binary, and those who don't
Underrated comment
And those who weren't expecting a ternary joke
There are 10 types of people, those who understand the hexadecimal system, and F the rest.
@@MuzikBikethats still 2 #binaryandseximalareslaynumbersystems
Wrong, there are |. types of people
The bit about the square root algorithm made me literally get up out of my chair, scream "what??" at my phone multiple times, and roam around my apartment for several minutes rethinking life
I shared your astonishment when I realized there's an algorithm for that. It just shows how the compactness of binary makes it so versatile.
When it is? I am too lazy to watch all of it
@@PinkeySuavothe final section of chapter three goes over the method. pretty neat
honestly, this video has now converted me into a true binary supporter
also i did not expect all of the arithmetic stuff with binary to be SO simple and easy to do
no way its the crystal garden golden person
i frogelined and peacelined
oh hey it's the |..|eg guy
I did not expected that usage of binary in human scale is not only possible, but pretty convenient too and it has a lot of advantages
re: chapter six. i'm not sure your naming scheme really does a great job here. it definitely feels like "ninety one" is less information to parse than "four one hex, two four, three", and in general, standard names tend to feel less cumbersome. perhaps there's a less mathematically elegant, but more linguistically practical, way of handling binary number naming.
"four one hex, two four, three" is more analogous to "nine ten one" like some languages do.
For me it's more natural with -s on the things that should be plural, so four one hexes two fours three. (like "nine tens one")
I think something that could help solve the cumbersomeness is basically doing base 16. So make all numbers less than hex compound words, so you can treat each as a single unit.
Like ninety one would be four-one hexes two-fours-three.
Compared to base 10 English, the digits are more syllables, but that happens in other languages. Like in greek the numbers up to 10 are all 2 or 3 syllables. With this system the numbers up to 16 are all 1, 2 or 3 syllables.
The fact that it's base 16 means it's actually less cumbersome than base 10 for large numbers.
It also keeps all the elegance, since it's just a conceptual reframing to help interpretability.
There's still a slight problem that, for example, "four-one hexes two-fours-three bytes hex one"
could be read as "four-one hexes" + "two-fours-three bytes hex one", you'd need to say "(four-one hexes two-fours-three) bytes hex one" somehow. I think this is just a problem that comes with the b^(2^n) system, you might have to switch to b^(kn), maybe 2^(4n) so powers of hex. Tho you could have the power names be like hex, byte, hex-byte, short, hex-short, byte-short, hex-byte-short, int and so on, so it'd be "four-one hex-bytes, two-fours-three bytes, hex, one".
@@d.l.7416 in written form, using the 2^(2^n) system, your latter example isn't technically ambiguous. but when spoken it's unclear what order of magnitude is being referred to at any point until the number ends, because at any point while telling you the number, someone could just say "short" and suddenly the number you were thinking of is 16 orders of (base 2) magnitude too small, etc.
my proposal for naming numbers is to simply read off the bits, using short syllables that can be flexibly strung together and said quickly. If 1 is pronounced like "wun" or "nun", and 0 like "oh", "wo", or "no", then a sequence like 1101 1001 becomes "wununowun wunowowun"; and you could optionally insert the magnitude words like hex between gaps. This mirrors the written form and retains its advantage of being able to group digits into whatever sized chunks are most convinient.
Hearing "nine ten one" in English is a little hard to parse, but in my native Hokkien where numbers work exactly like this, it's perfectly natural (九十一 káu-tsa̍p it). I think this just goes to show that it's all a matter of getting used to it. The "four one hex two four three" system is perfectly fine.
@@jfb- I think that would become very confusing very quickly. People already mishear fifteen (wunununun) and fifty (wununo nowuno) in decimal numbers, and you're expecting them to parse wununowun from wunonunun in fast speech (not to mention that both of those numbers are a mouthful). At the very least I think you should pick sounds with greater contrast. Maybe something like ko and mi, where the sounds of each differ in as many features as possible. You'd probably still have people getting mimikomi and mikomimi mixed up if you rattled off a couple of mikokoko-bit numbers, but it would be an improvement.
Back for a rewatch. This is levels of autism I strive for
fr lol
yess
on my third rewatch rn
I have to say. I was extremely skeptical at first, but the elegant way of writing binary numbers you came up with really sold me
Try it for a month and you will change your mind. It's a nice idea, but I can tell you from experience that it doesn't work in practice.
@@Salsmachevthis video's been out for like 3 days
it's just too bad they're completely unreadable at a glance
@@mrosskne I'm curious if you're saying this having used that notation for a while or not
@@starstufs I literally just said "at a glance"
In my humble opinion this video goes into the youtube's mathematical hall of fame.
A deep and new point of view shedding light on a topic that everyone can relate to yet few thought consciously before.
I have no other words than to thank you for your work.
npc ahh comment
@@mikechad27npc reply
Sadly, any way of speaking base-2 numbers will be non-portable. Some ancient cultures will say that int is 2^16, while modern ones say 2^32. And then there's the oft-forgotten cultures where byte (or char, as they'd say) is e.g. 2^31. Instead the names for the double powers of two should follow a simple and memorable system, e.g. 2^16 could be "int underscore least sixteen underscore tee". Since that's a mouthful you should also introduce "intmax underscore tee" as shorthand for "the biggest power of two I feel like thinking of right now".
Devastatingly nerdy comment, love it
just use the x86 names: byte, word, dword, qword / quad, and dqword / double quad. i can't remember what the next two were but i could've sworn it was ddqword and qdqword. super convenient and not confusing at all
@@rubixtheslime after qword is either oword (octa) or xmmword (128 bit simd register)
after oword is xword (hexadeca) or ymmword (256 bit simd register)
Based and Embeddedpilled
More like "the biggest power of two I feel like thinking of right now, except I must never change my mind or everything I've said before will break".
I'm incredibly surprised this video didn't mention that the GREATEST advantage binary has, is that it's a system we could ACTUALLY switch to without nearly as much hassle as any other system.
I'll be honest, I went into this video thinking "huh, interesting. Id like to see a new point of view", got to the twist reveal that it's about binary and went "ok, this is either a joke or I'm in for an interesting if unconvincing response", but now you've really convinced me. Holy cow, I had no idea what I was in for. The counting is fun too since I don't have to remember so much, and somehow these numbers are easier to understand with my dislexia too. They could also be made easier to understand for dyslexic people with a few simple tweeks so that's comforting. And on top of all that, binary numerals would be so fun to make fonts for, as you can pretty much make the symbols whatever you want as long as one is "less" in some way than the other. Hollow/full circle, down/up arrow, Mario/Luigi, literally infinate options lol.
Anyways awesome video, please make more! I'd absolutely love more specific video lessons on how to use binary with your numeral and naming systems!!
STOP DOING BINARY
"Hello, I would like 🙆🙅🙆 apples please!"
*They have played us for absolute fools*
@@jargontrueseer STOP DOING HEXIMAL
hello, i would like 📈📈🤖⬇✅🗣️💻 apples please
they have played us like FOOLS
The fact that the little sound effects match with the number on screen and the harmonic series is a very nice touch
I got converted to binary after seeing the square root algorithm, knowing how complex it is in base 10
gd guy
its funny cuz the main narrator kepe also made yBot, small world huh
@@g_vost WHAT
She truly made a channel and a dedicated trailer for the counting video essay. Dedication like that deserves my full attention and like!
Check the credits! The writer/editor has a feminine name and isn't the narrator
@@timperkin9 thanks! I fixed it in the original comment :) I never check the description or anything so I missed that one :(
Drama in the black background white text youtuber community
I really don't like the way that binary is proposed to be written in this video. The bottom connection thing is actually really nice, but the whole "short ticks for 0, long ticks for 1, and downward short ticks for the radix point" thing seems like it'd be really prone to accidental slip-ups and unnecessary ambiguity, especially without some sort of guide on the paper.
Agreed. A good way to distinguish them could be to write 0 like the lowercase Greek gamma and 1 like the cursive lowercase L. It would be fast, less prone to errors and easy to read.
@@polymloth I just sat down to try and find a system based on this comment, and this was what I came up with lol, connecting adjacent characters like cursive. This is a pleasing analog of chiral topology, something like over- and under-crossings in a knot diagram.
You can use word breaks to indicate digit groups, and a slash to indicate fractions as usual (I don't actually think having a positional notation for fractions gets you much idk).
I also think it would pay to write numbers little-endian.
@@duncanw9901 Exactly what I had in mind!
the thing is, if natural languages have no issue making this distinction (take the word יוון in Hebrew), this shouldn't be an issue with numbers.
we can also always take inspiration from Hebrew crossword solvers and turn the short tick to a short cross.
I think language is the part where I'm least impressed here.
There are studies suggesting that languages that express numbers in fewer syllables lead to people doing faster mental arithmetic.
A lot of languages have specific words not only for the singular digits 1-10, but also for some numbers into the teens. English has "Twelve", for example, instead of "Twoteen". French has a single digit words for eleven, twelve, thirteen, fifteen, and sixteen. And I think there's also some linguistic benefit to having words for numbers like twelve rather than "ten two" than just a historical linguistic artifact--small integers are just the numbers we will use the most, being able to express them quickly and unambiguously is inherently valuable, even if it makes the language harder to learn than Toki Pona. It's also worth noting that languages tend to have specific words for 20, 30, 40, 50, etc. In English: twenty, not "two ten"--now there's no syllable advantage here (although in some languages the word for 20 is one syllable) but there may still be a linguistic advantage to having a separate word--even if hearing is an issue, you'll never mistake "twenty" for "ten" or "two".
Rather than express number words in base 4, I think it would probably be advisable to have language work in...probably hexadecimal honestly? Either that or Octal, but I would lean hexidecimal for two reasons. First because it's 2^4, and 4 is a power of 2, so it'd be easier to deconstruct if you were already thinking in binary. Second having number names for numbers higher than 10 is fairly reasonable, cause a lot of languages already have that (1-12 in English and German, 1-16 in French and Spanish).
But honestly, this is all hypothetical, really. People are never going to switch number bases unless governments force it by changing money to a different number base. As long as people have 10s, 20s, and 50s in their wallet, they're going to think in decimal.
you have got to make a video for the better way to say numbers
Finaly managed to crack the text at 8:20 on my own.
The substitution is 啊 = A, 痹 = B, 雌 = C, 低 = D, 婀 = E, 付 = F, 佮 = G, 喝 = H, 乙 = I, 咳 = K, 刕 = L, 冪 = M, 妳 = N, 我 = O ,仳 = P, 儿 = R, 絲 = S, 偍 = T, 無 = U, 予 = V, 劸 = W, 牙 = Y.
And the actual text is that one comment from jan Misali's Ido video:
" YOU HAVE GOT TO BE ABOUT THE MOST SUPERFICIAL
COMMENTATOR ON CON-LANGUES SINCE THE IDIOTIC
B. GILSON.
DID I MISS THE ONE WHERE YOU SAID WHICH CONLANG
YOURE FLUENT IN AND READ AT LEAST THREE TIMES A
WEEK AND CAN READ NEW BOOKS IN EVERY WEEK OF
EVEN ONE YEAR OR LISTEN TO RADIO SHOWS IN EVERY
WEEK? NEW RADIO SHOWS? "
this is gold. thank you.
interestingly the characters pronounced in chinese are close to the sound of their corresponding letters
amazing dedication
Great video. Although the "speaking system" is quite flawed. So i made a new one.
Firstly, the biggest flaw is recursiveness. It's new and neat, but when you want to say a number to other person, it's better to convey number's magnitude right away. For example in decimal you would say "world population in 1975 was 4 billion and dot dot dot". However in binary it would be "three four three hex two BYTE two four one hex four three SHORT dot dot dot", and only when you say SHORT the person can sense the magnitude.
As a follow up, what if the person is writing the number down? For example when he hears "four int two..." how many zeroes should he put before writing down "two"? If the number is "four int two short" then 14 zeroes. If it's "four int two byte" then 22 zeroes. If it's "four int two byte short" then 6 zeroes.
Secondly, phrases are a bit bigger. A small number in decimal (255 - "two hundred fifty five") would be "three four three hex three four three": 22 symbols versus 37 symbols (or 6 syllables versus 7 syllables). We sometimes omit hundreds, so it's minus 2 syllables.
Thirdly, "speaking/writing system" interferes with the idea of grouping bits into groups of 2, 3 or 4 bits. System works with groups of 2 and 4, but does not with groups of 3. As it's said in video there is a learning curve, where person first learns arithmetic on group2 then group3 then (maybe) group4. But what if he considered group4 arithmetic too complex and stopped at group3 ? After he's done calculations on group3 he has no choice rather than regroup the whole thing and only then say the number out loud.
For the first problem I'd kinda go traditional method (millions, billions, trillions etc.)
For the second problem I'd compress numbers up to hexadecimal digits
For the third problem I don't know. Either group3 people will have to regroup, or make another speaking system for group3 representations (which is quite bad).
The digits
Imho it would be worse to use digit name, that we already use, so I gave new names, trying to reflect "binariness". Also these numbers should be fast and easy to pronounce and phonetically distinct from each other , because they will be used a lot in speech. I'm not a conlanger, but I tried my worst
wan du ti ro
rówan ródu róti ko
kówan kódu kóti kro
krówan kródu króti hes
Here ó is a stressed o. These words represent names for following hexadecimal numbers:
1 2 3 4
5 6 7 8
9 A B C
D E F 10
I used "hes" instead of "hex" for sixteen, because I think it's faster to pronounce it this way. For 0 we could use "zero"
I think "r" prefix for 4 and "k" prefix for 8 work quite good with binary.
With this system we can count up to 255 (in a similar way we count up to 999 in decimal without involving power names, such as thousands, millions, billions).
"hes" is used as a connecting word between quartets (similar to "hundred" in decimal). If the right half of number is zero, "hes" is omitted.
........ - zero
.....|.| - rowan
|||||||| - kroti-hes-kroti
...|.... - hes
..|..... - du-hes
|..|.||. - kowan-hes-rodu
Now for power names.
Let's call collection of 8 bits as bunches (not bytes cuz we will use this keyword). Bunch in decimal would be 3 digits.
Every bunch in decimal is followed by a power name (thousand, million, billion...)
If we do the same in binary with suggested names (byte, short, int, long, overlong, byteplex), these "power names" (technically power phrases) will be like that:
256 ^ 1 = byte
256 ^ 2 = short
256 ^ 3 = short byte
256 ^ 4 = int
256 ^ 5 = int byte
256 ^ 6 = int short
256 ^ 7 = int short byte
256 ^ 8 = long
(We suppose that most significant words come first)
Here I would suggest other naming system (which in general will have more syllables). However each power name will be represented with one word, pronouncing these names I think will be easier, since "int" "short", "long" are not pronounced well together. Also every name will end on "-yte", indicating, that it is indeed a power name:
256 ^ 1 = byte -> byte
256 ^ 2 = short -> plyte
256 ^ 4 = int -> fryte
256 ^ 8 = long -> ksyte
256 ^ 16 = overlong -> znyte
(I did not come up with an alternative for byteplex)
Now instead of "overlong long short byte" we would get "znyte ksyte plyte byte". But we need to combine these words. The rules are:
1) Last word gains prefix "o-"
2) Other words turn into prefix form
The prefix forms are:
plyte -> pil
fryte -> fer
ksyte -> kas
znyte -> zun
So by these rules "znyte ksyte plyte byte" will convert into "zun-kas-pil-o-byte", or "zunkaspilobyte". Here are first 14 power names:
byte
plyte
pilobyte
fryte
ferobyte
feroplyte
ferpilobyte
ksyte
kasobyte
kasoplyte
kaspilobyte
kasofryte
kasferobyte
kasferoplyte
And now one example with all of this: distance to the Sun in nanometers:
|... ...|||.. ...|.||. .||..||. ||..||.. .|.|.|.. .|.||... ..|..||. .|.|.||.
"ko ksyte wan-hes-kro ferpilobyte wan-hes-rodu feroplyte rodu-hes-rodu ferobyte kro-hes-kro fryte rowan-hes-ro pilobyte rowan-hes-ko plyte du-hes-rodu byte rowan-hes-rodu"
I'm pretty sure if we had developed binary & hexadecimal counting, we would not be translating numbers from base 10, or from metric, especially since considering the metric system was developed in a world where base 10 was well-estsblished. What if we had established a measuring system based on 100,000 [in hex) of the diameter of the earth through the poles? The "hex stick," if I may, would be about 16 cm in length. A little small, but still usable. As much as we like to make up words, I'm sure we'd still have million and billion, or an equivalent in hex language & a little different in magnitude. Mil & bil are kind of arbitrary labels, but if hex had an equivalent, you could again feel the magnitude. Let's just pretend for a bit that hun, thou, mil, & bil applied to the number in the 3rd, 5th, 7th, and 9th digit respectively.
Distance to the Sun would be about 37 milhex, if I may. A population of 4 billion would be EE milhex, which is just under 1 bilhex. Magnitude problem solved. 🙂
@@livingpicture Well yeah, we solved magnitude problem in our ways. But thou, mil, bil are names for 3*n hex digits (or nibbles) which is kinda arbitrary. I tried to follow the idea in the video: to establish names only for 2^n digits. The traditional approach (3*n digits) is probably easier to remember, but idk, I didn't learn my system :)
someone please like this comment later today so I remember to come back to this amazing comment
I kinda prefer this (but what do you guys think about this?):
0 = zero
1 = one
10 = 2 = two
100 = 4 = four
1000 = 8 = eight | 1111 = 15 = fifteen (there should be alternative names for 10[dec] to 15[dec])
1,0000 = 16 = *hex*
10,0000 = 32 = two hex | 1111,0001 = 241 = fifteen hex one
1,0000,0000 = 256 = *byte*
1111,0111,0101 = 3,957 = fifteen byte, seven hex five (fun fact: this is 3 syllables shorter than its decimal name)
1,0000,0000,0000 = 4096 = hexabyte (or hexbyte?)
1111,1110,0100,1001 = 65,097 = fifteen hexbyte, fourteen byte, four hex nine
1,0000,0000,0000,0000 = 65,536 = *"short"* (there's probably a better alternative name for this we can borrow from Computer Science)
This gets absurd...
10,0000,0000,0000,0000 = "two short"
1,0000,0000,0000,0000,0000 = "hex short"
1,0000,0000,0000,0000,0000,0000 = "byte short"
1111,0000,0000,0000,0000,0000,0000,0000 = "fifteen hexbyte short"
Maybe we should think of an alternative for "hexbyte"?
@@Gelatinocyte2we already have:
ten
eleven
twelve
draze
eptwin
fem
As a programmer seeing hex byte short int used as power names is both horrifying and amazing
You present your arguments well, however: I am nonbinary.
So I'll have to stick with Seximal, or maybe Balanced Ternary for the small number and negative advantages
Perhaps you would like base negative two, or negabinary? It uses alternating negative and positive powers of two so negatives can be written just like positives without the need of a minus sign.
This comment is pure perfection.
@@lemoneer7474 Wouldn't nonabinary be base 18?
I'm non-decimal. It's a shame no-one supports me! /jk
@@lemoneer7474base √-1 -1 (called i-1) works well too
I audibly laughed at 5 in the morning when you proposed “a stack” for 8^2, but it sounds only fair when you think about it
MINECRAF???!?!??!!
wow I didn't get it on the spot that's just super funny x'D
things heating up in the counting fandom
I think the proposed method on how to say the binary numbers has a couple of major drawbacks:
1) The recursive, non-linear conversion between words and symbols makes it hard to dictate, and non-trivial to write a dictated number down.
2) The symbolically easy doubling becomes unintuitive, e.g. 3 4 1 doubled becomes H 2 4 2
3) It can bury the most significant part towards the end, for example 3 4 2 H 3 4 2 B. You have to listen to all of the spoken numbers to make sure you're even in the right order of magnitude.
as far as I can see, these are all issues with existing spoken number systems. this system isn't much different than the one we use for decimal
i think 1 and 3 come from the fact that it's a 2^2^n system instead of a 2^kn system.
you could instead make it a 2^4n system (powers of hex), but name the powers using the 2^2^n system. So hex, byte, hex-byte, short, hex-short and so on.
like you'd say for example, three-fours-one hex-bytes three bytes hex two-fours, which is basically just how standard english base 10 words.
@@thebestwaytocount These issues are much more significant in binary since they show up much earlier on than in something like decimal
Recursive algorithms are simple algorithms. No one that has any experience reasoning about procedures thinks any different.
Take a cue from the world of programming, where our spoken numbers are base 16 but our math is base 2. If we used the numerals from this video it would make the conversion trivial.
Don't you love it when you have that one really strong opinion that no one else cares about, but then you stumble across an hour long video essay about it at 3 am
It's my favourite feeling tbh
> that one really strong that
I think the noun is missing.
Regardless, yes.
@@zairaner1489 oops
Me But One's Compliment Notation.
I always felt, instinctively, that binary "should" be the best base, but it just seemed too cumbersome to use in practice. Thanks for your excellent work.
I went into this with a strong preference for hex. I got pleasantly surprised by the suggestion of binary, and when you started grouping the bits it's essentially a multi base system around binary. Hex is really just groups of 4 bits, which is why it's good. I think the naming would sound more natural if it's done for groups of bits instead of what's suggested here
Yes! Exactly!
I feel like I've stumbled on a piece of forbidden knowledge.
The funniest thing was trying to come up with some counter-arguments while noticing I had already used a binary representation of octal (tri-octal to be more precise) to convert an alphabet, and already figured on my own how easy it was to apply a vigenaire cypher mentally on it thanks to how trivial it is to do arithmetics on it
This video is exactly 200 minutes in seximal. Cheeky.
Are minutes a seximal unit?
@@Arbarano(kinda sexagesimal)
1:07:16 That Toki Pona fact at the end... This man doesn't know mercy
i fully support the use of "stack" as an official name for 2^6
10^110
and i use nibble for 16
@@snailemoji96or hex
16:00 I thought I'd heard that somewhere; it's a "trick" used to store floating point numbers in the IEEE standard; the leading 1 in the mantissa is implied
i dont think so. they have to represent 0 too
They represent 0 by setting the exponent to its lowest possible value, at which point that leading 1 is treated as a leading 0 instead.
jan Misali said this in his floating point number video iirc
I like how we all do this research but will still all ever only use decimal casually
This feels like an induction into a religion. Now I see how things should be, and my duty is to convert the non-biners to the true light of one and zero.
In the end, it's not about "which system is generally better", but "which system is better after I came to a conclusion which were my criteria and priorities".
Nah. It's more about "which system am I already using".
However, I think binary wins when we start talking about how to TYPE these numbers out.
My jaw dropped when I saw that square root algorithm. Fuckin’ black magic 👍
TLDR: binary > every other base because binary < every other base
unary.
i would love to see some examples of this binary notation used in some common settings.
some examples
>in a car for speed and distance.
>prices and measurements in a grocery store.
>numbers in a video game.
>a deck of playing cards.
that batch of 4 trick seems like it would be very readable.
i would also like to see something like "babies first numbers video", that would really show how easy binary actually is.
a webpage for "try math in a new base" could be a nifty demonstrator for it's usability.
i think the symbols themselves need a direction notation, the underline may be enough but I'd like it more obvious.
I made a userscript that you can find on greasy fork called "convert to binary" that attempts to convert numbers on webpages to their binary representation. The only pitfall of it is that it doesn't accurately convert non-integer numbers (like it'll turn 10.32 into lılı.lııııı)
First speed limit sign you see that says 10110 is basically the end of the emperiment.
@@jibbjabb43 well you wouldn't be using Arabic numerals for the task, you would use binary combs like the ones showed in the video.
@@ARockRaider It doesn't look any better. The issue here is both conversion and size. It's pretty silly to suggest that using rather indistinguishable characters is somehow better here. There is *some* ability to parse a misread sign simply becuase of possible interpretation, but that's because we're operating within a presumed 'field' of possible speeds.
It's also more important that I used a number like 22, which I can defend, and you instead focused on what the digits look like. Which tells me you either didn't take my critique seriously or you are too predisposed to the idea of it working without considering it's real-world applications.
@@jibbjabb43 I had assumed that you picked a number to look the most outlandish, i was pointing out that you wouldn't use actual 1s and 0s and that you would be using a notation that makes the numbers every bit as clear.
that you picked 22 makes my assumption of an intentionally outlandish number very clear, on top of that the exact speed would be adapted to the number system 20 for example would be i .i.. and as a group with proper underlines would be very different from 40 or i. i...
neither look anything like 30 or i iii.
but i expect that you wouldn't be using multiples of 5 for speed-limits, rather you would pick a new batch that always stays neat and round.
I was gonna make a joke comment saying “my favourite way to count is binary” but then you actually open with binary I love this
My first reaction was "wait, binary??", especially as i watched the jan Misali's videos before, but then it turned into "oooh, thats how we can do it", and then "it's beautiful as heck"
I have no idea why youtube didn't recommend this to me sooner this is the rabbithole I live for
Actually, I have a more general counterargument:
Redundancy is good, actually. All of these arguments demonstrate that binary is the most efficient base in a similar way to how Ithkuil is the most efficient language. Sure, that might be true on a technical level, but if you miss more or less any information, it's harder to recover. For example, something weird happens with the HDMI on my TV, so the edges of the screen get cut off when I try playing a Switch game on it. But because all of the Hindu-Arabic numerals are fairly distinct, I was still able to follow the timer in the Star raids in pokémon, despite the top half of the number being cut off. Meanwhile, if the top half of your binary numbers got cut off, that'd be it. There would be no way to distinguish numbers anymore. Or similarly, consider seven-segment displays. They're actually *horribly* inefficient, since you can display 128 distinct figures with them, and we only use 10. But because we use so few, depending on which segment goes out, you can still distinguish most numbers. And even in a case like the middle segment going out and making 8 and 0 indistinguishable, context clues help. If you see 1-"thing that looks like it's supposed to be a 9" change to 10, you can infer that it's probably counting down and is probably supposed to be 18. But if it changes to "thing that's probably a 2"-0, you can infer that it's counting up and that it *is* supposed to be a 0.
Or similarly, finger counting. Functionally, "binary" finger counting is actually duotrigesimal. No one's going to actually think of each finger as its own digit, so you're effectively expecting people to learn 32 different hand shapes. So if you're just using hand counting to show someone a number, like a kid going "I'm this many", it's going to be less efficient compared to just holding up a number of fingers and subitizing. Or similarly, if you're actually counting up or down, with both seximal and jisanbeop, you only ever need to change either 1 finger at a time or all 5 fingers. (And with all 5 fingers, it's also always either resetting the hand or changing 4 fingers to a thumb) Meanwhile, with binary finger counting, you might have to change any number or combination of fingers. And that's actually so well known of a problem that Frank Gray came up with the idea of reordering binary numbers so you only ever have to change 1 bit at once all the way back in 1947. (With the idea itself going back to at least the 1870s)
I actually agree with most of your points, but have a few nuanced points to make.
1) This one's simple, actually using binary doesn't mean blocking out half the digit would leave it unreadable. Binary is actually pretty good about that, you can *always* tell if you can see part of the digit.
2) I use decimal for readability purposes, but I find binary a better system for doing simple counting tasks. Remember that you could use different systems for different purposes, it's done in some cultures in other areas, such as speaking one language and writing another. (Looking at you, Switzerland...)
3) While your argument about finger counting has some merit, I actually think that's only a problem for children. Since we're actually using a *unary* system for that task, binary finger counting is strictly better but also more complex. (Seeing each finger as a digit with two states, the *number of digits* doesn't have anything to do with the counting system - that's just a physical limitation.) It's too complicated to teach to children without a whole course around counting systems, which doesn't make sense at that stage of life. For adults in the mathematics or comp sci sphere though, it's easy enough to adopt and therefore we probably should do so on an individual basis.
Not sure what the video itself seeks to argue, as I'm not about to watch someone try to convince me of the merits of a counting system I already use, so don't take any of this as an argument toward the creator's points. I'm simply arguing for what I believe to be best in my own experience.
@@impishlyit9780
1. Eh, not necessarily. One of the video's weaker arguments has to do with number length. It basically makes a *font* based argument, where if you change binary numbers to tall (1) and short (0) lines, the horizontal width of numerals becomes comparable to other bases. And that really *does* run into that issue. Or more broadly, I forgot to include this bit for contrast, but imagine a binary display. If any one segment goes out, you cease to be able to distinguish anything in that place. At a minimum, you'd need to do something like pairs of dots, where only one can be on, for you to be able to lose a segment and still be able to distinguish numbers. Meanwhile, even though only the top left and bottom right segments lack minimal pairs, you can still generally make things out on a 7-segment display, even if one of the segments goes out. You'd need 2-3 to go out before it starts really impacting your ability to read things.
2. No counterargument here.
3. It's also more than just kids. For example, how often do you *actually* need to count on your hands? It's typically smaller things, like how you might silently count down from 5. And while I'll grant that people probably count down specifically from 5, in part because we have 5 fingers on each hand, unary feels a lot easier for that. In a way, it's sort of like sorting algorithms. Merge sort is more effective for really big lists, but it also requires a lot of overhead. So for shorter lists, the less asymptotically efficient insertion sort winds up running faster overall. The main time I can think of that it's potentially useful to display bigger numbers would be something like days, but there are also systems like the Medieval Arabic hand numbering system that do it more easily than finger binary. It's actually even more efficient than finger binary. It encodes one digit on your thumb and index finger and one digit on your other three fingers, so it can go up to 9999 on two hands, as opposed to 1024.
But overall, my main criticism of the video really is that it successfully argues that binary is the least redundant numbering system possible, but not that redundancy is something we need to be avoiding
You should into your TV's settings and turn on either "overscan" or "just scan", or set the aspect ratio to "full" or "just scan". Turning on "game mode" might also fix this.
"Alright buddy, you've pissed me off."
*counts to 20 in binary using my hands*
counts to I .I.. using my hands*
You extend your pinky and middle finger?
I really appreciated jan misali's introduction to the advantages of seximal, and have considered myself a convert for many years. You present a strong case here, and it's certainly convenient that your choice happens to be binary. One of the two systems that I have to know anyway. I'd like to see separate (short) videos on some of the topics here. How to write binary numbers. How to speak binary numbers. How to divide binary numbers. Etc.
i love the binary multiplication table because it's also the logical AND gate
Some notes as I go:
- I see you don't want to be able to write the numbers 0 and 1, if the leading digit doesn't matter for information.
- You seem to be neglecting the human tendency to ossify details. A binary system wouldn't stay a binary system for long. Within a few generations, eople would start writing shorthand octal or hex with a couple strokes rathet than four or five, and those would become the basic unit. That grouping trick is actually a negative when you look at language evolution.
- "What's the Most Commonly Used Prime Which is Incompatible with This Particular Base?" got a laugh out of me, good work on that one!
- Your repeating symbol and your grouping marks would be super easily confused in hasty handwriting. That's not an inherent binary problem though, just one with your notation.
- 2⁴ should be called nybble or nyb. It's cuter and sounds better in practice.
- ... I both love and hate you for pointing out to me that stack is a number name. It is though, saying something like "three stacks twelve" is perfectly normal in minecraft contexts.
- I think seximal is still more aesthetically pleasing, to be honest. And that matters because the only context switching away from decimal will ever happen is art. ... though balanced ternary might win there, it's just the right amount of weird to be really fun.
i love this comment so much. Agreed with everthing you said 😂
For your first point: I do agree that they handled this poorly. But I think all is not lost. Instead, you can think of what they are talking about as a consequence of the fact that simply knowing the bit length of a number tells you the value of its most significant bit (which has to be 1). The “length” of the number does not give you so much information in other bases, this is definitely true.
As for your second point, so long as people don’t forget where these ossified forms come from and they are visually similar to unossified forms and can be easily decomposed into them, then I don’t think this is actually an issue.
Since the measure of information per digit is considering all infinitely many numbers and the number of single digit numbers is finite, the amount of extra information carried by that first digit tends towards zero in the limit of considering all numbers since a finite portion of an infinite space is always 0% of the total space.
I think this means that when considering information on the long scale this simplification is fine, since the context of the discussion was already the long term efficiency of a base and not its efficiency writing small numbers.
@@keldwikchaldain9545 1 and 0 are kind of important numbers though.
Man really about to convince me the invention of every number after 1 was a mistake
43:29 "Traditionally, to notate a recurring fraction, the entire recurring segment is marked, but that doesn't make a lot of sense: It's simpler to just mark where it starts."
I think your opinion comes from the age of typing and digital text representations. In hand-writing, marking the recurring segment makes more sense for the following reasons:
(1) Often, you will get this repeating decimal from just long division until you enter a cycle, and adding a line of the repeating part is something you can do after you've written the number, which doesn't require you to erase anything or rewrite the number, and also separates the repeating segment from any digits of the next repetition you may have written before realizing it was repeating. This doesn't apply if you're rewriting the number somewhere else, but if you're copying a hand-calculated quotient with a long repeating decimal, you'll want to use some mark like the over-bar to mark the repeating segment of the decimal anyway, so any other notation like the "r" you're using will just be another notation to learn IN ADDITION to something like the over-bar. Thus, I think this is the most important reason.
(2) In hand writing, writing a line over several numbers is not significantly harder or more time consuming than writing an r before them. It's actually a slightly simpler shape than an "r", though the length and precision required mean I'll just say it's about as hard overall.
This is different from typing or digital text, where putting bars over numbers requires special characters or text formatting, both things that are much more of a pain to deal with than just typing "r", both just in typing it, and in the sneaky problem that things like this sometimes won't render properly on other people's computers or on printers. (With typewriters, it usually requires some kind of tricks involving typing over the same text I suspect.)
That being said, one downside of the over-bars specific to hand-writing is that, due to the imprecision of handwriting, it can often be unclear to readers exactly which numbers the over-bar is over. In typing, this would only really be a problem if your method of creating over-bars was bad, and using preceding "r" would get rid of this issue in handwriting (as would using parentheses or circling, although those aren't quite as convenient to add on to an already written sequence of digits as an over-bar is).
In my school they taught us to write repeating digits usung brackets:
1/7 = 0.(142857)
This eliminates the precision problem as brackets are easier to write and mark exactly where the repetition begins and ends
Amazing. Genuinely, i was amazed multiple different times. I'm a computer science major myself, I had already discovered the bit about multiplication, but the section on factoring? I had never thought to do that, and it blew my mind. Division? So much better than base 10. Your notation is beautiful, I'm definitely adopting it for when I do binary work, and I'm geniunely strongly considering switching to binary in my everyday life.
missed opportunity to title it "The best way two count"
smart!
As someone who played with numerals a lot as a teen, and who's done a lot of programming, I find it VERY telling that we programmers who work with it often will ALWAYS display it as hexadecimal.
The 'best base' is really about what is the best base for humans, and minimizing how it pushes against our mental limits.
I really liked your new binary-based symbols for hexadecimal numbers. I wonder if a society based on binary math and those symbols would work better and learn math easier.
I think the binary naming in this video got really awkward and long to speak. Instead, use the groupings and name those. I think a society would need to use octal or hexadecimal for communication, while doing arithmetic in binary.
I'm entirely convinced about the mathematic side. But I think that goes without saying as the arguments were quite conclusive.
As a programmer myself, during my education (specifically when learning assembly), I always found working with the binary numbers far more intuitive / less cumbersome than working with the hex numbers. Like, at all points, I would think about it in the binary expansion, but then every time I had to actually write the number in the program I would have to group the binary by 4s and convert to hex, its just a mess.
This is not to say hex is bad per se, but that it is necessarily just a layer of translation/abstraction over the actual numbers, which are binary. "x5555" is obscure, needs translation. "101 101 101 101" is the real number, the thing you actually need to work with. Anyways..
Well let’s see. The majority of programming is done staring at a text rendered with a monospace font. Why of course it would be quite a nightmate to juggle literals 32+ characters wide. There’s no real compressibility unless the programming language allows macros or something to use a custom number literal format (or the IDE used allows to work with proportional fonts in comfortably).
I suspect there can be more mundane reasons other than this one.
@@05degrees Yeah, suppose we start with a system like the proposed. I think it wouldn't take much for people to start comming up with symbols and meanings that more readly represents the common quantities and groupings.
Me at the beginning:
Worth watching, but unlikely.
Me at the middle:
Fantastic arithmetic, but I mostly say rather than play, and binary is long.
Me at the end:
I'm convinced.
I love the notation system you created and the grouping shorthand, it’s very elegant and also makes the fundamental patterns you discuss visually obvious at a glance, without having to translate into numbers, even for someone totally new to the notation without having built base specific intuition. I could easily learn to do a lot of that math with your binary notation without converting into or out of decimal, which makes an excellent argument for it’s naturalness and simplicity
Felt compelled to decipher the text at 8:20.
It reads:
"You have got to be about the most superficial
commentator on con-langues since the idiotic
B. Gilson.
Did I miss the one where you said which conlang
you're fluent in and read at least three times a
week and can read new books in every week of
even one year or listen to radio shows in every
week? New radio shows?"
Was a fun challenge, thanks!
Congratulations! You have been given the award for the most uses of "Fermat's Little Theorem" in a video that is not about Fermat's Little Theorem or about modular arithmetic!
I really loved the video, but I think, as many commenters have pointed out, that the legibiliy of your line system could be improved. Some might even compare it to a certain Minecraft optimised number notation system.
My suggestion would be to change out the short line for a small circle. This not only de-clutters the lines by separating them more cleary, but also prestents a great opportunity for ligatrues utilising the existing latin alphabet and also reducucing stroke number. or for l. and and for .l maybe for l.l, and maybe for .l. Keep in mind the arcs would be proportionally smaller in written notation
I'll flesh this out further and come back to this comment later, I'm expecting some grouping and priority conflicts, but there might be a nice way to deal with them.
I take some umbrage with the “efficient finger counting” argument, because finger counting is primarily a tool for teaching children. Binary finger counting (and even seximal finger counting) is more complex than simply counting the number of fingers raised. Bases are an arbitrary construct and humans don’t think in terms of them, so a child will never intuitively understand binary finger counting the way they understand “base 10” finger counting (though i would argue it’s more accurately described as base 1 finger counting)
Many cultures have counted in dozenal or hexadecimal on one hand. Many Asian cultures to this day count to ten on one hand and find the western way to count on fingers jarring. Your argument is eurocentric.
I’d argue the mentioned systems are still ultimately base-1 counting systems. They use 12 or 16 naturally ordered positions of the hand. Counting up to 12 with the dozenal thumb-thing is still just moving your thumb through a “number line” of hand positions. It still retains the simple linearity of western counting that’s lost with binary counting.
well actually maybe base infinity would be more accurate but my point is that neither 10 finger counting nor 12/16 one-hand counting have to actually deal with the things that make bases complicated (i.e. multiplying by powers of the base according to the position of the digit)
@@considerthehumbleworm During the Middle Ages doing arithmetics with your fingers was common enough that almost every book on mathematics had a chapter on it. For example in _'Liber Abaci'_ by _Leonardo Fibonacci_ finger counting is the first chapter. Systems to count up to 100.000 were around for quite some time.
Conflating finger counting to what _we_ today in the west teach to toddlers is a mistake. Drawing conclusions from oneself about others rarely works out.
Finger counting is what we teach to toddlers tho. I think it’s a bad idea to offer binary finger counting as a replacement in those contexts. It’s useful to be able to count higher than only 10 in other contexts but it only muddies the water for learning basic arithmetic. Honestly, teaching toddlers systems that go up to 8 or 16 is probably preferable to 10 to ease the transition from “number line counting” to using actual positional binary (even binary finger counting, eh?). That transition’s a bit of a given with 10 finger counting and base 10, so it’s worth considering how that works with binary. Also we don’t have to hate each other because we disagree about numbers lol
The best base is base 10
Get it? Not base "ten", but base "one zero" where the base of that number is whatever you want. For me that's base "two" too.
don't explain the joke
explain the joke
@@mrosskne 10 is 2 in binary
@@mrosskne 10 is 6 in base 6
@@mrosskne10 is 10 in base 10
The thing that fascinates me the most is that base 2 is every base 2ⁿ at the same time, so it is capable of using properties from all of them.
Amazing video.
I think many of your ideas have basis. But i dont think that it works quite as well in practice. I am a fan of base 6, because of jan Misali's video, but as a programmer and computer scientist, i have to acknowledge the advantage of binary for raw computation strength. There is a good reason we use it in computers, and computation is a place where numbers shine. But unfortunately for us humans, we are not easily wired for base 2. It's hard to say how the real world efficiency of human calculations would turn out if we completely switched to binary, but i think that our brains are slightly better wired to understand larger bases. As someone who has poked around with basic arithmetic in both binary and seximal, i can say that binary is easier algorithmically, but if i had to do a lot of math, i would prefer base 6. Part of this is the practical problem of we need to write down the numbers we are doing math with and i feel like bigger numbers are better for cramps. I think it would be interesting to do legitimate experiments teaching people variou bases, and see who in the long run uses the bases most effectively. Seximal, for me, just seems to be a nice balance between small bases, which are easier for math, and big bases, which are easier for brains. Curious to see if jan Misali responds, but i still think i am on the seximal side for the best base for humans.
i ask him in his new super mario game video and no hes not gonna respond
What a high quality video! I wonder if I’m really slow or most could keep up without extensive pausing and replaying. Congratulations on such a well made video regardless.
Not just you, lots of parts seemed a little confusing
Programmers and hardware engineers have been doing this forever, using 8 or 16 as the "compressed" written format, but dropping back to bits for actual arithmetic, especially when making machines to do so. Anyone in those groups worth their salt can convert from 0-F their binary quartets and back intuitively, and when learning to do that, often use fingers in ways that resemble your grouped bits. The main problem with binary as a written or spoken system is that its hard to read at a distance, and the octal/hexidecimal are "error correcting" in that a smudge on a piece of paper or a dent in a sign makes simple tally marks unreadable, but leaves letters and hindu-arabic numerals mostly in tact.
no matter what you think, base 10 is always the best (as long as you read it in your base of choice, of course)
got us in the first half
This really is exceptional content. As a nerd I really appreciate the work that was put into this video. Great work. My only note is the pacing was too fast for me to grasp some of the details that were claimed to be “immediately obvious”.
that's what the footnotes are for! this video would've been astronomically long otherwise :)
@@thebestwaytocount Says the uploader of an over I hour long video responding to an I ..I. minute long one
@@theramendutchman Dude an hour and a half is very short, stop watching tiktok
@@matheuscabral9618 First, there's no need to make assumptions and attack people. Please refrain from it and maintain a civil discussion.
Secondly, my point is that they said their video would've been longer, which is unexpected to me seeing how it responds to a 10 minute video; this video is already so much longer than the video it responds to!
@@theramendutchman eh
"just use only three fingers on each hand" leads to communication issues. The great thing about addition only decimal finger counting is no matter how you count you always get the same result. It is unambiguous and inclusive to everyone who has any fingers. But yes it also means that you can only communicate very few numbers at one time. Basically decimal counting is TCP (sacrificing speed and efficiency for making sure it is as unambiguous as possible within the system) while binary counting is UDP (so prioritizing speed and data density while risking the wrong message being sent)
Babe wake up found new banger channel
I tried the sqrt algorithm with 225 and I feel like a forbidden secret has been unlocked to me
though I do have one question, how would this algorithm work with non-perfect squares and would it work with complex numbers as well?
@@nukollodda For non-perfect squares, the algorithm continues past the decimal point to whatever degree of accuracy is required. As for how to adapt it to accept complex inputs and outputs, that's above my current level of understanding.
I want to point out that another word, while less common, for 2^4 is "nibble" if we're using "byte" for 2^8. Hex is fine, but I want more excuses to use "nibble", thanks.
came here to point out "Nybble" as well :)
I think 'word' instead of short is better too.
@@minerscale personally I'd have to disagree here. Word only exists for its counterparts Dword and Qword. It would perhaps be better to either go with short, int, long or switch all of them to Word Dword Qword
16:14 this is exactly what floating point does. It doesnt store the leading one to save a bit, which allows the number to have a bit more precision
18:14
What happens, is that you'd stop being able to count it easily by sight at a distance - counting "how many fingers are extended" doesn't care about positioning, binary counting does.
And the issue with this, is that when using hand signals to convey numbers, you're probably doing that _because your voice can't reach the other_. Most situations where that happens are scenarios in which stopping for a couple seconds to reorient & count the exact numbers for yourself is an awkward use of time.
As an example: if I am on a construction zone with operating machines, and I need the handyman to replanish the third team with beams, screaming "BRING BEAMS TO THREE!" will probably be only partially heard, and after hearing the expected "WHAT?", being able to just raise three fingers while hollering "BEAMS TO THREE!!!" adds a layer of important redundancy, as the handyman will be instantly able to note that one word is 3, and the other is the object to bring, allowing them to process it better across the other sounds.
And yes, this is also true with binary, but with that system you'd be forced to use and read for the specific fingers which hold specific meanings, which can easily mean that the handyman needs to ask clarification _again_ because either they didn't catch if the fingers raised were for 2 and 4, or 1 and 2, or they caught that, but not what to bring.
It's already hard enough with all the redundancy we use, and needing to clarify and reclarify everything is stressful and awkward for everyone involved.
In this sense, using binary for finger counting is strictly worse.
Seximal, or well, the other counting methods named came about because of that need to balance redundancy and information density.
Sometimes, being able to alternate between asking for 2 of something and 60 of something is important, and between the thumb being quite ubiquitous and only needing to pay attention to "which hand are they raising", reading fingers at a distance using only 4 parts, is plausible.
Using 10 parts, making order important for every finger, is simply too awkward to use.
However, I will concede some other thing that's reasonable: if instead of an specific number you are asking for a general amount, then having each finger imply a different order of magnitude is useful, albeit there's already an order-independent system for that in which the amount of fingers raised is how many 0 are after a 1.
So yeah, binary is really damned good for counting... In paper and computers. It isn't good for hand counting tho, but honestly it doesn't need to be - hand symbols can simply not change alongside written ones.
i dont regret how i spent these IııIııı minutes
oh my god this is the first math video in a long time that has made me feel like this
14:49 I can't speak for jan Misali, but I think he probably just meant that if you write binary with Hindu-Arabic numerals, the numbers get unwieldy in size, and base 4 in Hindu-Arabic numerals is a good way to compress it.
15:36 This fact isn't what makes binary the most efficient, it just makes it the most efficient by a long shot. Even if we didn't consider the fact that the leading digit can't be 0, by replacing b-1 with b in the formula, binary would still be more efficient than base 3.
side note: I don't know how we're measuring which is better rigorously, even though it's clear intuitively. Because the graph for base 3 does sometimes dip below the graph for base 2, like for example between 8 and 9. Is it the one which is the lowest for the most numbers, which has the lowest average, or what?
16:27 I wouldn't be so hard on seximal. jan Misali admits that in seximal, numbers are written longer, and the justification isn't radix economy, it's that the square base is small enough to be used for compression, it's only too large to do basic arithmetic.
In fact, I think it doesn't make all that much sense to use radix economy as an argument at all, our brains don't simply look at the representation of a number and take a specific amount of time to process each digit based on the log of what they expect the digit to be. Although it would be fun to see a study comparing each base in its own writing system and the speed of writing or reading numbers.
Personally as programer I chose binary with hexadecimal compression because it is the best of both worlds. Your number to long, just start compressing into hexadecimal, need to do math decompress it. It just allows better writing efficacy and in low level programming you just do base two operations on hex nums anyways.
I am also a programmer, I love the idea of having two different ways of writing numbers, I previously only thought about hexadecimal and even made my own writing system for it wich also uses a sub-base of 4, 2*2 = 4, and 4*4=16, I love the symmetry
Binary for math operations and hexadecimal for showing numbers!
I LOVE how you play the corresponding musical interval whenever you mention a fraction
I feel like this system could be improved by using unique labels in spoken language for every permutation of 4 bits, effectively treating it like hexadecimal when talking. You already concede that spoken binary would create some very long number labels and offer quartery as a compromise, so you might aswell embrace the pseudo-hexadecimal nature of grouping 4 bits together by adding a horizontal line at the bottom by covering these groupings with a single label each.
Exactly! And since lower numbers don't have decimal based names (compare "eleven" to "twenty-one") we can start by borrowing those. In spanish we have unique names up to 15 (it's "quince" as opposed to "diecicinco") so we already have the firt 4 digits covered. From that point, I'd say the next 4 digits should have a prefix (so, I.II. would be something like hexasix), but we would need to find something that rolls of the tongue (I love stack for 64, but it sounds clunky as a prefix)
After watching, you’ve successfully convinced me to use base 12
There is no base 12 only base 10 😂
every base is base 10
which base is the "12" in?
I love how the sounds are so entangled with the meanings, impressive.
"Single hand is chiral". My favorite sentence that says so much while just saying that hand is a hand :)
The music argument in the early chapter is interesting. Our notation is specified in fractions of powers of two - but they are fractions of four beats per measure. We also work just as often in three (triplets). However, we count in groups of 4 or 6, and often 8 (dance) and 12 (triplets over 4 beat measure). When dealing with irregular numbers like 5 or 7, the rhythm is usually done in syncopated emphasis of groups of three and two. For example, 5 is usuall 3+2 or doubled at 3+3+2+2; 7 is usually 3+2+2. Syncopated division of 8 is common with 3+3+2 (typical of latin and jazz) and 12 with 3+3+2+2+2 in flamenco compas. You further get the complication of one instrument playing a phrase in triplet over the rest of the band playing in duple.
The final pattern is that phrases tend to also be groups of four measures, a prime example being the three sections in a twelve bar blues. This is super internalized, and my brain stops and pays attention if a phrase isnt that multiple.
4-bar phrases and 16-bar sections are common enough that when I'm counting measures of rest or something, it usually feels most natural to count in a base 4 system.
On each hand I use my 4 non-thumb fingers and with 2 hands I can count up to 16.
For example:
1 = ...|
2 = ..||
3 = .|||
4 = |||| (1-4 had implicit |... right)
5 = ...| left, ||.. right
6 = ..|| left, ||.. right
7 = .||| left, ||.. right
8 = |||| left, ||.. right
9 = ...| left, |||. right
etc.
On both hands I start from my index finger, but with each one visualized palm-facing down, that results in the left hand counting right-to-left and the right hand counting left-to-right, but the point is that it feels right in the music.
The "best" base is the one which lies at the intersection between maximal efficiency (cramming in as much info as possible), maximal legibility (being able to get that info back out), and maximum versatility (being useful in multiple numerical arenas). While binary is objectively the most efficient way of storing data, it isn't the most legible or versatile, so that's just the starting point for finding our magic intersection. Legibility peaks around the concept of subitization, the ability for the brain to quickly count numbers, which begins to decrease after 4 and is gone for most people by 8. This means that any number longer than 4 digits is going to be harder to read, objectively, with separators as a bandaid patch. Thus we definitely don't want any common numbers to exceed 4 digits, so binary isn't necessarily the best choice on this front. The last point is versatility: can it express simple fractions simply, how easily does it convert to binary, can you count on your fingers with it simply, etc.
At a guess based (heh) on all this, the intersection is probably around base-4, which means no one gets to be happy and DNA actually got the right answer somehow.
Phone numbers, IP addresses, credit card numbers, social security numbers, etc: these kinds of things usually try to group digits into groups of at most 4. If they tried to make groups longer than than that, they would be harder to remember.
For decimal we tend to use groups of 3 so each group is about a thousand. But assuming we can handle groups of 4 just fine, and assuming we want a group to still get to around a thousand, a nice base would probably be around the 4th-root of a thousand? Base 6 looks pretty good for that. But a thousand was kind of arbitrary. With base 4 in groups of 4 you can get up to 256, and while that's not a thousand, it's still probably big enough for most people's everyday purposes. Like, it's larger than a Dunbar number so it's probably good enough, right?
Part of the argument for binary radix economy was that if a digit represents less information, then you can get away with more of them in a grouping, but I'm not convinced that's true. I still think grouping by 4 is best. In music, we can deal with measures of 4/4, 4 bar phrases, etc. just fine, but when time signatures get more complicated than that we tend to group them into sizes 4 or smaller, like grouping 5 into 2+3, 6 into 3+3, 7 into 2+2+3, 8 into 4+4, 9 into 3+3+3, etc.
if you had groupings of groupings, though, what could that look like?
Imagine if the regular groupings presented in this video are like music notation for 8th notes in groups of 4.
Maybe groupings of groupings could be like 16th notes!
With this notation, you get binary with all the benefits of base 4 (and 8 and 16), so if you say base 4 is the best, then base 2 is the best
this is remarkably well-researched and thought out. just gotta show this to my mathematician friends to explain to me the bits i dont understand
This is an official petition for this channel to teach us math using binary
I'm signing it using binary code that is sent to RUclips's servers from my PC to add this comment here
How many likes is your goal?
I say at least 10100 is good
YES
nah pinary is better
@@WhitePikm1nphinary
I think the whole argument that non-binary systems are "more wasteful" because they might have an extra leading zero is pretty enormously flawed and relies on the assumptions that A) pattern recognition and readability isn't a thing, and a series of two symbols that repeat over and over is at all functional.
By this argument, and the whole ""efficiency"" radix argument, you could also make the same deduction, that all language should just be written in binary strings. After all, what is the English alphabet but a series of base 26 numbers?
Good point, except the alphabet isn’t for numbers, but for words and/or sounds. You’re not performing arithmetic with words. Also, no one is ignoring pattern recognition. The video simply argues that it can, in fact, apply to binary, especially with the right notation and proper parsing methods for the bits.
17:43 this is incorrect. While you are thinking in decimal, each finger only carries one numeral of information. It either "doesn't" exist, or it is a 1. You count the 1s to get the number. We have a name for that number system and it isn't decimal. It's unary. Tally marks. Base 1. Of course binary is going to be way more efficient, because you are doubling the amount of information that each finger can carry, you go from being able to count to 10 (or 20 if you're european) with 10 fingers (1:1) to 1023 (102:1) or 31 with one hand, because every single finger added doubles the number you can count to, whereas in unary, every added finger only increments that number by one. Surely if you actually gave each finger 10 different positions corresponding to the 10 digits of base 10 and somehow were able to reliably use, hold, and distinguish all of those positions simultaneously without the tendons in your fingers binding against each other like a shitty typewriter, you could count to 9,999,999,999, which is way more than anyone would ever need for everyday counting purposes (though so is 1023).
Finger binary is super useful though. I use it all the time at the risk of people thinking I'm making rude gestures or gang signs (18 = heavy metal, 19 = I love you, 4 and 5 = fuck you, 3 and 7 = finger gun)
and I do happen to know a way to calculate square roots. So a square root of N is a number G such that G * G = N, we can formalize this:
G^2 = N
G^2 - N = 0
f(G) = G^2 - N
f'(G) = 2G
g(G) = (G^2 - N) / 2G
g'(G) = G - g(G)
Keep computing g'(G) on any initial guess number and it will eventually converge on the answer. It converges faster if the initial guess is closer to the true answer, as g(G) is essentially computing the relative "error" of the guess, and thus will oscillate past the true answer and slowly converge on it. If N = 9 and you start with G = 1, then you will get 1 -> 5 -> 3.4 -> ~3.1 and you can see how it converges to 3. But let's say you're trying to compute sqrt(2), you can take guess values based on the closest known perfect squares, which are 1 and 4, whose square roots are 1 and 2. So the answer must be between 1 and 2, so you start with G = 1.5 and it converges to 1.414 fairly quickly. Same with 10, you know the closest surrounding perfect squares are 9 and 16, whose square roots are 3 and 4, so the answer must be between 3 and 4. Choose 3.5 as your initial guess and it quickly converges to ~3.16228 within 3 generations.
Here is that algorithm implemented in unix dc:
[lGd*lN-lG2*/lGr-sG]sg
to use, set the precision, G and N registers and then run the macro a few times. Here's what that might look like:
[lGd*lN-lG2*/lGr-sG]sg
20k 10sN 3.5sG lgxlgxlgxlgxlgx lGp
3.16227766016837933200
People call me insane for using dc, but it's fucking awesome once you "get it". I even used it to implement a working base 64 decoder and encoder, which I will concede is insane, but it was fun.
But yes, approximating square roots by hand is a pain in the ass. That sqrt(9) with a starting guess of 1? I did that by hand. Really pushed the limits of my mental math abilities when I had to divide 2.56 by 6.8 and make a multiplication table for 6.8 by adding 6.8 to itself over and over in my head. Look at this shit
sqrt(n) = x such that x*x = n
x^2 = n
x^2 - n = 0
f(x) = x^2 - n
f'(x) = 2x
g(x) = f(x)/f'(x)
g(x) = x^2 - n / 2x
guess - g(guess) -> ans
9 -> g(x) = x^2 - 9 / 2x
1 -> g(1) = -8 / 2 -> -4
1 - (-4) = 5
5 -> g(5) = 25 - 9 / 10 -> 1.6
5 - 1.6 = 3.4
3.4 -> g(3.4) = 3.4^2 - 9 / 2(3.4) -> 11.56 - 9 / 6.8 -> 2.56 / 6.8 -> 0.3
3.4 - 0.3 = 3.1
3.1 -> 3.001 -> 3.00001 -> 3
(3 + .4) * (3 + .4)
9 + 2(.4*3) + .4*.4
11.56
0.3060706
6.8 | 2.56
25
20.4
4.66
46
40.8
5.26
52
47.6
4.46
44
40.8
3.26
1 6.8
2 13.6
3 20.4
4 27.2
5 34.0
6 40.8
7 47.6
8 54.4
9 61.2
10 68.0
17: im on the phone
@@plopgoot5458???
@@BradenBest They're talking about this gesture images.radio.com/aiu-media/handgesture-2058165b-9063-44a0-8c42-94bb308b602b.jpg
Base 4 + 3i is the optimal base.
i don't know aabout that, but base -1+i is buded the most beutiful base.
I personally prefer base 2pi-i*sqrt(97) because I hate rational numbers (especially integers), but this is an interesting choice as well
Grouping "bits" by up to for 4 is also pretty practical in a human sense, since around 4 is the natural range of human "subitizing" i.e telling how many of a thing there are at just a glance
I propose a much worse way to count, base -29
Great idea, but I prefer base 2.5 myself