For ratios, you could create a new system. I'm not even remotely an expert, so I'm gonna coin this (most likely already named) system "Nested dots" (since calling them decimal points is like calling Dozenal "Duodecimal"): For DEC1/4th, you write ".IIII". DEC3.25 would be "III.IIII". So what about three fourths? DEC3.75 would be "III.IIII.IIII.IIII", so literally three fourths. Just don't try to write IP adresses in this notation, you'll run into issues.
No the fair enough was regarding the important of fourths. Quarters isn't more or less proper than fourths; one is used in the US and one is used in the UK. jan Misali is clearly from the US and thus says fourths
@@disgustof-riley8338 we use quarters for some things in the US. Most notably for our money (25¢ piece is a quarter). But also quarter gallon (though this is shortened to "quart"), or in divisions of a year (Q1 2023 = Jan-Mar 2023). -If I had to guess as to why we use fourth when talking about fractions in math in the US, I would say that it has to do with keeping them lined up with how we enumerate lists in writing (first, second, third, fourth) - using quarter in that context would make no sense.- EDIT: I just thought about it for a bit and realize we don't use "second" for fractions either, we use half. So I retract my guess. I mean, fourth clearly comes from enumeration terms, but that doesn't answer the "why". It honestly might just be to avoid confusion with money. Or perhaps something to do with how we measure things in inches/feet.
@@rateeightx I know it's a joke but you can't have a base i, but I don't know if it's possible to have an usable imaginary base at all ? interesting idea
@SQ38 But there could be 3 forms of threeven-ness, just like there's 2 forms of even-ness (I know the actual word is parity but who cares). There's numbers that are divisible by 3, numbers that are just above a multiple of 3, and numbers that are just below a multiple of 3. In addition to threeven (3n), I'll call these morven (3n+1) and lessven (3n-1) because I'm coming up with these names on the spot and I lack imagination. Anyways, another cool thing about seximal is that the 6 digits correspond with all possible combinations of evenness and threevenness, so you an easily tell both by the last digit of any number 0=even & threeven 1=odd & morven 2=even & lessven 3=odd & threeven 4=even & morven 5=odd & lessven
I am really starting to like seximal, it's weird I never considered it. Some of my reasons are: - I liked binary and balanced ternary as bases from a fundamental standpoint, and 2*3 = 6. - Standard dice are 6 sided, which reflects the fact that there are 6 directions in 3D space. - 1+2+3 = 6, so my only reason for liking decimal (1+2+3+4=10) works for seximal too.
@@gamerrfm9478 That would be unary, which was mentioned in the video, actually! Tally marks are a unary counting method. Unfortunately, it can't really represent anything but nonzero integers, making it almost completely impractical.
TheGreenNinja Sorry! I must’ve gotten it wrong! I was simply making a joke on its uselessness and I fully acknowledge how terrible of a system it would be.
If you want less jokey (and more universal!) measures, try powers of the Planck units. For example, six to the niftieighth¹ power Planck lengths is _shockingly_ close to foursy-four² centimeters! Give it some fitting name and base units around it. ¹ forty-fourth ² twenty-eight It's also about nine tenths of a foot.
You have no idea how much I want to see Planck units adapted to a reasonable set for everyday usage. For example, 1 nano-c is a little under 1 foot/second, and surprisingly close to 1 km/h. For a seximal alternative, 1/6^10 (one nif-biexianth) c is 4.958m/s. Combining these two units as they are isn't great since the only give about nif thirsy two (56.47) milliseconds. Some fine tuning will be necessary to find powers that work for all the main units.
If you use a positional base, like this video is all about. Base 1 has 1 digit: 0, and it has only one number: 0. - The "unary" base is not a positional base, but a bijective base. A bijective base doesn't have 0, so bijective base 10 is 1-2-3-4-5-6-7-8-9-A. In positional base 1; 00 isn't a different number from 0, just like in any other positional base. So you can only write 0.
There are versions of Unary that are usable and historically used, but it does not work with positional systems AT ALL. They're basically just tally marks: position and order doesn't matter, just count the number of "1"s to get the number. Most that were actually used, like roman numerals, had special symbols for large groups of tally marks to make counting faster, and once you do that you can add special rules based on the order in which they appear, but fundamentally a "pure" usable unary system would only care about the number of 'ticks' and nothing else. Also can only represent ratios as ratios, since using a normal positional radix point anything on the other side of the radix point would just be more 1s--though cultures that used such systems usually had pretty simple notation for writing ratios, like |||:||||| for decimal 0.6 so just a different way of thinking about it and still perfectly usable as long as the numbers are small...which they never do.
10:18 Pros: - arithmetic is SUPER easy, like holy shit. - square roots exist as doable functions Cons: - fractions are red, red is bad - numbers get long fast which may or may not be because the zero is fat
"Senary" is what that one person who desperately avoids suggestive language uses, but when you realize it puts more focus on the suggestion they're trying to get away from.
@@Anonymous-df8it Because as a listener you're going to think "Why that obscure term? Is this person really trying this hard not to say 'sex'?" I grew up in a country with a language where "six" and "sex" are commonly pronounced the same. A few rare people try to pronounce "six" in a different way which doesn't at all fit with the region, so it's artificial and forced, and everyone can immediately tell. But as for "senary" and "seximal", at least most people in the real world will think one is a nerd no matter what one calls it.
Hi! These are the (extremely weird) bases you did't talk about. -Golden ratio base (having the golden ratio as base) en.wikipedia.org/wiki/Golden_ratio_base -Factorial base (impratical since the base will change according to position. also needs infinitely many symbols) en.wikipedia.org/wiki/Factorial_number_system -Base with sign digit (Balanced Ternary is the well-known example of these bases) en.wikipedia.org/wiki/Signed-digit_representation -Negative bases (base -2,-3,-10 etc.) en.wikipedia.org/wiki/Negative_base -Quater-imaginary base (base 2i when i² is -1) en.wikipedia.org/wiki/Quater-imaginary_base P.S. I'm not a native English speaker. So apologies for any grammartical error in advance.
I use a weird shorthand based on seximal as shorthand for times of day. A day has exactly 400 (aka 240) 14 (aka 10) minute chunks. 0xx is some time in the first 6 hours 3xx is some time in the final 6 hours. The 2nd digit specifies which of the 6 hours, the final digit specifies which increment of 14 (10) minutes it is. 3:40 in the morning is 034. 13:50 in the afternoon is 215. 18:20 is 302. 23:40 is 354. I do this so nobody else can understand my notes when I die. Using dozenal would be so much more effective.
Since you decided to argue with fraction lengths, I wrote a program to add up the lengths of the periods of all unit fractions from 1/2 to 1/144 for the bases in question (2-20). Result: The best base by far is 16, followed by 4 and 9. When going higher, 36 and 25 take second and third place. Then I decided to test convenience by removing multiples of all primes > 11. Now the best base becomes 15, followed by 10, 6 and 9. Considering more fractions, base 18 here pulls ahead and gets second place. Just for fun removing multiples of 11 too, the best now are 15 and 6. Interestingly, with other variations of the parameters, 55 got first place twice. Dunno what's up with that.
It's cool that you wrote a program to check it. Fraction lengths isn't the key thing that matters for divisibility tests, though. What makes for easy divisibility testing: - If the number is not a prime power, all its prime power factors should have easy divisibility tests. - If the number is a prime power and its prime factor is shared by the base, it's always easy test divisibility. - If the number is a prime power and it is coprime to the base, then check the period length. If the period is length 1, the divisibility test is easy. Otherwise, the divisibility test is hard, with maybe 11 as a special exception. Also, going up to 1/144 seems way too high. Even 1/19 is getting pretty high.
@@blueblimp The things you name are closely related to fraction period lengths though. In base b, the fraction 1/n has period length ord_m(b) (order of b modulo m -> why worst-case period is phi(n) ) with n=g*m where g is the greatest divisor of n containing only prime factors of b. - Your first point is basically the fact that gcd(p,q)=1 implies ord_(p*q)(b) | ord_p(b)*ord_q(b) - The second point is equivalent to ord_1(b) = 0 (no period) - The last point just simplifies matters to ord_m(b) easy, else hard I agree that 144 may be too high. On the other hand, the ranking is not stable when going too low. There is definitely a mistake I made: I basically weighted all the lengths equally. The result become much better for 6, 12 and 10 when weighing the periods (of the fractions 1/n) inversely proportional to n or n^2.
Great idea! Though you should adjust the scoring of your program to weigh the lengths of a certain fraction by its value (i.e. the reciprocal of the denominator), instead of uniform weights. After all, that's the probability that a given random integer contains the considered factor, so a good measure for the relevance of divisibility by that factor. It's intuitively obvious that the further you move out to larger denominators, the less important they get. I'd be interested in the updated result!
It matters whether they are reccurring or not. Also, the higher you go, the more decimals there will be after the period, but the LESS weight it should have, because you are less likely to see it in math. So the weighting system should actually be reversed somehow. I think the lower bases would perform better when this is done.
I think a good metric would be to look at all the primes p_1, p_2, ... up to some stopping point. There's no need to test non-primes since everything is governed by the primes anyway. If l_n is the length of the period of the base expansion of 1/p_n, then calculate Σ (l_n/p_n)^2 and see which base minimizes this value. On a side note, there's a chance that this sum would converge if taken over all primes. I would be curious to know if it does and what that means.
If anyone wants to build a silly comunity that uses seximal, a regularised calendar, esperanto, Dvorak-style keyboards, and all the improved systems we can think of, let me know! I'd love to meet other utopists :)
In my fictional universe there's a colony on Mars which uses Esperanto, with the Shavian/Sxava writing system, on a Darian calendar. The first two are the result of a political decision to "fix our mistakes and build a better world together".
hexadecimal remains my favourite simply because it's SO useful in computing, and having everyone learn it since birth would make it easier for people to grasp how code works.
I'm a computer science student. I mentally convert every mention of binary numbers into decimal numbers, pretending that computers work in decimal. It works almost every time. Here is an example using the floating-point number system: A float has 1 sign bit: 0 for + and 1 for −. Lets convert that to decimal: 1 digit: 0 for +, 1 to 9 for −. The exponent is a binary number using 8 bits. It has a bias of 127, meaning you subtract 127 from it to get the actual value of the exponent. This is used to create a number of the form 2^(exponent), ranging from 2^-127 to 2^128. And now in decimal: 3 digits, with a bias of 499. This creates a number of the form 10^(exponent), ranging from 10^-499 to 10^500. The mantissa has 23 bits. It is almost allows preceeded by an implicit 1, to create a number of the form "1.(mantissa)". In decimal we have to make one small adjustment: The only thing we can guarantee in bases other than binary, is that there is a 0 to the left of it. This does not really allow for an implicit extra digit, but that has exceptions anyways. So, in decimal, the mantissa is just a 17 digit number between 1 (inclusive) and 10 (exclusive). Using this system, I can perfectly understand every topic like "precision problems with floats" or "subnormal numbers" or "how to represent NaNs", without actually having to ever think about binary. Binary just isn't ACTUALLY as useful to the average programmer as people say. Its infinitely more important to somebody who designs computer hardware. A coder just needs to know what the limits are, like "a byte is 8 bits and goes from 0 to 255", which are just regular numbers that I literally just wrote in decimal. "What's the decimal value of DEADBEEF" is a question that nobody has ever actually needed the answere to.
You've convinced me, seximal is awesome. It's great for both large integers and fractions. It seems like people often forget one of them when promoting bases
In my number system I just invented, Hectimal/base 100, needs half the digits compared to decimal! The words for 1/3 in Hectimal is zero point thirty-three recurring.
you could use dozenal to write the time (in minutes) by just 3 symbols. there are 12 2hr periods in a day. the first symbol could be used to tell which 2hr time period it is. now that we know which 2hr period we're in, divide that into 12 10min periods (for a total of 120min). the second symbol tells us which 10min period it is. the third symbol tells us which minute of that 10min period we're at. for example: time: 9: 30 (am) that's in the fifth 2hr interval. So, first digit=4 it's in the third 10min interval. So, second digit=2 now, we have to increment the time by 0min, so third digit=0 Finally, time=420
A variation on Quaternary I started using when counting measures of rest in music: Since so much of music is based on 4/4 time, and so many musical phrases are based on groupings of 4 measures or other multiples of 4 such as 8-bar periods, 12-bar blues, etc. I started counting base 4 on my hands a lot when performing music. I usually start with my left hand: 1 = left index finger, 2 = left index & middle fingers, 3 = left index through ring, 4 = left index through pinkie. Then I start using my right hand for the next place: 5 = right index & left index, 6 = right index & left middle, 7 = right index & left ring, 8 = right index & left pinkie And I continue using combinations like that: 9 = right middle & left index, ten = right middle & left middle, eleven = right middle & left ring, dozen = right middle & left pinkie and so on. It's a different way of doing it, like the base itself wouldn't be written as 10, it's still 4. But the number one after the base is 11. Counting looks like 1, 2, 3, 4, 11, 12, 13, 14, 21, 22, 23, 24, 31, and so on. There is no 0 in this system, because beat numbers and measure numbers in music use one-based-indexing and not zero-based-indexing.
Not hard-wired, just trained for very nearly your entire life. If you actually try to use a different base for a long enough time, you will find it natural. It is incredible how quickly your brain can gain intuition for something if you let it. Just a few weeks ago, I was struggling to remember to use my Caps Lock key for Escape (I recently changed it to do that), and then a few days ago, I found myself doing it automatically on a machine that didn't have that set up. Similarly, when I get into (left-to-right) seximal for a bit (which I do every once in a while), I get pretty good at not messing it up. Also, verbal anything is weird and hard. 43 makes a lot more sense than the spoken "thirsy-four", especially when he still calls 41 "ten".
@@angel-ig Because writing numbers left-to-right is more consistent with how we write everything else, and you get to add, subtract, and multiply from left-to-right, instead of having to teach kids to do it in reverse.
@Ángel I.G. What do you mean backwards? No one writes numbers from the right to left. Maybe the Japanese. Or are you really telling me that Americans are taught to do this and this has been true for decades? That's inhumane honestly
@@TheAlison1456 I wrote "43" (4 and 3 sixes) in left-to-right seximal, as opposed to the normal right-to-left way everyone does in English. That's the backwards number he was talking about. Also "41" (4 and six).
My favorite is Base 60, it easily beats decimal, dozenal and seximal in terms of fractions because it is divisible by all numbers 1 to 6. You can avoid having to use 60 digits by writing each digit as two decimal digits like on clocks.
Prime numbers ending in either 5 or 1 is interesting. I have noticed that numbers divisible 6 tend to be near prime numbers and like to hang out in the middle of double primes :)
i have a plan to gradually shift everything into seximal. the secret weapon is the mindset that we don't need to change everything at once. i begin with just one step: using seximal in my next videogame, most notably its scoring system.
The thing about binary and hexadecimal is that where they are used, at least at a level humans interpret, ratios don't matter. They are used cardinally. Oh and binary can be used to represent a string of Boolean statements and the hand counting thing is awesome and I have found it practical at times. The more I use those 10 bases, the more beautiful I find them and the more I hate base A.
i feel deeply moved by your "it's fun to get silly sometimes... theoretically change the things we take for granted, even if we know it could never actually happen"
Odd bases are actually good, because its always true that the number that is 1 more and 1 less from the base is even, and the factors of this numbers have easy divisibility test/ simple periodic expansions. so with that said the best base is 15 since is handle well all number until 11
I’m so sad that no one mentioned the better way to finger count! You can use your thumb as a representation of 5, so 1-4 are counted normally, 5 is just the thumb, and then 6-9 are the positions for 1-4 paired with the thumb. I was taught this method in elementary school and it’s so useful, because it allows you to could to 99, if you use your left hand as the tens digit.
heximal has fun associations: hex, hecate, heka. Matches hexagon, consistent with hexadecimal. Maybe ancient magic cults used to use that as their base.
Always been a huge fan of base 6, great video! Btw have you ever considered how the numerical system affects the way time periods are perceived? In base six we wouldn't think in centuries or millennia, but the six-equivalents (whatever quirky name you want to call them). For example we would be in year 13203, we would consider year 10000 (1296) a big deal, and we would experience a millennium fear every 216 years. Also the stages of human life would be perceived differently (or would they): 0-6 (infancy) 6-12 (pre-adolescence) 13-18 (~teens), 18-24 (~college years) 24-30 (young adulthood) 30-36 (100 years landmark) Pretty neat!
Now we just have to hope my great grandma lives to 108 instead of the 106 she already has. Three nif is another form of "centenarian" that makes sense in seximal.
These two videos literally changed my mind about numbers from 'boy howdy, I sure hate base ten but base twelve seems impractical' to 'boy howdy, why did we even use base ten instead of base six' in the length of time it took me to vacuum out a rental car.
Hi there! I really enjoyed both this video and it's first part. If you're interested in exploring other numeral systems, I would like to suggest that of Kizh, (formerly referred to as "Garbieleño Chumash") the language indigenous to where I am from. It is a quinary system, and I understand that such a prime-based system makes representing fractions more difficult, but there are very interesting other features in Kizh's numeral system, including "kavyaa’" which operates as "X almost twice", or "X + (X-1)" (ex: wachaa’ kavyaa’ is "four almost twice" or "4 + (4-1)", which is 7. In a quinary system, this would be written as "12", of course.) I would be happy to link you to the resources through which I learned of Kizh's numeral system, as well as a document I made exploring and explaining this system to those who are familiar with base 10. (I am by no means an expert in mathematics, numeral systems, or Kizh itself, but this explanation was part of my final project for a course I took titled "North American Indigenous Languages", instructed by Dr. Marriane Mithun, in the Linguistics department at UCSB.) Thank you for your time and labor, and I hope you have a great day!
The Myst series of games uses a base twenty-five numbering system, but the digits are designed in a way that you only need to remember 5 symbols. (Explaining the system actually spoils some of the puzzles in Riven so I recommend playing that game if curious)
While before you had convinced my seximal was bestimal, the "primes end with 1 or 5" bit is like THE COOLEST PART!!! I was literally just watching a video about how prime numbers act that way the other day!
I am just going to through this out there: our perception of a 10 base system is more about measurements i.e. decimal system and works extremely well for that. But for other maths, we only speak in a ten based system because, that is what the English language bases its numbers on. There are numerous other languages that use other number based systems and sometimes combinations of number systems for larger groupings of numbers. In addition, we are using our written numbers 0 to 9, when again , many languages use completely different characters to represent numbers. In the field of mathematics, we have more standardization of mathematical notation. Base this or that somewhat becomes a moot point in this area of topic as the accepted notation in mathematics is universal. To attempt to change our counting base system is basically mostly about quantity, and nothing more. Quantity, measurements( i.e. lenghth, area, volume, temperature etc), and mathematical formula format of notation all serve very different purposes. As does chemistry has its own standardize format of notation, and so forth.
For bases greater than 10 I understand changing how we say the number because we have to but 6 is less than 10 so there is no reason we can’t say the number normally so one hundred (base 10) is two hundred and forty four (base 6)
Only for the weak minded. We have language to speak, and if we decide to speak words to represent *digits* instead of quantities, then surely we could speak using decimal.
(2:40) Why create a completely new system when you still can use metric? You just need new prefixes. so deca-, hecto-, kilo-, which are 10¹×, 10²×, 10³×, would instead be: 6¹×, 6²×, 6³× the same with deci-, centi-, mili-, currently being 10⁻¹×, 10⁻²×, 10⁻³×, would be 6⁻¹×, 6⁻²×, 6⁻³× new names and letters can be used, to avoid any confusion. Using the prefixes from Conlang Critic; nifti- is 6⁻²×, so a niftimeter (nm) is 0.01₆ meters. unti- is 6⁻⁴×, so an untimeter (um) is 0.0001₆ meters. biti- is 6⁻⁸×, so a bitimeter (bm) is 0.00000001₆ meters. feta- is 6²×, so a fetameter (Fm) is 100₆ meters. grand- is 6⁴×, so a grandmeter (Gm) is 10000₆ meters. - Additional prefixes would still be needed, as you do have GW (gigawatt), GB (gigabyte). But this is a good start. So in computing, you have byte (B), fetabyte (FB), grandbyte (GB) ≈ 1.30 KB or 1.27 KiB, so you need larger prefixes.
@@diribigal Kilogram should rather be renamed to gram if we switch base; so if the kilo- prefix isn't used, just dropping it altogether works. Another option is to make up a new name and define it the same as 1 kg.
I'm using seximal in my conculture because I wanted a base other than ten, and six works out as well as I thought dozen would. It's especially useful because I have exactly six vowels, and six consonants. So I can reserve one consonant for digit syllables :D
As a computer scientist and hobbyist CPU-builder, I quite like hex, but am readily willing to admit my bias on that front. I really just want a nice hex calculator that isn’t just a decimal calculator with a hex display mode that I need to constantly tell it use
The multiplication table being easier is a big plus, so many kids leave school without knowing the basic times tables. For sub 10 multiplications ignoring reverses in decimal there are 36 to learn, in seximal there are only 10, learning these would be trivial.
Base pi is obviously the best. It's better than base e because you get another digit. You can simply and easily represent the circumference of circles. Plus, as an added benefit, the human race by definition knows all digits of pi.
Hot (?) Take: Complexity of floating point representations of rational numbers is irrelevant to the merit of a base, because rational numbers are most simply represented as... ratios (fractions). Each base is equally efficient at representing irrationals. With that in mind, I'll inject my taste: The only remaining criteria relevant to me are digital utility, finger countability, written efficiency, and digit entropy (as a source of mental labor). Digital utility prescribes a power of two. Finger countability somewhat favors quartal; you can count from 0 to 4^2 - 1 = 15 without thumbs or pinkies. Octal fingers can count from 0 to 4•8 - 1 = 31, requiring pinkies and thumbs, but with certain schemes you flip people off with 7
my favorite base is base 30 labeling it in decimal terms and base 50 in seximal. i just like primes and thirty/fifsy is 2 * 3 * 5 so it's the first three primes. I haven't really thought through it though, and after watching this I would have to do something like the Babylonians did.
((5*12)-1)*6+(2*5)+1(+1/4) -system for the lunisolar calendars is no way to count but rather accurate. 2*lunations times 6 and add 2 lunar weeks of five and six days. Solday resets the cycle every 4 years and every 128 years there's a gap. Moon uses 59-day system so some complications are unavoidable.
alright, if you want to get silly with it, base -1.5 is pretty fun to play around with, i spent a half hour or so trying to do working out to figure out the pattern for counting, along with how you have to do the basic four math operators
I like base 420. Yes, it's a Funny Number™, but it also has some pretty cool properties. It's divisible by all numbers between 1 and 7. Here are the reciprocals of the first thirsy integers in base 420. 1/2 = .210, 1/3 = .140, 1/4 = .105, 1/5 = .084, 1/6 = .070, 1/7 = .060, 1/8 = .052:210, 1/9 = .046:280, 1/10 = .042, 1/11 = .[038:076:152:305:190:381:343:267:114:229], 1/12 = .035, 1/13 = .[032:129:096:387:290:323] (thirteenths are actually more convenient than elevenths, weird), 1/14 = .030, 1/15 = .028, 1/16 = .026:105, 1/17 = .[024:296:197:271:321:074:049:172:395:123:222:148:098:345:370:247], 1/18 = .023:140 Fifteen of those are terminating.
I think dozenal is the best, but given how easily you can convert dozenal and seximal you could even use both of them at the same time. The only thing that matters is that both are better than decimal
1:21 Not taking seximal seriously probably comes from people associating anything with the letters "s, e, x" in that order to the activity they never partook in: eroticly touching another animal, whether it's done mutually or not (humans are animals btw). In reality, sex is rarely associated to that and is much more often associated with the type of reproductive organ an animal has or the number 6.
There are tricks you can do to composite numbers without new symbols. D'ni is base 25 but actually kinda base 5 with every other digit on its side. And yes I only know this because of that one puzzle in Riven :P
I count to 99 on my fingers without trouble. Right index, middle, ring, pinkie are one each, Right thumb is five. Left index, middle, ring, pinkie are ten each, left thumb is fifty. As for economy, base 2 and 4 can each perform a 64-way selection for a cost of twelve (six binary digits at a cost of 2 each, or three quaternary digits at a cost of 4 each) but for the same cost base 3 can do an 81-way selection (four ternary digits at a cost of 3 each).
Ngl, the nice coincidence that makes niftable work is super nifty (hah). If anything tips me over from being a dozenal advocate, it may be this (after all the other things balanced things closer, having not previously considered base6 before thinking it would be a pain given smaller base - but that was nicely covered 👍)
hey, at least it's better than dozenal and decimal. dozenal: 1/5 = 0.(2497) decimal: 1/7 = 0.(142857) seximal: 1/15 = 0.(0313452421) yes, seximal is bad at elevenths, but dozenal's worst fraction is fifths and decimal's worst fraction is sevenths, and seximal is already good at representing fifths and sevenths, therefore, seximal is better at fractions than dozenal and decimal.
Balanced number systems (such as balanced ternary) are superior, they properly extend to the negatives. 1. If you have a method to add numbers, in a balanced number system you can also use it to subtract. 2. Rounding and cutting off the number at a digit is the same. 3. The multiplication table is a quarter the size. Therefore balanced seximal is the best number system, with digits -3,-2,-1,0,1,2,3 (ambiguities such as 6 + (-3) = 3 can be resolved by convention to enforce rule 2.)
I think it's really cool to examine this stuff in practical terms. Like, yeah, fractions are really good indicators for how good a base is, but as a history nerd my mind immediately went to how we would count years and periods of history. It's a given that we talk in centuries as periods in human history, like the 20th century as the "century of extremes" or the 19th as the "century of industrialization". We have a term for the "long 19th century" from the French revolution in 1789 to the first world war in 1914. We speak about decades as periods of culture, like the 80s as the decade of video games and neon and synthetic music, the 2000s as the decade of fledgling social media and emo. All of this would work completely differently in seximal. A "sexade" (=6 dec, 10 sex years) doesn't work as a parallel to a decade (=10yrs dec, 14yrs sex) because it's so damn short. A "niftury" (from CENtury) or perhaps “niffy” (it’s actually CENTURy, after all) (=36 dec, 100 sex years) is way longer, and the average human would only live through 2-3 niffies compared to humans living through up to ten decades. The next step up is six nifs, also called “tarumba” (=216 dec, 1000 sex) Ironically, since a century is so damn long, a tarumbaium (from millennium? maybe?) works better as a way to think of historical periods compared to centuries than regular niffies (or sexades) compared to decades. A lot happens in a tarumbaium, but a lot happens in a century too, and under closer scrutiny, neither hold up as a serious period of analysis. The (currently) last step up is unexian (1296 dec, 10000 sex) years, which I guess can just be called an unexian or unexiant or something. (I’m not sorry for the shoddy nomenclature lmao) Since the current year in seximal is 13211, or one unexian thirsy two nif seven, we’d be currently living in: - The second unexian - The tenth tarumbaium - The nif thirsy third niffy, or I guess the third niffy of the tenth tarumbaium OR MAYBE the thirsy third niffy of the second unexian - we’d also be living in the sixties, I think? Followed by the dozenies/dozzies? - the current tarumbaium began in the year 1944 dec which is a pretty neat cutoff point all things considered. The previous began in the year 1728 dec which means the adoption of radar is part of the same time period as the discovery of Uranus. Wild.
Wow. I’m a bit of an idiot. I’ve watched quite a few of your videos and am familiar with how different bases work and all that, but I still left a comment on the original video saying that you miscounted the number of factors of 10 and 12. Whoops.
I'm a fan of base 2 for the following reasons: *Addition and subtraction are easy in any base system, but for binary multiplication is very easy. There is no need to memorize multiplication tables, it's simply repeated addition with the bits shifted to match the pattern of the number you're multiplying. *Where base 2 really shines in my opinion is in long division. Because each bit is only either a yes or a no, calculating each bit in the binary expansion of a fraction boils down to a comparison between two integers and at most a single subtraction. *In any other base, you always have to either one of these two things: do repeated multiplications to find the smallest integer multiple less than the number you're trying to divide do repeated subtractions until the number you're trying to divide fails the comparison Both of these are very annoying when dividing by a number with more than about two or three places, while binary sidesteps this issue entirely. Also this has to be done for every single place in the expansion which gets very tedious very fast. In binary you can just keep going as long as you like. *This is counterbalanced by the fact that for the same level of precision at say 3 decimal digits, you need about 10 bits, but in my experience (yes I experimented with this) doing division with a high number of bits is much easier than doing division with moderate amount of digits just because each step of the process is a simple yes or no question. *From a philosophical point of view, I'm a fan of the fact that as the smallest usable base, it has objective merit as the best possible with its advantages. With base 6 or 12 or really any other base, you can always make the argument that you should have more factors to further help with ratios and really the only counter argument is "that's too many digits for us to memorize," but this is a subjective judgment with no mathematical justification. Not that much of this matters, because after you're out of middle school you're never going to do long division again.
Watch Matt Parker doing long division with 20-digit-numbers every year on pi day. All you need to do is make a times-table of the number you want to divide by (meaning you multiply it by every digit in your base, which is 1 to 9 in decimal) and then at every step of the division, you need to find the largest of those numbers that fits, and then subtract it. So, after the initial multiplication step, which happens literally only once, every iteration is just up to 10 comparisons followed by a single subtraction, which isn't actually that much different from what you described.
I read something about monks who'd created what was effectively a base-10,000 (10^4) numbering system that was actually readable, by using composite figures as digits. Each figure began as a single vertical line, and having one of nine markings each representing 1-9 (or no marking for 0) attached to a corner of the line (top left, top right, bottom left, bottom right). Effectively just an efficient way of compressing decimal, but cool nonetheless. Sadly I don't remember what the markings were nor how it was read, but if anyone else knows what I'm talking about and remembers, please do tell.
In decimal, to check if a number is divisible by seven, you can actually multiply the last digit by five and add the rest of the number (divided by ten). For example: 105 is divisible by 7 because 5 (the last digit) * 5 + 10 = 35, and 35 is divisible by 7. Therefore, 49 doesn't look prime because 9 * 5 + 4 = 49, which is clearly divisible by seven because 9 * 5 + 4 = 49, which is divisible by seven.
Base 16 is really only good at fractional powers of two, but it is super good at it. Cut something in half eight times and you're at HEX0.01 vs DEC0.00390625. We cut things in half all the time, way more than cutting into thirds or fifths or whatever. Having your base be not only a square, but a square of a square probably leads to some cool properties too (not sure what though). You're starting to convince me that seximal is among the best choices for a base though, given its simplicity and broader utility with fractions.
6 is good but I'm a daredevil that don't mind bases like 120!!! It fits my enthusiasm for the cosmos but I now have my eyes on 24 and am questioning its potential properties for a simpler system. Either way notice how all the best number bases are multiples of 6 and 12 ? It is that way for a reason!!!
Other fun bases: Tetravigesimal (24) - It's like Dozenal but good at eighths. Duotrigesimal (32) - Another power of 2 base Balanced Nonary (-4~4) - Balanced ternary with a little more room. Base64 (Gee, I wonder) - A favorite of encryption nerds, and enough digits for the decimal numbers, all the letters in a case-sensitive manner, and / and +. Captchary - A method of writing in Base 64 favored by filthy Homestucks. Goodbye / and +, hello ? and !. Eulary (e) - The number with the lowest radix economy. Nicemal (69) - It handles thirds and twenty-thirds well. Aside from that it's just for the lulz. Negadecimal (-10) - Negative bases are interesting since, like balanced bases, you can write any positive or negative number without worrying about the minus sign, but unlike balanced bases, you don't have to either use an odd number or forego zero. Derfmal (11?) - Undecimal shifted weirdly, with the added digit (derf, represented by a 4 flipped vertically) being between 5 and 6 rather than after 9. People who've watched iCarly know about this one. Quater-imaginary (2i) - Represents every complex number in strings of just 0, 1, 2, and 3. Quadrigentitredecimal (413) - Does sevenths and fifty-ninths well. Aside from that it's just a nerdy joke. Cannabicimal (420) - Haha blaze it. Also has a lot of factors. Leetimal (1337) - U 4M N0082, 1 R L337 Platimal (5040) - It's an antiprime so it has to be good, right? Tetranonagesimal (94) - Enough for every single thing you can type on a normal QWERTY keyboard, without using spaces or Alt Code characters, of course: 012356789ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwxyz!@#%^&*()`-=[]\;',./~_+{}|:"?
Centesimal is interesting, because there's actually a centesimal proposal for Toki Pona! zero = ala one = wan two = tu five = luka twenty = mute And each centesimal digit is made like a regular Toki Pona number out of these. For example, 69 is mute mute mute luka tu tu. The ":" is "ali", which refers to the number 100. Thus, 42069 is tu tu ali mute ali mute mute mute luka tu tu (while it would begin with 420 'ali's in the regular system). To make it easier to break large numbers down, you can probably separate them like so: 4,20,69. There isn't a standard for non-integers or negative numbers yet, but nasin nanpa pona does look quite promising.
What about my new number system, where we start at one being 1, then every prime number is written as p subscripted with the order of the prime (recursively using the numbering system so far), and composite numbers are written by stringing together the prime factors, with superscripts for exponents. So counting goes 1, p_1, p_(p_1) ,p_1^p_1, p_(p_(p_1)), p_1p_(p_1) and so on. Pros: -multiplication is made very simple, just count how many of each prime factor you have -divisibility checks become incredibly easy, and prime numbers are numbers with no suberscripts and only one p written outside of a subscript. -using negative exponents allows us to express every fraction without any infinitely repeating symbols Cons: -absolutely terrible
honestly, i wasn't fully paying attention on the prime factor joke and rewatched it because i was like "wtf did i hear that correctly? that's such a stupid comparison" then i also heard your tone and realised you were joking xD
All that a heximal system would need to work properly with the Metric system is a new set of prefixes. Such a system would be very easy to adopt. In fact, we recently adopted a set of new prefixes and it worked fine: en.wikipedia.org/wiki/Binary_prefix . Just base the names of the new prefixes on the know prefixes but with a consistent twist, as was done with the binary prefixes. E.g.: "kihe", "mehe", gihe", etc How hard would those prefixes be to learn for someone who already knows the base 10 prefixes?
OK here me out
Unary
1 = |
2 = ||
3 = |||
4 = ||||
5 = |||||
All fractions are equally unimportant
also zero apparently lol
r/woooosh
Huh, this only has two likes!
columbus8myhw yea the guy speaking about base IIIIIIIIIIII is very stupid
For ratios, you could create a new system. I'm not even remotely an expert, so I'm gonna coin this (most likely already named) system "Nested dots" (since calling them decimal points is like calling Dozenal "Duodecimal"):
For DEC1/4th, you write ".IIII".
DEC3.25 would be "III.IIII".
So what about three fourths? DEC3.75 would be "III.IIII.IIII.IIII", so literally three fourths.
Just don't try to write IP adresses in this notation, you'll run into issues.
"(also you should say quarters it's more proper)"
"Fair enough."
_proceeds to call it fourths_
There's only one logical name for the thing between thirds and fifths.
@@TheEvilCheesecake
You make a good point: wholes, halves, thirds, half halves, fifths, third halves, septenths, half halve halves... Etc
No the fair enough was regarding the important of fourths. Quarters isn't more or less proper than fourths; one is used in the US and one is used in the UK. jan Misali is clearly from the US and thus says fourths
@@disgustof-riley8338 we use quarters for some things in the US. Most notably for our money (25¢ piece is a quarter). But also quarter gallon (though this is shortened to "quart"), or in divisions of a year (Q1 2023 = Jan-Mar 2023).
-If I had to guess as to why we use fourth when talking about fractions in math in the US, I would say that it has to do with keeping them lined up with how we enumerate lists in writing (first, second, third, fourth) - using quarter in that context would make no sense.-
EDIT: I just thought about it for a bit and realize we don't use "second" for fractions either, we use half. So I retract my guess. I mean, fourth clearly comes from enumeration terms, but that doesn't answer the "why". It honestly might just be to avoid confusion with money. Or perhaps something to do with how we measure things in inches/feet.
@Ethan Matzdorf I'm from the US and I've never heard people use Q1 for jan-mar like that
i love how you constantly switch between base 10, base 10 and base 10
@ainielyabut6191true, needs more comments
HEY
I use base 36,
@@Thiegocolla77you mean base 10?
Yeah- "all bases are base 1-0." ("Base Neutral method for base names")
Base-5040 is clearly the best.
SO MANY FACTOOOORS
Base infinity is far better.
Base i Is Best!
@@rateeightx I know it's a joke but you can't have a base i, but I don't know if it's possible to have an usable imaginary base at all ? interesting idea
@@lpu_n.4926 So... Base Pi?
I love the term "threeven" so much
@SQ38 Probably
@@Miju001 But what if the number can be written as 3k+2?(k is a natural number) throden?
@@EsperantistoVolulo I think it'd still be throdd
@SQ38 But there could be 3 forms of threeven-ness, just like there's 2 forms of even-ness (I know the actual word is parity but who cares). There's numbers that are divisible by 3, numbers that are just above a multiple of 3, and numbers that are just below a multiple of 3. In addition to threeven (3n), I'll call these morven (3n+1) and lessven (3n-1) because I'm coming up with these names on the spot and I lack imagination.
Anyways, another cool thing about seximal is that the 6 digits correspond with all possible combinations of evenness and threevenness, so you an easily tell both by the last digit of any number
0=even & threeven
1=odd & morven
2=even & lessven
3=odd & threeven
4=even & morven
5=odd & lessven
It'd make a good character name
I am really starting to like seximal, it's weird I never considered it. Some of my reasons are:
- I liked binary and balanced ternary as bases from a fundamental standpoint, and 2*3 = 6.
- Standard dice are 6 sided, which reflects the fact that there are 6 directions in 3D space.
- 1+2+3 = 6, so my only reason for liking decimal (1+2+3+4=10) works for seximal too.
On the other hand, every even base cannot be balanced : (
Base 15 (1+2+3+4+5) here I come
Primary base (1=1)
@@gamerrfm9478 That would be unary, which was mentioned in the video, actually! Tally marks are a unary counting method.
Unfortunately, it can't really represent anything but nonzero integers, making it almost completely impractical.
TheGreenNinja Sorry! I must’ve gotten it wrong! I was simply making a joke on its uselessness and I fully acknowledge how terrible of a system it would be.
I was sold when you said "niftimeter" tbh
Its so nifty!
Its so nifty!
Its so nifty!
It’s so nifty!
Nif tens. It's so nifty!
If you want less jokey (and more universal!) measures, try powers of the Planck units. For example, six to the niftieighth¹ power Planck lengths is _shockingly_ close to foursy-four² centimeters! Give it some fitting name and base units around it.
¹ forty-fourth
² twenty-eight
It's also about nine tenths of a foot.
I totally had the same idea! I might still have my notes somewhere.
"and *base* units around it"
Was that intentional?
it's just nif eight, not niftieight
@@torreywhiting5402 probably not actually, but hilarious once you pointed it out
You have no idea how much I want to see Planck units adapted to a reasonable set for everyday usage. For example, 1 nano-c is a little under 1 foot/second, and surprisingly close to 1 km/h. For a seximal alternative, 1/6^10 (one nif-biexianth) c is 4.958m/s. Combining these two units as they are isn't great since the only give about nif thirsy two (56.47) milliseconds. Some fine tuning will be necessary to find powers that work for all the main units.
Bacteria be out here counting in base 1.
If you use a positional base, like this video is all about. Base 1 has 1 digit: 0, and it has only one number: 0. - The "unary" base is not a positional base, but a bijective base. A bijective base doesn't have 0, so bijective base 10 is 1-2-3-4-5-6-7-8-9-A.
In positional base 1; 00 isn't a different number from 0, just like in any other positional base. So you can only write 0.
So 1 in base 1 is just repeating 0, except instead of repeating off to the right, it repeats to the left...
Actually, no, that's still dumb.
@@Liggliluff you can also write -0 which can be a different thing in some contexts
There are versions of Unary that are usable and historically used, but it does not work with positional systems AT ALL. They're basically just tally marks: position and order doesn't matter, just count the number of "1"s to get the number. Most that were actually used, like roman numerals, had special symbols for large groups of tally marks to make counting faster, and once you do that you can add special rules based on the order in which they appear, but fundamentally a "pure" usable unary system would only care about the number of 'ticks' and nothing else. Also can only represent ratios as ratios, since using a normal positional radix point anything on the other side of the radix point would just be more 1s--though cultures that used such systems usually had pretty simple notation for writing ratios, like |||:||||| for decimal 0.6 so just a different way of thinking about it and still perfectly usable as long as the numbers are small...which they never do.
10:18
Pros:
- arithmetic is SUPER easy, like holy shit.
- square roots exist as doable functions
Cons:
- fractions are red, red is bad
- numbers get long fast which may or may not be because the zero is fat
"Senary" is what that one person who desperately avoids suggestive language uses, but when you realize it puts more focus on the suggestion they're trying to get away from.
???
@4ourevermore Why does it put more focus on that word?
Well, if you search up that base, people will think you have misspelled "scenery".
@@Anonymous-df8it Because as a listener you're going to think "Why that obscure term? Is this person really trying this hard not to say 'sex'?"
I grew up in a country with a language where "six" and "sex" are commonly pronounced the same. A few rare people try to pronounce "six" in a different way which doesn't at all fit with the region, so it's artificial and forced, and everyone can immediately tell.
But as for "senary" and "seximal", at least most people in the real world will think one is a nerd no matter what one calls it.
@@waluigi-time But senary is the normal name, is it not?
Hi! These are the (extremely weird) bases you did't talk about.
-Golden ratio base (having the golden ratio as base) en.wikipedia.org/wiki/Golden_ratio_base
-Factorial base (impratical since the base will change according to position. also needs infinitely many symbols) en.wikipedia.org/wiki/Factorial_number_system
-Base with sign digit (Balanced Ternary is the well-known example of these bases) en.wikipedia.org/wiki/Signed-digit_representation
-Negative bases (base -2,-3,-10 etc.) en.wikipedia.org/wiki/Negative_base
-Quater-imaginary base (base 2i when i² is -1) en.wikipedia.org/wiki/Quater-imaginary_base
P.S. I'm not a native English speaker. So apologies for any grammartical error in advance.
Where's base pi?
@@zyaicob pi is so close to 3 it whould be the ternary system with very slight changes
Oh god the quater imaginary bases are killing me.
But now i want to use one, fuck.
Are you okay?
It’s funny how people who say English isn’t their first language have better grammar than native English speakers
Can we just take a second and a half to appreciate the name "suboptimal"?
6 seconds
@@norude I think you mean '10'
@@leave-a-comment-at-the-door 0b110
17* seconds
Still waiting for base-5040, or as I like to call it platimal.
We just need to come up with all those symbols and number names is all.
Platimal, brought to you by the power of cheating: each digit is a decimal number 0 through 5039 with a - between digits for easier reading.
1,000,000,000,000 = 7-4087-3018-2080
1,000,000,000 = 39-1852-3520
1,000,000 = 198-2080
1,000 = 1000
100 = 100
10 = 10
1 = 1
1/2 = 0.2520
1/3 = 0.1680
1/4 = 0.1260
1/5 = 0.1008
1/6 = 0.840
1/7 = 0.720
1/8 = 0.630
1/9 = 0.560
1/10 = 0.504
1/11 = 0.458-916-1832-3665-2290-4581-4123-3207-1374-2749 recurring
1/12 = 0.420
1/13 = 0.387-3489-1163 recurring
1/14 = 0.360
1/15 = 0.336
@@bernhardschmidt9844 You have done gods work
@@Gareon155 Does DEC5040 = a plat?
@@shadowyzephyr no, but it was Plato's favorite number. Fetaheptavigesimal is fun.
"it's fun to get silly sometimes" is a motto i wanna live my whole life by
I use a weird shorthand based on seximal as shorthand for times of day. A day has exactly 400 (aka 240) 14 (aka 10) minute chunks. 0xx is some time in the first 6 hours 3xx is some time in the final 6 hours. The 2nd digit specifies which of the 6 hours, the final digit specifies which increment of 14 (10) minutes it is. 3:40 in the morning is 034. 13:50 in the afternoon is 215. 18:20 is 302. 23:40 is 354.
I do this so nobody else can understand my notes when I die. Using dozenal would be so much more effective.
how does this comment only have 5 likes after 3 yrs(i actually understood how it works and can use it slowly)
This is fucking cool as hell
Since you decided to argue with fraction lengths, I wrote a program to add up the lengths of the periods of all unit fractions from 1/2 to 1/144 for the bases in question (2-20). Result:
The best base by far is 16, followed by 4 and 9. When going higher, 36 and 25 take second and third place.
Then I decided to test convenience by removing multiples of all primes > 11. Now the best base becomes 15, followed by 10, 6 and 9. Considering more fractions, base 18 here pulls ahead and gets second place.
Just for fun removing multiples of 11 too, the best now are 15 and 6.
Interestingly, with other variations of the parameters, 55 got first place twice. Dunno what's up with that.
It's cool that you wrote a program to check it. Fraction lengths isn't the key thing that matters for divisibility tests, though.
What makes for easy divisibility testing:
- If the number is not a prime power, all its prime power factors should have easy divisibility tests.
- If the number is a prime power and its prime factor is shared by the base, it's always easy test divisibility.
- If the number is a prime power and it is coprime to the base, then check the period length. If the period is length 1, the divisibility test is easy. Otherwise, the divisibility test is hard, with maybe 11 as a special exception.
Also, going up to 1/144 seems way too high. Even 1/19 is getting pretty high.
@@blueblimp The things you name are closely related to fraction period lengths though.
In base b, the fraction 1/n has period length ord_m(b) (order of b modulo m -> why worst-case period is phi(n) ) with n=g*m where g is the greatest divisor of n containing only prime factors of b.
- Your first point is basically the fact that gcd(p,q)=1 implies ord_(p*q)(b) | ord_p(b)*ord_q(b)
- The second point is equivalent to ord_1(b) = 0 (no period)
- The last point just simplifies matters to ord_m(b) easy, else hard
I agree that 144 may be too high. On the other hand, the ranking is not stable when going too low.
There is definitely a mistake I made: I basically weighted all the lengths equally. The result become much better for 6, 12 and 10 when weighing the periods (of the fractions 1/n) inversely proportional to n or n^2.
Great idea! Though you should adjust the scoring of your program to weigh the lengths of a certain fraction by its value (i.e. the reciprocal of the denominator), instead of uniform weights. After all, that's the probability that a given random integer contains the considered factor, so a good measure for the relevance of divisibility by that factor. It's intuitively obvious that the further you move out to larger denominators, the less important they get.
I'd be interested in the updated result!
It matters whether they are reccurring or not. Also, the higher you go, the more decimals there will be after the period, but the LESS weight it should have, because you are less likely to see it in math. So the weighting system should actually be reversed somehow. I think the lower bases would perform better when this is done.
I think a good metric would be to look at all the primes p_1, p_2, ... up to some stopping point. There's no need to test non-primes since everything is governed by the primes anyway. If l_n is the length of the period of the base expansion of 1/p_n, then calculate
Σ (l_n/p_n)^2
and see which base minimizes this value.
On a side note, there's a chance that this sum would converge if taken over all primes. I would be curious to know if it does and what that means.
If anyone wants to build a silly comunity that uses seximal, a regularised calendar, esperanto, Dvorak-style keyboards, and all the improved systems we can think of, let me know! I'd love to meet other utopists :)
A seximal keyboard does free up 4 digits for other symbols :)
@@misotanniold787 esperanto is more edgy
In my fictional universe there's a colony on Mars which uses Esperanto, with the Shavian/Sxava writing system, on a Darian calendar. The first two are the result of a political decision to "fix our mistakes and build a better world together".
Nah toki pona
And our circle constant, Tau, is close to 10
hexadecimal remains my favourite simply because it's SO useful in computing, and having everyone learn it since birth would make it easier for people to grasp how code works.
Exactly. Would also help people understand how powers work, making higher level mandatory math easier to adjust to
I'm a computer science student. I mentally convert every mention of binary numbers into decimal numbers, pretending that computers work in decimal. It works almost every time. Here is an example using the floating-point number system:
A float has 1 sign bit: 0 for + and 1 for −. Lets convert that to decimal: 1 digit: 0 for +, 1 to 9 for −.
The exponent is a binary number using 8 bits. It has a bias of 127, meaning you subtract 127 from it to get the actual value of the exponent. This is used to create a number of the form 2^(exponent), ranging from 2^-127 to 2^128. And now in decimal: 3 digits, with a bias of 499. This creates a number of the form 10^(exponent), ranging from 10^-499 to 10^500.
The mantissa has 23 bits. It is almost allows preceeded by an implicit 1, to create a number of the form "1.(mantissa)". In decimal we have to make one small adjustment: The only thing we can guarantee in bases other than binary, is that there is a 0 to the left of it. This does not really allow for an implicit extra digit, but that has exceptions anyways. So, in decimal, the mantissa is just a 17 digit number between 1 (inclusive) and 10 (exclusive).
Using this system, I can perfectly understand every topic like "precision problems with floats" or "subnormal numbers" or "how to represent NaNs", without actually having to ever think about binary.
Binary just isn't ACTUALLY as useful to the average programmer as people say. Its infinitely more important to somebody who designs computer hardware. A coder just needs to know what the limits are, like "a byte is 8 bits and goes from 0 to 255", which are just regular numbers that I literally just wrote in decimal. "What's the decimal value of DEADBEEF" is a question that nobody has ever actually needed the answere to.
Well, this fits my theory of why people prefer certain number systems. "This system is the best because it's so much easier for me personally!"
@@BaldorfBreakdowns when did i say that this is easier for me?
@@swedneck You said it makes it easier for coding, which I presume is something you do, based off of your comment.
I'm going into computer science, but if you'd asked me before, I would've already said that I favor base 16. IT'S SO GOOD
You've convinced me, seximal is awesome. It's great for both large integers and fractions. It seems like people often forget one of them when promoting bases
In my number system I just invented, Hectimal/base 100, needs half the digits compared to decimal!
The words for 1/3 in Hectimal is zero point thirty-three recurring.
It ain’t conlangs... but it’ll do
For now
Then I’ll need sacrifices
@@pqbdwmnu the reverse card from UNO
you could use dozenal to write the time (in minutes) by just 3 symbols.
there are 12 2hr periods in a day.
the first symbol could be used to tell which 2hr time period it is.
now that we know which 2hr period we're in, divide that into 12 10min periods (for a total of 120min).
the second symbol tells us which 10min period it is.
the third symbol tells us which minute of that 10min period we're at.
for example:
time: 9: 30 (am)
that's in the fifth 2hr interval. So, first digit=4
it's in the third 10min interval. So, second digit=2
now, we have to increment the time by 0min, so third digit=0
Finally, time=420
A variation on Quaternary I started using when counting measures of rest in music:
Since so much of music is based on 4/4 time, and so many musical phrases are based on groupings of 4 measures or other multiples of 4 such as 8-bar periods, 12-bar blues, etc. I started counting base 4 on my hands a lot when performing music.
I usually start with my left hand: 1 = left index finger, 2 = left index & middle fingers, 3 = left index through ring, 4 = left index through pinkie.
Then I start using my right hand for the next place: 5 = right index & left index, 6 = right index & left middle, 7 = right index & left ring, 8 = right index & left pinkie
And I continue using combinations like that: 9 = right middle & left index, ten = right middle & left middle, eleven = right middle & left ring, dozen = right middle & left pinkie
and so on.
It's a different way of doing it, like the base itself wouldn't be written as 10, it's still 4. But the number one after the base is 11. Counting looks like 1, 2, 3, 4, 11, 12, 13, 14, 21, 22, 23, 24, 31, and so on. There is no 0 in this system, because beat numbers and measure numbers in music use one-based-indexing and not zero-based-indexing.
Waited an entire month just to get a video on some fucking numbers
I think my brain is hard-wired in decimal, because once you start using terminology for a different base, I have a stroke
Not hard-wired, just trained for very nearly your entire life. If you actually try to use a different base for a long enough time, you will find it natural. It is incredible how quickly your brain can gain intuition for something if you let it. Just a few weeks ago, I was struggling to remember to use my Caps Lock key for Escape (I recently changed it to do that), and then a few days ago, I found myself doing it automatically on a machine that didn't have that set up. Similarly, when I get into (left-to-right) seximal for a bit (which I do every once in a while), I get pretty good at not messing it up. Also, verbal anything is weird and hard. 43 makes a lot more sense than the spoken "thirsy-four", especially when he still calls 41 "ten".
@@MCLooyverse Why do you write numbers backwards?
@@angel-ig Because writing numbers left-to-right is more consistent with how we write everything else, and you get to add, subtract, and multiply from left-to-right, instead of having to teach kids to do it in reverse.
@Ángel I.G. What do you mean backwards? No one writes numbers from the right to left. Maybe the Japanese.
Or are you really telling me that Americans are taught to do this and this has been true for decades?
That's inhumane honestly
@@TheAlison1456 I wrote "43" (4 and 3 sixes) in left-to-right seximal, as opposed to the normal right-to-left way everyone does in English. That's the backwards number he was talking about. Also "41" (4 and six).
My favorite is Base 60, it easily beats decimal, dozenal and seximal in terms of fractions because it is divisible by all numbers 1 to 6. You can avoid having to use 60 digits by writing each digit as two decimal digits like on clocks.
Base 30 does the same thing.
Prime numbers ending in either 5 or 1 is interesting. I have noticed that numbers divisible 6 tend to be near prime numbers and like to hang out in the middle of double primes :)
i have a plan to gradually shift everything into seximal. the secret weapon is the mindset that we don't need to change everything at once.
i begin with just one step:
using seximal in my next videogame, most notably its scoring system.
The thing about binary and hexadecimal is that where they are used, at least at a level humans interpret, ratios don't matter. They are used cardinally. Oh and binary can be used to represent a string of Boolean statements and the hand counting thing is awesome and I have found it practical at times. The more I use those 10 bases, the more beautiful I find them and the more I hate base A.
Base C is better
"You should refer to them as quarters. It's more proper"
*proceeds to continue referring to them as fourths*
i feel deeply moved by your "it's fun to get silly sometimes... theoretically change the things we take for granted, even if we know it could never actually happen"
Odd bases are actually good, because its always true that the number that is 1 more and 1 less from the base is even, and the factors of this numbers have easy divisibility test/ simple periodic expansions. so with that said the best base is 15 since is handle well all number until 11
I’m so sad that no one mentioned the better way to finger count! You can use your thumb as a representation of 5, so 1-4 are counted normally, 5 is just the thumb, and then 6-9 are the positions for 1-4 paired with the thumb. I was taught this method in elementary school and it’s so useful, because it allows you to could to 99, if you use your left hand as the tens digit.
chisanbop
heximal has fun associations: hex, hecate, heka. Matches hexagon, consistent with hexadecimal. Maybe ancient magic cults used to use that as their base.
Always been a huge fan of base 6, great video!
Btw have you ever considered how the numerical system affects the way time periods are perceived? In base six we wouldn't think in centuries or millennia, but the six-equivalents (whatever quirky name you want to call them). For example we would be in year 13203, we would consider year 10000 (1296) a big deal, and we would experience a millennium fear every 216 years.
Also the stages of human life would be perceived differently (or would they):
0-6 (infancy)
6-12 (pre-adolescence)
13-18 (~teens),
18-24 (~college years)
24-30 (young adulthood)
30-36 (100 years landmark)
Pretty neat!
Now we just have to hope my great grandma lives to 108 instead of the 106 she already has.
Three nif is another form of "centenarian" that makes sense in seximal.
@@meta04 I hope she does too! My best wishes
base infinity
every number has a unique digit
These two videos literally changed my mind about numbers from 'boy howdy, I sure hate base ten but base twelve seems impractical' to 'boy howdy, why did we even use base ten instead of base six' in the length of time it took me to vacuum out a rental car.
Hi there! I really enjoyed both this video and it's first part. If you're interested in exploring other numeral systems, I would like to suggest that of Kizh, (formerly referred to as "Garbieleño Chumash") the language indigenous to where I am from. It is a quinary system, and I understand that such a prime-based system makes representing fractions more difficult, but there are very interesting other features in Kizh's numeral system, including "kavyaa’" which operates as "X almost twice", or "X + (X-1)" (ex: wachaa’ kavyaa’ is "four almost twice" or "4 + (4-1)", which is 7. In a quinary system, this would be written as "12", of course.) I would be happy to link you to the resources through which I learned of Kizh's numeral system, as well as a document I made exploring and explaining this system to those who are familiar with base 10. (I am by no means an expert in mathematics, numeral systems, or Kizh itself, but this explanation was part of my final project for a course I took titled "North American Indigenous Languages", instructed by Dr. Marriane Mithun, in the Linguistics department at UCSB.) Thank you for your time and labor, and I hope you have a great day!
What's the best letter?
"E"
Wait.
What are you thinking about?
Base *E*
I love how base dozen five is called suboptimal
We really are being passive aggressive in math and i am here fo rit
you should create an entire channel on seximal. I would love to see that.
I laugh at these decimal systems from my throne of fractions
The Myst series of games uses a base twenty-five numbering system, but the digits are designed in a way that you only need to remember 5 symbols.
(Explaining the system actually spoils some of the puzzles in Riven so I recommend playing that game if curious)
yay another video!
While before you had convinced my seximal was bestimal, the "primes end with 1 or 5" bit is like THE COOLEST PART!!! I was literally just watching a video about how prime numbers act that way the other day!
I am just going to through this out there: our perception of a 10 base system is more about measurements i.e. decimal system and works extremely well for that.
But for other maths, we only speak in a ten based system because, that is what the English language bases its numbers on.
There are numerous other languages that use other number based systems and sometimes combinations of number systems for larger groupings of numbers.
In addition, we are using our written numbers 0 to 9, when again , many languages use completely different characters to represent numbers.
In the field of mathematics, we have more standardization of mathematical notation. Base this or that somewhat becomes a moot point in this area of topic as the accepted notation in mathematics is universal. To attempt to change our counting base system is basically mostly about quantity, and nothing more.
Quantity, measurements( i.e. lenghth, area, volume, temperature etc), and mathematical formula format of notation all serve very different purposes. As does chemistry has its own standardize format of notation, and so forth.
For bases greater than 10 I understand changing how we say the number because we have to but 6 is less than 10 so there is no reason we can’t say the number normally so one hundred (base 10) is two hundred and forty four (base 6)
Are You Saying On Base 6 We Should Say "Two Hundred & Fourty Four" When We Mean (Decimal) 100? That'd Be Really Confusing...
Only for the weak minded. We have language to speak, and if we decide to speak words to represent *digits* instead of quantities, then surely we could speak using decimal.
"Base 1" - Unary
0: 0
1: 00
2: 000
3: 0000
4: 00000
...and so on
The fractions are equally unimportant
(2:40) Why create a completely new system when you still can use metric? You just need new prefixes.
so deca-, hecto-, kilo-, which are 10¹×, 10²×, 10³×, would instead be: 6¹×, 6²×, 6³×
the same with deci-, centi-, mili-, currently being 10⁻¹×, 10⁻²×, 10⁻³×, would be 6⁻¹×, 6⁻²×, 6⁻³×
new names and letters can be used, to avoid any confusion.
Using the prefixes from Conlang Critic; nifti- is 6⁻²×, so a niftimeter (nm) is 0.01₆ meters. unti- is 6⁻⁴×, so an untimeter (um) is 0.0001₆ meters. biti- is 6⁻⁸×, so a bitimeter (bm) is 0.00000001₆ meters. feta- is 6²×, so a fetameter (Fm) is 100₆ meters. grand- is 6⁴×, so a grandmeter (Gm) is 10000₆ meters. - Additional prefixes would still be needed, as you do have GW (gigawatt), GB (gigabyte). But this is a good start. So in computing, you have byte (B), fetabyte (FB), grandbyte (GB) ≈ 1.30 KB or 1.27 KiB, so you need larger prefixes.
A wrinkle is that the kilogram is the standard SI unit of mass, so do we have niftikilograms, niftigrams, or both? If both that's an imperial pain.
@@diribigal Kilogram should rather be renamed to gram if we switch base; so if the kilo- prefix isn't used, just dropping it altogether works. Another option is to make up a new name and define it the same as 1 kg.
I'm using seximal in my conculture because I wanted a base other than ten, and six works out as well as I thought dozen would. It's especially useful because I have exactly six vowels, and six consonants. So I can reserve one consonant for digit syllables :D
As a computer scientist and hobbyist CPU-builder, I quite like hex, but am readily willing to admit my bias on that front. I really just want a nice hex calculator that isn’t just a decimal calculator with a hex display mode that I need to constantly tell it use
The multiplication table being easier is a big plus, so many kids leave school without knowing the basic times tables. For sub 10 multiplications ignoring reverses in decimal there are 36 to learn, in seximal there are only 10, learning these would be trivial.
*@[**12:41**]:*
Base-21, unvigesimal, is decent at fractions for an odd base as well... which is also mostly because it is a threeven base.
Base pi is obviously the best. It's better than base e because you get another digit. You can simply and easily represent the circumference of circles. Plus, as an added benefit, the human race by definition knows all digits of pi.
Hot (?) Take:
Complexity of floating point representations of rational numbers is irrelevant to the merit of a base, because rational numbers are most simply represented as... ratios (fractions). Each base is equally efficient at representing irrationals.
With that in mind, I'll inject my taste:
The only remaining criteria relevant to me are digital utility, finger countability, written efficiency, and digit entropy (as a source of mental labor).
Digital utility prescribes a power of two. Finger countability somewhat favors quartal; you can count from 0 to 4^2 - 1 = 15 without thumbs or pinkies. Octal fingers can count from 0 to 4•8 - 1 = 31, requiring pinkies and thumbs, but with certain schemes you flip people off with 7
Here’s my top 10 favourite bases:
Base 10
Base 110
Let me know your thoughts
Thank you
my favorite base is base 30 labeling it in decimal terms and base 50 in seximal. i just like primes and thirty/fifsy is 2 * 3 * 5 so it's the first three primes. I haven't really thought through it though, and after watching this I would have to do something like the Babylonians did.
((5*12)-1)*6+(2*5)+1(+1/4) -system for the lunisolar calendars is no way to count but rather accurate. 2*lunations times 6 and add 2 lunar weeks of five and six days. Solday resets the cycle every 4 years and every 128 years there's a gap. Moon uses 59-day system so some complications are unavoidable.
alright, if you want to get silly with it, base -1.5 is pretty fun to play around with, i spent a half hour or so trying to do working out to figure out the pattern for counting, along with how you have to do the basic four math operators
He's been jan Misali before, and one day he shall be jan Misali again, he will be more jan Misali than we can even imagine.
I like base 420. Yes, it's a Funny Number™, but it also has some pretty cool properties. It's divisible by all numbers between 1 and 7.
Here are the reciprocals of the first thirsy integers in base 420.
1/2 = .210, 1/3 = .140, 1/4 = .105, 1/5 = .084, 1/6 = .070, 1/7 = .060, 1/8 = .052:210, 1/9 = .046:280, 1/10 = .042, 1/11 = .[038:076:152:305:190:381:343:267:114:229], 1/12 = .035, 1/13 = .[032:129:096:387:290:323] (thirteenths are actually more convenient than elevenths, weird), 1/14 = .030, 1/15 = .028, 1/16 = .026:105, 1/17 = .[024:296:197:271:321:074:049:172:395:123:222:148:098:345:370:247], 1/18 = .023:140
Fifteen of those are terminating.
I think dozenal is the best, but given how easily you can convert dozenal and seximal you could even use both of them at the same time.
The only thing that matters is that both are better than decimal
The measurement arguments are my favourite.
1:21 Not taking seximal seriously probably comes from people associating anything with the letters "s, e, x" in that order to the activity they never partook in: eroticly touching another animal, whether it's done mutually or not (humans are animals btw). In reality, sex is rarely associated to that and is much more often associated with the type of reproductive organ an animal has or the number 6.
My brain can’t stop reading this as “Sexual Responses” every time this pops up on my feed
There are tricks you can do to composite numbers without new symbols. D'ni is base 25 but actually kinda base 5 with every other digit on its side. And yes I only know this because of that one puzzle in Riven :P
I absolutely love hexadecimal.
Base16 is awesome when writing big numbers. Also, 1024 rounding error makes justice.
Base imaginary is clearly the best. i => 1, 1 => -i, -i => -1, and -1 => i
I count to 99 on my fingers without trouble. Right index, middle, ring, pinkie are one each, Right thumb is five. Left index, middle, ring, pinkie are ten each, left thumb is fifty.
As for economy, base 2 and 4 can each perform a 64-way selection for a cost of twelve (six binary digits at a cost of 2 each, or three quaternary digits at a cost of 4 each) but for the same cost base 3 can do an 81-way selection (four ternary digits at a cost of 3 each).
Seximal is best? Nuh-uh. Every number in existence is a factor of infinity.
Go Infinitimal!
Long live the Empire!
Ngl, the nice coincidence that makes niftable work is super nifty (hah). If anything tips me over from being a dozenal advocate, it may be this (after all the other things balanced things closer, having not previously considered base6 before thinking it would be a pain given smaller base - but that was nicely covered 👍)
Seximal is the best base. Especially when it comes to fractions.
Eleven: Let me introduce myself...
No one cares about Eleven
hey, at least it's better than dozenal and decimal.
dozenal: 1/5 = 0.(2497)
decimal: 1/7 = 0.(142857)
seximal: 1/15 = 0.(0313452421)
yes, seximal is bad at elevenths, but dozenal's worst fraction is fifths and decimal's worst fraction is sevenths, and seximal is already good at representing fifths and sevenths, therefore, seximal is better at fractions than dozenal and decimal.
Balanced number systems (such as balanced ternary) are superior, they properly extend to the negatives.
1. If you have a method to add numbers, in a balanced number system you can also use it to subtract.
2. Rounding and cutting off the number at a digit is the same.
3. The multiplication table is a quarter the size.
Therefore balanced seximal is the best number system, with digits -3,-2,-1,0,1,2,3 (ambiguities such as 6 + (-3) = 3 can be resolved by convention to enforce rule 2.)
I think it's really cool to examine this stuff in practical terms. Like, yeah, fractions are really good indicators for how good a base is, but as a history nerd my mind immediately went to how we would count years and periods of history.
It's a given that we talk in centuries as periods in human history, like the 20th century as the "century of extremes" or the 19th as the "century of industrialization".
We have a term for the "long 19th century" from the French revolution in 1789 to the first world war in 1914.
We speak about decades as periods of culture, like the 80s as the decade of video games and neon and synthetic music, the 2000s as the decade of fledgling social media and emo.
All of this would work completely differently in seximal.
A "sexade" (=6 dec, 10 sex years) doesn't work as a parallel to a decade (=10yrs dec, 14yrs sex) because it's so damn short.
A "niftury" (from CENtury) or perhaps “niffy” (it’s actually CENTURy, after all) (=36 dec, 100 sex years) is way longer, and the average human would only live through 2-3 niffies compared to humans living through up to ten decades.
The next step up is six nifs, also called “tarumba” (=216 dec, 1000 sex)
Ironically, since a century is so damn long, a tarumbaium (from millennium? maybe?) works better as a way to think of historical periods compared to centuries than regular niffies (or sexades) compared to decades.
A lot happens in a tarumbaium, but a lot happens in a century too, and under closer scrutiny, neither hold up as a serious period of analysis.
The (currently) last step up is unexian (1296 dec, 10000 sex) years, which I guess can just be called an unexian or unexiant or something.
(I’m not sorry for the shoddy nomenclature lmao)
Since the current year in seximal is 13211, or one unexian thirsy two nif seven, we’d be currently living in:
- The second unexian
- The tenth tarumbaium
- The nif thirsy third niffy, or I guess the third niffy of the tenth tarumbaium OR MAYBE the thirsy third niffy of the second unexian
- we’d also be living in the sixties, I think? Followed by the dozenies/dozzies?
- the current tarumbaium began in the year 1944 dec which is a pretty neat cutoff point all things considered. The previous began in the year 1728 dec which means the adoption of radar is part of the same time period as the discovery of Uranus. Wild.
Dude, a negative one digit🤯😀😀😀😀
Ah yes, base 5040, also called *_fifmiltetrogesimal,_* my favourite!
or fetaheptavigesimal, FHV for short
@@egon3705 _ᵖˢˢᵗ, ⁱ ᵈᵒⁿᵗ ᵘˢᵉ ᵃˡᵍᵒʳʰʸᵗʰᵐˢ_
Wow. I’m a bit of an idiot. I’ve watched quite a few of your videos and am familiar with how different bases work and all that, but I still left a comment on the original video saying that you miscounted the number of factors of 10 and 12. Whoops.
I'm a fan of base 2 for the following reasons:
*Addition and subtraction are easy in any base system, but for binary multiplication is very easy. There is no need to memorize multiplication tables, it's simply repeated addition with the bits shifted to match the pattern of the number you're multiplying.
*Where base 2 really shines in my opinion is in long division. Because each bit is only either a yes or a no, calculating each bit in the binary expansion of a fraction boils down to a comparison between two integers and at most a single subtraction.
*In any other base, you always have to either one of these two things:
do repeated multiplications to find the smallest integer multiple less than the number you're trying to divide
do repeated subtractions until the number you're trying to divide fails the comparison
Both of these are very annoying when dividing by a number with more than about two or three places, while binary sidesteps this issue entirely. Also this has to be done for every single place in the expansion which gets very tedious very fast. In binary you can just keep going as long as you like.
*This is counterbalanced by the fact that for the same level of precision at say 3 decimal digits, you need about 10 bits, but in my experience (yes I experimented with this) doing division with a high number of bits is much easier than doing division with moderate amount of digits just because each step of the process is a simple yes or no question.
*From a philosophical point of view, I'm a fan of the fact that as the smallest usable base, it has objective merit as the best possible with its advantages. With base 6 or 12 or really any other base, you can always make the argument that you should have more factors to further help with ratios and really the only counter argument is "that's too many digits for us to memorize," but this is a subjective judgment with no mathematical justification.
Not that much of this matters, because after you're out of middle school you're never going to do long division again.
Watch Matt Parker doing long division with 20-digit-numbers every year on pi day.
All you need to do is make a times-table of the number you want to divide by (meaning you multiply it by every digit in your base, which is 1 to 9 in decimal) and then at every step of the division, you need to find the largest of those numbers that fits, and then subtract it.
So, after the initial multiplication step, which happens literally only once, every iteration is just up to 10 comparisons followed by a single subtraction, which isn't actually that much different from what you described.
after watching these videos I can now convert numbers into base 6
you talked about Kaktovik Iñupiaq, a digit system like this could be used for niftimal to get the best of both worlds, i guess
But, base 7 does sevenths really well
Wake up people! convert to base seven today!
it's an odd base and odd bases aren't very good. besides, seximal is already good enough at sevenths. we don't need single digits for sevenths.
I read something about monks who'd created what was effectively a base-10,000 (10^4) numbering system that was actually readable, by using composite figures as digits. Each figure began as a single vertical line, and having one of nine markings each representing 1-9 (or no marking for 0) attached to a corner of the line (top left, top right, bottom left, bottom right). Effectively just an efficient way of compressing decimal, but cool nonetheless. Sadly I don't remember what the markings were nor how it was read, but if anyone else knows what I'm talking about and remembers, please do tell.
Cistercian number system
Jan Misali, the only youtuber who talks fast enough. I don't have to speed up his videos!
WAIT IS HEX THE ONE USED FOR COLORS?!
This is the most elaborate joke ive ever seen
In decimal, to check if a number is divisible by seven, you can actually multiply the last digit by five and add the rest of the number (divided by ten). For example: 105 is divisible by 7 because 5 (the last digit) * 5 + 10 = 35, and 35 is divisible by 7.
Therefore, 49 doesn't look prime because 9 * 5 + 4 = 49, which is clearly divisible by seven because 9 * 5 + 4 = 49, which is divisible by seven.
Base 16 is really only good at fractional powers of two, but it is super good at it. Cut something in half eight times and you're at HEX0.01 vs DEC0.00390625. We cut things in half all the time, way more than cutting into thirds or fifths or whatever. Having your base be not only a square, but a square of a square probably leads to some cool properties too (not sure what though).
You're starting to convince me that seximal is among the best choices for a base though, given its simplicity and broader utility with fractions.
The best numbering system is base infinity because then you can write every number with it's own digit! 🙂🙂🙂🙂🙂🙂🙂🙂🙂🙂🙂🙂🙂🙂🙂🙂🙂🙂
6 is good but I'm a daredevil that don't mind bases like 120!!! It fits my enthusiasm for the cosmos but I now have my eyes on 24 and am questioning its potential properties for a simpler system. Either way notice how all the best number bases are multiples of 6 and 12 ? It is that way for a reason!!!
“The numbers mason! What do they mean?!” “No seriously I have no clue whats going on.”
Other fun bases:
Tetravigesimal (24) - It's like Dozenal but good at eighths.
Duotrigesimal (32) - Another power of 2 base
Balanced Nonary (-4~4) - Balanced ternary with a little more room.
Base64 (Gee, I wonder) - A favorite of encryption nerds, and enough digits for the decimal numbers, all the letters in a case-sensitive manner, and / and +.
Captchary - A method of writing in Base 64 favored by filthy Homestucks. Goodbye / and +, hello ? and !.
Eulary (e) - The number with the lowest radix economy.
Nicemal (69) - It handles thirds and twenty-thirds well. Aside from that it's just for the lulz.
Negadecimal (-10) - Negative bases are interesting since, like balanced bases, you can write any positive or negative number without worrying about the minus sign, but unlike balanced bases, you don't have to either use an odd number or forego zero.
Derfmal (11?) - Undecimal shifted weirdly, with the added digit (derf, represented by a 4 flipped vertically) being between 5 and 6 rather than after 9. People who've watched iCarly know about this one.
Quater-imaginary (2i) - Represents every complex number in strings of just 0, 1, 2, and 3.
Quadrigentitredecimal (413) - Does sevenths and fifty-ninths well. Aside from that it's just a nerdy joke.
Cannabicimal (420) - Haha blaze it. Also has a lot of factors.
Leetimal (1337) - U 4M N0082, 1 R L337
Platimal (5040) - It's an antiprime so it has to be good, right?
Tetranonagesimal (94) - Enough for every single thing you can type on a normal QWERTY keyboard, without using spaces or Alt Code characters, of course: 012356789ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwxyz!@#%^&*()`-=[]\;',./~_+{}|:"?
Base 9 actually looks like a good idea, as 1 recurring digit is completely acceptable for a half
Centesimal is interesting, because there's actually a centesimal proposal for Toki Pona!
zero = ala
one = wan
two = tu
five = luka
twenty = mute
And each centesimal digit is made like a regular Toki Pona number out of these. For example, 69 is mute mute mute luka tu tu.
The ":" is "ali", which refers to the number 100. Thus, 42069 is tu tu ali mute ali mute mute mute luka tu tu (while it would begin with 420 'ali's in the regular system).
To make it easier to break large numbers down, you can probably separate them like so: 4,20,69.
There isn't a standard for non-integers or negative numbers yet, but nasin nanpa pona does look quite promising.
I have an idea for a new series: Something like Conlang Critic but based on bases!
I still don’t know how seximal or dozenal work, I’m beginning to think that I don’t understand base-10.
What about my new number system, where we start at one being 1, then every prime number is written as p subscripted with the order of the prime (recursively using the numbering system so far), and composite numbers are written by stringing together the prime factors, with superscripts for exponents. So counting goes 1, p_1, p_(p_1) ,p_1^p_1, p_(p_(p_1)), p_1p_(p_1) and so on.
Pros:
-multiplication is made very simple, just count how many of each prime factor you have
-divisibility checks become incredibly easy, and prime numbers are numbers with no suberscripts and only one p written outside of a subscript.
-using negative exponents allows us to express every fraction without any infinitely repeating symbols
Cons:
-absolutely terrible
addition would be an incredible unbelieveable ugly nightmare...
honestly, i wasn't fully paying attention on the prime factor joke and rewatched it because i was like "wtf did i hear that correctly? that's such a stupid comparison"
then i also heard your tone and realised you were joking xD
All that a heximal system would need to work properly with the Metric system is a new set of prefixes. Such a system would be very easy to adopt. In fact, we recently adopted a set of new prefixes and it worked fine: en.wikipedia.org/wiki/Binary_prefix .
Just base the names of the new prefixes on the know prefixes but with a consistent twist, as was done with the binary prefixes. E.g.: "kihe", "mehe", gihe", etc How hard would those prefixes be to learn for someone who already knows the base 10 prefixes?