How do we name number bases?

Everyone is familiar with the names of certain number bases. The base-ten system we all use daily is commonly called decimal. Computer scientists, then programmers, then pretty much anyone else who uses computers as tools know that bases 2, 8, and 16 are called binary, octal, and hexadecimal, respectively. Neugebauer, Georges Ifrah, and Menninger refer to bases 20 and 60 as vigesimal and sexagesimal, respectively. Some are familiar with the term duodecimal referring to base 12, with some fans of that base calling it “dozenal”. Once we move away from this set of bases, we no longer have the luxury of a steadfast lexicon of base names. We might call base 24 “quadrovigesimal” or “tetravigesimal”, for instance.

Linguistic purists pine for “sexadecimal” or maybe “sexadenary” versus “hexadecimal”, but this endeavor is Anglophone, not Latin nor Greek. The Latin and Greek number particles are familiar to most English-speakers to serve as meaningful modules that would make up a system of base names.

  The Lamadrid Base Name System applied on this website
Common Lexicon
r Base Name
2 Binary
8 Octal
10 Decimal
12 Duodecimal
16 Hexadecimal
20 Vigesimal
60 Sexagesimal
Bases 1-9
1 unary
2 binary
3 ternary
4 quaternary
5 quinary
6 senary
7 septenary
8 octal
9 nonary
Unit Particle
1 un-
2 duo-
3 tri-
4 tetra-
5 penta-
6 hexa-
7 hepta-
8 octo-
9 ennea-
Decade Particle
10 dec-
20 viges-
30 triges-
40 quadrages-
50 quinquages-
60 sexages-
70 septuages-
80 octoges-
90 nonages-
Hundred Particle
100 centes-
200 ducentes-
300 trecentes-
400 quadringentes-
500 quingentes-
600 sescentes-
700 septingentes-
800 octingentes-
900 nongentes-

The names used on this website are not actually part of a common lexicon like the words “decimal” and “binary”, because this site discusses uncommon number bases. If we are to have names, and we are naming a system that already features existing instances of names, then we will need to devise a framework that includes these existing names.

The system used here mates a Greek unit name with a Latin decade and hundred name to produce the name of a number base. This system is called the Lamadrid system of radix names. In personal correspondence in 2012, Lamadrid suggested the unit and decade system; I’ve extended the system through the hundreds and thousands in a manner similar to what Lamadrid suggested. This furnishes names for bases we would most often consider, and does not alter the presently highly circulated base names.

For bases 1-9, we use the ending -ary, thus “unary”, “binary”, “ternary”, etc. We make an exception for the word “octal”, but systematically we could use the word “octonary”. Base 10 retains the name “decimal”, and the -imal suffix is used hereafter. Thus the teen bases start “tridecimal”, “tetradecimal”, “pentadecimal”, “hexadecimal”, etc. After base 20 “vigesimal”, we continue the pattern of putting units ahead of decades: “unvigesimal”, “duovigesimal”, etc. We don’t involve the Latin practice of subtracting one or two from the decade (like *“undevigesimal” for base 19). Instead we simply say “enneadecimal”. This is because these are not Latin words; the construct that subtracts units from decades tends to confuse, indeed the . These are English names of number bases.

When we reach base 100, we use the Latin hundreds, thus “centesimal”, then “uncentesimal”, “duocentesimal”, …, “centodecimal” for base 110. The units will always precede the decades, but the subsequent multiples of decimal powers will precede both. So base 111 is “centoundecimal”, 112 = “centoduodecimal”. For bases 1000 and above, we use the particles “milles”, “dumilles”, etc. Thus, base-2520 is “dumillequingentovigesimal”. At a certain point, the names become cumbersome and we resign to write “base-55440”. The system is not perfect but will be applied consistently in the material on this website.

Note that nobody is compelled to use the names of bases presented here. These names are presented since this website concerns number bases, and we will need handles for these bases in our discussions. Certainly Lamadrid and I would encourage the use of these names as the English names of number bases, as they do fit into a framework that yields solutions that dovetail with extant base names.

Visit the basic page describing number base names here.