what should we call the other bases?

most people use decimal, "base ten". this is usually written "base 10". the thing is, <10> only means ten IN base ten. in fact, every base calls itself "base 10". as a result, it's more practical to use words to refer to bases, hence the name "decimal", which comes from the Latin word for "ten".

for small bases, this works fine because there's basic Latin words for the numbers they're based on. decimal (base ten), nonary (base nine), octal (base eight), septimal (base seven), seximal (base six), quinary (base five), quaternary (base four), trinary (base three), binary (base two), and even the highly impractical unary (base one) all have easy to parse names derived this way.

the problem is that Latin uses base ten, so bases larger than ten end up with names that put a bit too much of an emphasis on their relationship with decimal: undecimal, duodecimal, tridecimal, etc. people who like base twelve like to call it "dozenal" instead of "duodecimal" for this exact reason. these names are simply too biased in decimal's favor. ideally, every base should have a unique name that reflects its properties, rather than trivial information about its size.

names for the first nif bases

1: unary
2: binary
3: trinary
4: quaternary
5: quinary
10: seximal
11: septimal
12: octal
13: nonary
14: decimal
15: undecimal (un- means "plus one")
20: dozenal
21: baker's dozenal (my friend Kate came up with this name and it's very good)
22: biseptimal (two times seven)
23: triquinary (three times five)
24: hex (already a popular name for this base; short for "hexadecimal")
25: suboptimal (not a great base)
30: triseximal
31: untriseximal
32: vigesimal (already a popular name for this base)
33: triseptimal
34: bindecimal (two times eleven; "biun-" becomes "bin-")
35: unbindecimal
40: tetraseximal
41: pentaquinary
42: bindozenal
43: trinonary
44: tetraseptimal
45: untetraseptimal
50: pentaseximal
51: unpentaseximal
52: tetroctal
53: trindecimal
54: bisuboptimal
55: pentaseptimal
100: hexaseximal

generalizing this for larger bases

the basic roots that exist in this system are:

  • unary (base one; doesn't combine with other roots)
    • (u)n- (plus one)
  • binary (base two)
    • bi- (times two)
  • trinary (base three)
    • tri- (times three)
  • quaternary (base four)
    • tetr(a)- (times four)
  • quinary (base five)
    • pent(a)- (times five)
  • seximal (base six)
    • hex(a)- (times six)
  • septimal (base seven)
    • hept(a)- (times seven)
  • octal (base eight)
    • oct(o)- (times eight)
  • nonary (base nine)
    • enn(a)- (times nine)
    • dozenal (base twelve)
      • doz(a)- (times twelve)
    • vigesimal (base thirsy two / twenty)
      • icosi- (times thirsy two / twenty)

    prefix forms of prime numbers are formed with hen- and -sn(a)-, as in "hendecasna-" for "times eleven".

    there are also a few special case roots that follow different rules.

      • decimal (base ten)
        • dec(a)- (times ten)
        • multiples of ten are formed as though this name were "gesimal"
      • baker's dozenal (base dozen one)
        • hendozasn(a)- (times dozen one)
        • multiples of dozen one are formed as though this name were "undozenal"
      • hex (base dozen four)
        • no unique prefix form; just tetratetr(a)-
      • suboptimal (base dozen five)
        • hentetratetrasn(a)- (times dozen five)

      using this set of roots, any given large base can be given at least one unique name. preferably, the name should be derived from the two numbers that are closest together that multiply to form the base number, with the larger one last. so, even though there's plenty of ways to name base nif foursy, the ideal name is "hexagesimal", because it's six times ten.

      alright, here's a bunch more names for big bases.

      120 (DEC48): hexoctal (six times eight)
      320 (DEC120): decadozenal (ten times twelve)
      500 (DEC180): dozatriquinary (twelve times dozen three)
      1040 (DEC240): tripentahex (dozen three times dozen four)
      1400 (DEC360): trihexavigesimal (thirsy times thirsy two)
      3200 (DEC720): tetrahexapentaseximal (foursy times fifsy)
      3250 (DEC840): tetraheptapentaseximal (foursy four times fifsy)
      1 5400 (DEC2520): pentennaheptoctal (113 times 132)
      3 5200 (DEC5040): heptadecoctononary (154 times 200)
      110 4400 (DEC55440): hendecasnatriheptatripentahex (1036 times 1040)

      other names for these bases can be made with these roots, some of which are shorter than these "ideal names". if you want to use a large base, it's probably better to have a short name than it is to have a mathematically symmetric name. for example, "hendecasnatriheptatripentahex", when broken down, really means "eleven times three times seven times three times five times sixteen", which can be rearranged into "seven times eight times nine times ten times eleven", or "heptoctennadecundecimal". it's still pretty long, but it's shortened by enough that it makes a difference.

      EDIT: here's a txt file with the first nif unexian base names in this system

      say, which of these bases are the GOOD ones?

      okay so that was a lot of just like, boring definitions. here's all the bases that anyone ever uses, and quick summaries of what I think of them:

      • BINARY: bases don't get much smaller than this, so it's really bad when it comes to compactness.
      • TRINARY: according to math, this is the most "economic" base, whatever that means.
      • QUATERNARY: four is a highly composite number, and it's right between two primes, so it's really almost as good as seximal, just a bit smaller.
      • QUINARY: a good example of how prime bases aren't very good.
      • SEXIMAL: [see rest of site]
      • OCTAL: sometimes used for binary compression, but it isn't really all that good at that. like, quaternary is better at compressing binary even though it's a smaller base.
      • DECIMAL: yeah, you know this one. it's VERY okay.
      • DOZENAL: better than decimal, but honestly it's kinda overrated.
      • TRIQUINARY: surprisingly good for an odd base.
      • HEX: everyone's favorite way to compress binary, and for good reason!
      • VIGESIMAL: like decimal but worse at threes.
      • TETRASEXIMAL: honestly, if you're fine with the six extra symbols, better than dozenal.
      • TRINONARY: great with threes, but cannot do fives.
      • TETROCTAL: another power of two, and this one is the WORST ONE!!
      • HEXAGESIMAL: we're getting into some impractically large bases here, but nif foursy has enough factors that it almost makes up for it. it's between two primes though, so it doesn't have any benefits beyond sheer compositeness alone.
      • DECADOZENAL: my favorite large base. it's good for most of the reasons seximal is good. honestly, decadozenal deserves its own page on here.