Ranking Every Base 1 Through 4*4

Before we get started, I'm trying out a new website background technique! I made this in blender. I couldn't quite figure out how to get the formatting how I wanted, but let me know if you like it or not.

This is an in order list from worst to best bases, from the range 1 to 4*4! (base neutral way to write dozen four). There will be the name of the base, followed by the base number in seximal, decimal, and then binary. In that order. If there's less than three, it means some of them have been merged for brevity.

Unary (Base 1)

Ah, unary. Tally marks. Every integer is expressed as a series of 1s equal in length to the integer it's representing. This does have some advantages, that math is really easy, and that it's very visual, but that's about it. It's not actually totally unusable if you cheat a little bit and use math. For example, you can write the largest known prime as millions of decimal digits, or you can write it as 2^136,279,841 - 1_DEC. So we can write numbers like the number of degrees in a circle as 11¹¹¹ * 111¹¹ * 11111. Better than ten nif ones, but still worse than 1400_SEX, or 360_DEC.

This is basically prime factorizations trying to be a positional numbering system. Though you can write the number in other ways that use composites, but I think they're all less efficient. Maybe I'll write a program to find the best unary representation of a number. Still by far the worst base in this range.

Elevenary (Base 15, 11, 1011)

This base is prime and near decimal. Only notable quality is being good with 5s. As bad as possible with 3s, which are probably around ~4 times more important. Good size I suppose. The worst prime base on this list, but quinary and trinary are cutting it close with how small they are. They're ahead because they're small primes so they're good at small primes, by definition.

Quinary (Base 5, 101)

Prime base. Almost decent size. As bad as possible with threes, which is terrible. Quinary is one more than 4 so it's not actually too bad with fours, but still no excuse for fumbling halves and thirds.

Trinary (Base 3, 11)

Prime base. Super small, and it doesn't get away with being prime and small like binary does. Binary gets away with it by being the smallest and even. Trinary is neither. At least it's good with thirds, the second simplest fraction. Not good with 5 or 7. Also, people say it's the most informationally efficient base, being closest to e, but I think that's wrong. Binary is the most informationally efficient. I do not have the mathematical prowess to defend this claim, unfortunately.

Baker's Dozenal (Base 21, 13, 1101)

This is a prime base. A bit big, but still in the usable range. This prime base piggy backs off of dozenal, making it almost usable. Bad with fives, okay with sevens. A worse version of septimal.

Septimal (Base 11, 7, 111)

Base 11, it gets 2 and 3 from 10, but what prime base isn't one more than a multiple of 2 besides binary? Pretty bad with 5, but there is a test with the powers of 2 as 5 is b-2. Obviously good with seven.

Nonary (Base 13, 9, 1001)

Base three squared! Nonary is good with thirds and nineths, and is sorta decent with powers of two because it succeeds 1000. One eighth, fourth, and half are all one digit repeating. Pretty good for an odd base. Also decent with fifths, just being <0.17>. This base is also a good size. Sevenths aren't terrible either, being just <0.125>. This is because square bases usually have pretty good fractions. They have finite or no cyclic numbers. This is also a pretty good sized base.

It doesn't handle sevens very well, nor does it handle five as well it should. 5 being supported by b+1 is kind of crappy. 7 is supported by b-2, so there is a divisibility test with powers of 2, but multiplying by 2 is even harder in odd bases.

Triquinary (Base 23, 15, 1111)

Base 3 * 5. This one isn't a terrible odd base, as it's composite. There's actually only two odd composites on this list, and this is the better one. Triquinary is obviously good with thirds and fifths, but because of biseptimal it's also good at seven. There's an easy divisibility test for 2, 3, 5, AND 7. Pretty solid! Also, it's one less than a power of two so it's good with those as well. A bit large.

Octal (Base 12, 8, 1000)

By far the worst power of two base in this range. And the worst even base on this list. Its neighbor gives it sevenths, which aren't terribly useful. It's terrible at thirds and fifths, and isn't a meta power of two. Over all just kind of awful. Easily the worst even base on this list.

Biseptimal (Base 22, 14, 1110)

Base 2 * 7. As the prime factorization suggests, it's only good with twos and sevens. Its predecessor is thirteen so it gets really screwed over here. One less than thirpen so it's as bad as possible with threes, but that's not terrible. Also not terrible with fives/fifths. To test divisibility by three or five, flip the sign of every other digit and add the digits together. So like 27 in biseptimal, -2 + 7 is 5, so the number is divisible by 5. Or add pairs of digits. That's probably easier.

Dozen three really helps biseptimal be usable. Biseptimal is great with 2 and 7, and okay with 3 and 5. But because 3 is so small, having b+1 be a multiple of 3 is as terrible as it can be.

Dozenal (Base 20, 12, 1100)

Ah, base three times four. Great with twos, threes, and fours, at the cost of everything else. It's large too. Like seximal, but bad. I'm very torn on whether to put this in front or behind of decimal. They're so similar in quality they're not worth comparing, in my opinion.

Decimal (Base 14, 10, 1010)

Yeah, this one. I put it in fifth place, which is actually pretty high, and that's because this base is genuinely pretty good. I hate on it a lot but that's just because it's what I'm used to so it's extremely dull. In truth, it's a pretty good base. Good with twos and fives, but also good with threes and nines using nonary. Just barely covers eleven by being before it and- oops. It forgot sevenths. Yeah, it's not a great base, but it is one of the best in the usable range. A bit big for my tastes, but still a decent size.

Quaternary (Base 4, 100)

Base four! This is like tessimal but smaller. It has no cyclic numbers, and has relatively simple fractions. It's a bit too small for my liking so I'm putting tessimal ahead. I agree with jan Misali's claim that quaternary is "really almost good as seximal, just a bit smaller".

Tessimal (Base 24, 16, 1,0000)

Coming in at third place, these last few bases are the ones I actually like. Tessimal is very similar to decimal, in that it's not great on its own but becomes great using its predecessor. Tessimal piggy backs off of triquinary to become usable. It has good thirds and fifths. And obviously, is very good with the powers of two. One half is <0.8>, one quarter is <0.4>, one eighth is <0.2>, and of course one tesseth is <0.1>. Because it's a square base, it has few cyclic numbers. I think it has none but I wasn't able to find info. Also, because it's two more than biseptimal, it's okay with sevens too. Not as good as base 10 with sevens of course, but passable. If you really like/need to use powers of two, this is the base for you. That's why we use two tessimal digits to compress bytes into human readable numbers!

I can't not address the size of this base though. Being the biggest on the list, it is quite big. But this base cheats by being a power of two. You can think of every digit as four bits, and then it doesn't feel as large anymore. Adding to this, it does do one seventh as <0.249>, but once you realize that it's just 0.'001 in binary it becomes really easy to calculate. Great base.

Binary (Base 2, 10)

Binary! One of my favorites, I put it in second place. (Isn't that funny? I put decimal in fifth place, binary in second place, and quaternary in fourth place?) For a few months, I was torn on whether I liked binary or seximal better. Of course there was this amazing video arguing for binary, but something didn't sit right. Yes, judging it purely on how many strokes it takes to write small numbers is shallow, but it's not pointless. Numbers do get long fast, even if it is maybe worth it.

There's all the cool stuff about it being small making it maneuverable, and yes that's true, but it still has to do that, while seximal just works. It's really cool that I can test for divisibility by 37_DEC really easy, but do I ever need to? No, not really. Having to add up pairs of bits for three and quads of bits for fives is just a bit cumbersome.

In conclusion, binary is no doubt a great base, but it is still binary. It kinda takes effort to use, and usually repeating fractions are just kind of a deal breaker sometimes. I don't want to type in "0.01010101" for one third. Though I do think it's unfair that this base is thought of as having so many cyclic numbers while quaternary has none. It's just a slightly more efficient form of quaternary, so they should be treated the same. A digit length of 4 in binary should be thought of as the same worth as a digit length of 2 in quaternary.

Binary is one of those things that is fun to play around with, but ultimately, whenever I don't think in decimal, I find myself thinking in seximal. It's conventional and easy to use. Binary is okay at everything, while seximal is really good at the first few primes. And honestly that's just better. Still really cool how informationally efficient it is. If you like binary you may want to check out my binary systemplex. Oh, and lastly, binary finger counting is by no doubt my favorite finger counting method. Seximal's is nice, but binary is just wayy better. If you can't move all of your fingers individually I totally get it though.

Seximal (Base 10, 6, 110)

Uh, I kind of spoiled all of this in the previous base's entry. Anyway, seximal is the most convienent general purpose base. It's the perfect size, I wouldn't want a digit more or a digit less (though I may be biased), and it prioritizes the most important two primes first. Then it prioritizes the third most important prime, and then it prioritizes the fourth most important prime last. It does things in the correct order. Decimal could learn from this, putting fifths foolishly above thirds. Like binary, seximal being small is one of its strengths.

Seximal's biggest flaw is that it can't handle powers of two and eleven very well. The divisibility test for eight is a bit cumbersome, you have to multiply the nifs column by 4, the sixes column by -2, and add those two with the ones column, and check if that's a multiple of 12. Eleven and onward aren't very useful so I don't really care, but, there is a divisibility test for every prime in seximal. Which, that's really the biggest thing binary had going for it, so, yeah.

The divisibility test is not easy, but neither is the binary one. Basically, every prime is one more or one less than a multiple of 10. This means that in base 10, there's a divisibility test for every prime. First find the multiple of 10 it's one off from, say for 15, 2. Then, take the last digit, multiply it by that number, and add it to the rest of the number. Or subtract it, if the prime is one more than a multiple of 10. "The rest of the number" is a bit vague, so defining it more clearly:

Say the prime we want to test is p. Find what multiple of 10 p is one off from, call that k, and the number we want to test n. Take the last digit of n, multiply it by k, and then add it to floor(n/10). Or, if p = (10k)+1, subtract it.

If p = (10k)+1, then calculate floor(n/10) - n%10. If p = (10k)-1, calculate floor(n/10) + n%10 instead. Then set n to that new number and repeat until you recognize a multiple of p. Sometimes you'll get 0, and that counts.

So yeah. Seximal does really well with the first 14 numbers, and has an eh test for every prime. Powers of 2 are nice, but binary is so small that math becomes cumbersome with all the carrying. Sure you can add multiple bits at once, but just this is my tierlist and subjectively I like base 6 more. Really, people's preference on bases is a spectrum. People who care about all primes, or just the powers of two, like binary the best, people who like all primes up to five or seven prefer seximal, and those who just really don't give a shit about numbers larger than 4 like dozenal the best.

Base 2 is pretty good, but it just can't compete with 10.

Chart I Made

This chart shows how these four bases do with the first dozen one numbers (except tessimal/bioctal I extended it all the way to tess). It combines fractions and divisibility tests into one alphabetic score. And I didn't drop E like most people do. Just A-F + S. It's kind of consistent, too. S is a factor of the base, A is a factor of a power of the base (usually, the size of the base can change this). B is a factor of b-1, C is some multiple of a factor of b-1, or a factor of b+1, or a factor of a power of the base. D is a factor of b-2, or two difficult numbers together. E usually means "almost completely unsupported but a factor of b-2 or something"

Conclusion

In conclusion, we should all count in decimal. It's what everyone uses so it's the easiest to "switch" to, and it's already pretty good. The only thing better than perfect is standardized. But seximal comes in close second.