# 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: elevenary

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 ("un-" means "plus one")

32: vigesimal (already a popular name for this base)

33: triseptimal

34: bielevenary

35: unbielevenary

40: tetraseximal

41: pentaquinary

42: biker's dozenal (the "ba" in "baker's dozenal" is removed when it comes after a prefix)

43: trinonary

44: tetraseptimal

45: untetraseptimal

50: pentaseximal

51: unpentaseximal

52: tetroctal

53: trielevenary

54: bisuboptimal

55: pentaseptimal

100: niftimal

## generalizing this for larger bases

the basic roots that exist in this system are:

- binary (base two)
- bi- (times two)

- trinary (base three)
- tri- (times three)

- quaternary (base four)
- tetra- (times four)

- quinary (base five)
- penta- (times five)

- seximal (base six)
- hexa- (times six)

- septimal (base seven)
- hepta- (times seven)

- octal (base eight)
- octo- (times eight)

- nonary (base nine)
- enna- (times nine)

- decimal (base ten)
- deca- (times ten)
- when used with a prefix, the suffix form is "-gesimal".

- dozenal (base twelve)
- doza- (times twelve)

- elevenary (base eleven)
- leva- (times eleven)

- baker's dozenal (base dozen one)
- baker- (times dozen one)
- when used with a prefix, the suffix form is just "-ker's dozenal".

- hex (base dozen four / sixteen)
- tesser- (times dozen four)

- suboptimal (base dozen five)
- mal- (times dozen five)

- vigesimal (base thirsy two / twenty)
- icosi- (times thirsy two)

- niftimal (base nif)
- feta- (times nif)

- centesimal (base two nif foursy four / one hundred)
- hecto- (times two nif foursy four)

in order to name any prime base after suboptimal, use the prefix "un-" and then name of the base before it. the multiplicative prefix forms of these numbers are formed with "hen-" and "-sna-", as in "hentrihexasna-" for "times thirsy one".

to name any other base, find the two numbers that are closest together that can be multiplied together to get the base, and use the multiplicative prefix form of the smaller one and then the base form of the larger one. for example, nif foursy (DEC60) is equal to six times ten, and no two other numbers that multiply to equal nif foursy are closer together. the multiplicative prefix form of six is hexa-, and the base form of ten is decimal, but it becomes -gesimal when it has a prefix. that means that base nif foursy is called "hexagesimal".

in addition to these roots for simple bases, you might also want to talk about more complicated bases. here's all the roots that are involved in those.

- unary (base one)
- also known as "tally marks"!

- nullary (base zero)
- a base that, if it had any digits, would only be able to write 0⁰, which is undefined.

- nega- (negative base)
- "negadecimal" and "negabinary" are already popular names for bases negative ten and negative two, respectively, so it makes sense to have nega- as a formal part of this system for any negative base.

- vot- / vöt- (reciprocal base)
- this is included to make it possible to talk about fractional bases, such as base one tenth, votdecimal, which is used in the constructed language Vötgil. this can be combined with multiplicative prefixes to form names like "bivottrinary" for base two thirds.

- -nary
- can be added after the English name for any number smaller than six to refer to base that number, as in "phinary" for base phi. this becomes just "-ary" whenever the number name ends with a consonant.

- -imal
- can be added after the English name for any number larger than six to refer to base that number, as in "tauimal" for base tau. this becomes just "-al" whenever the number name is more than one syllable.

when combining roots together, you sometimes end up with sequences of vowels that sound awkward when you try to pronounce the whole thing as one word. the way to deal with this is with two extra rules:

- if a root that ends with <a> or <o> is followed by a root that begins with a vowel, the vowel at the end of the first root is removed.
- if a root that ends with <i> is followed by a root that begins with <i> or <u>, the vowel at the beginning of the second root is removed.

every positive integer base also has a shortened name, which can be used whenever you quickly need to specify what base you're using. the shortened name is preferably the first three letters in the base's name, but if that conflicts with a smaller base's name, you can use just the first three consonants instead. if that also conflicts, then use either as many letters from the base's full name as is necessary to distinguish it from all smaller bases OR as many letters from just the consonants in the base's full name as is necessary, if that results in a shorter name.

## 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 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.

if you want to figure out what base is best for your own personal needs, you can use BASE OFF, a program I made that's available for download on the resources page.