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You are at the section Base Numbers

Base Numbers

Definition

A number base is the number of digits or combination of digits that a system of counting uses to represent numbers. A base can be any whole number greater than 0. The most commonly used number system is the decimal system, commonly known as base 10. Base 16 is commonly used for coding in computer programs as one hex number easily represents a cluster of four binary digis, or base 2.

A radix, or base, is the number of unique digits including the digit zero that is used to represent a number base system. Examples: 0 and 1 represent the digits for the binary system; 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9 represent the digits for the decimal system.

Base Names of Number Systems

The first name of the number base in bold is a generally accepted name of the base, although other names associated with it may or may not be lexicographically accepted widely.

0 - nullary (it can represent just the value zero. Not a very useful base. There is just nothing. The radix would have no digits!)

1 - unary (it's basically a tally system with just one numeral such as a stick, rock, marble, the digit zero or whatever else that could be representing a digit, otherwise known as the base one system.)

2 - binary / dual (used internally by nearly all computers)

3 - ternary / trinary

4 - quaternary (2 x 2)

5 - quinary / quinternary

6 - senary / heximal / seximal (2 x 3)

7 - septenary / septimal

8 - octal / octonary / octimal (2 x 2 x 2)

9 - nonary / nonal (3 x 3)

From this point on, the two-digit number combinations in the base number names will be repeated every 100 names including the last two digits in the range of 100-109, which are not seen in the range of 0-9 (modified versions of the names of bases 0-9 begin with the range of bases 100-109.)

The decades and units portions of the names of the base names (the last two digits) are colored red. Those two red pairings will be repeated every 100 names

One update: except for unary and unarsimal, I made all of the particles that begin with un into uni so it flows better when you speak it. The alternate spellings are next to them.

10 - decimal / denary / biquinary (most widely used by modern civilizations) (2 x 5)

11 - unidecimal / undecimal / undenary / elevenary

12 - duodecimal / dozenal / duodenary / biseximal (2 x 6)

13 - tridecimal / tredecimal / tridenary / triskadecimal

14 - tetradecimal / quadrodecimal / quadadecimal / quattuordecimal / biseptenary / biseptimal (2 x 7)

15 - pentadecimal / quindecimal / triquinary (3 x 5)

16 - hexadecimal / hex / sedecimal / sexadecimal (widely used for compact notation of binary data) (4 x 4, 2 x 2 x 2 x 2)

17 - heptadecimal / septendecimal / suboptimal

18 - octodecimal / triseximal / binonary (2 x 9, 3 x 6)

19 - enneadecimal / nonadecimal / decennoval / untriseximal

20 - vigesimal (2 x 10, 4 x 5)

21 - univigesimal / unvigesimal / unovigesimal / triseptimal (3 x 7)

22 - duovigesimal / bielevenary (2 x 11)

23 - trivigesimal / triovigesimal / unbielevenary

24 - tetravigesimal / quadrovigesimal / tetraseximal (hours in a day) (4 x 6)

25 - pentavigesimal / pentaquinary (related to base 5) (5 x 5)

26 - hexavigesimal / sexavigesimal

27 - heptavigesimal / septemvigesimal / trinonary (3 x 3 x 3)

28 - octovigesimal / tetraseptimal (4 x 7)

29 - enneavigesimal / novovigesimal / nonavigesimal / untetraseptimal

30 - trigesimal / pentaseximal (5 x 6)

31 - unitrigesimal / untrigesimal / unotrigesimal / unpentaseximal

32 - duotrigesimal / tetroctal (4 x 8)

33 - tritrigesimal / trielevenary (3 x 11)

34 - tetratrigesimal / bisuboptimal (2 x 17)

35 - pentatrigesimal / pentaseptimal (5 x 7)

36 - hexatrigesimal / hexatridecimal / sexatrigesimal / niftimal (use of all letters with one case and digits, related to base 6) (6 x 6)

37 - heptatrigesimal

38 - octotrigesimal

39 - enneatrigesimal

40 - quadragesimal

41 - uniquadragesimal / unquadragesimal

42 - duoquadragesimal

43 - triquadragesimal

44 - tetraquadragesimal

45 - pentaquadragesimal

46 - hexaquadragesimal

47 - heptaquadragesimal

48 - octoquadragesimal

49 - enneaquadragesimal (related to base 7) (7 x 7)

50 - quinquagesimal

51 - uniquinquagesimal / unquinquagesimal

52 - duoquinquagesimal

53 - triquinquagesimal

54 - tetraquinquagesimal

55 - pentaquinquagesimal

56 - hexaquinquagesimal

57 - heptaquinquagesimal

58 - octoquinquagesimal

59 - enneaquinquagesimal

60 - sexagesimal / hexagesimal (seconds in a minute and minutes in an hour)

61 - unisexagesimal / unsexagesimal

62 - duosexagesimal (using both cases of all 26 letters plus the digits 0-9)

63 - trisexagesimal

64 - tetrasexagesimal / quadrosexagesimal (related to base 8)

65 - pentasexagesimal

66 - hexasexagesimal

67 - heptasexagesimal

68 - octosexagesimal

69 - enneasexagesimal

70 - septagesimal

71 - uniseptagesimal / unseptagesimal

72 - duoseptagesimal

73 - triseptagesimal

74 - tetraseptagesimal

75 - pentaseptagesimal

76 - hexaseptagesimal

77 - heptaseptagesimal

78 - octoseptagesimal

79 - enneaseptagesimal

80 - octogesimal

81 - unioctogesimal / unoctogesimal (related to base 9) (9 x 9)

82 - duoctogesimal

83 - trioctogesimal

84 - tetroctogesimal

85 - pentoctogesimal (not pentaoctogesimal)

86 - hexoctogesimal

87 - heptoctogesimal

88 - octooctogesimal

89 - enneoctogesimal

90 - nonagesimal / nonogesimal / novagesimal

91 - uninonagesimal / unnonagesimal

92 - duononagesimal

93 - trinonagesimal

94 - tetranonagesimal

95 - pentanonagesimal

96 - hexanonagesimal

97 - heptanonagesimal

98 - octononagesimal

99 - enneanonagesimal

With bases 100-199, all systems should start with the cente- particle. May be modified as needed for certain words. We begin the first 10 of this hundred by using modified names of the base 1-9 systems in the proposed names beginning with 101

To make it easier for reading the long names, I made the hundreds portion of the base dark red.

The decades and units portions of the long names (the last two digits) are colored red. Those two red pairings will be repeated every 100 names

100 - centesimal / centimal

101 - centeunarsimal (proposed)

102 - centebinarsimal (proposed)

103 - centeternarsimal (proposed)

104 - centequaternarsimal (proposed)

105 - centequinarsimal (proposed)

106 - centesenarsimal (proposed)

107 - centeseptensimal (proposed)

108 - centeoctasimal (proposed)

109 - centenonarsimal (proposed)

Here are some suggested alternatives to the range of 101-109:

101 - centeunisimal (proposed-alternative)

102 - centeduosimal (proposed-alternative)

103 - centetriosimal (proposed-alternative)

104 - centetetrasimal (proposed-alternative)

105 - centepentasimal (proposed-alternative)

106 - centehexasimal (proposed-alternative)

107 - centeheptasimal (proposed-alternative)

108 - centeoctosimal (proposed-alternative)

109 - centeenneasimal (proposed-alternative)

With bases 110-199, we simply add the cente- prefix to each of the 10-19 names.

110 - centedecimal (proposed)

111 - centeunidecimal (proposed)

112 - centeduodecimal (proposed)

113 - centetridecimal (proposed)

114 - centetetradecimal (proposed)

115 - centepentadecimal (proposed)

116 - centehexadecimal (proposed)

117 - centeheptadecimal (proposed)

118 - centeoctodecimal (proposed)

119 - centeenneadecimal (proposed)

120 - centevigesimal / centovigesimal

121 - centeunivigesimal (related to base 11) (11 x 11)

122 - centeduovigesimal (proposed)

123 - centetrivigesimal (proposed)

124 - centetetravigesimal (proposed)

125 - centepentavigesimal (related to base 5) (5 x 5 x 5)

126 - centehexavigesimal (proposed)

127 - centeheptavigesimal (proposed)

128 - centeoctovigesimal (related to base 2) (27)

129 - centeenneavigesimal (proposed)

130 - centetrigesimal (proposed)

131 - centeunitrigesimal (proposed)

132 - centeduotrigesimal (proposed)

133 - centetritrigesimal (proposed)

134 - centetetratrigesimal (proposed)

135 - centepentatrigesimal (proposed)

136 - centehexatrigesimal (proposed)

137 - centeheptatrigesimal (proposed)

138 - centeoctotrigesimal (proposed)

139 - centeenneatrigesimal (proposed)

140 - centequadragesimal (proposed)

141 - centeuniquadragesimal (proposed)

142 - centeduoquadragesimal (proposed)

143 - centetriquadragesimal (proposed)

144 - centetetraquadragesimal / centotetraquadragesimal (related to base 12) (12 x 12)

145 - centepentaquadragesimal (proposed)

146 - centehexaquadragesimal (proposed)

147 - centeheptaquadragesimal (proposed)

148 - centeoctoquadragesimal (proposed)

149 - centeenneaquadragesimal (proposed)

150 - centequinquagesimal (proposed)

151 - centeuniquinquagesimal (proposed)

152 - centeduoquinquagesimal (proposed)

153 - centetriquinquagesimal (proposed)

154 - centetetraquinquagesimal (proposed)

155 - centepentaquinquagesimal (proposed)

156 - centehexaquinquagesimal (proposed)

157 - centeheptaquinquagesimal (proposed)

158 - centeoctoquinquagesimal (proposed)

159 - centeenneaquinquagesimal (proposed)

160 - centesexagesimal (proposed)

161 - centeunisexagesimal (proposed)

162 - centeduosexagesimal (proposed)

163 - centetrisexagesimal (proposed)

164 - centetetrasexagesimal (proposed)

165 - centepentasexagesimal (proposed)

166 - centehexasexagesimal (proposed)

167 - centeheptasexagesimal (proposed)

168 - centeoctosexagesimal (proposed)

169 - centeenneasexagesimal (proposed) (13 x 13)

170 - centeseptagesimal (proposed)

171 - centeuniseptagesimal (proposed)

172 - centeduoseptagesimal (proposed)

173 - centetriseptagesimal (proposed)

174 - centetetraseptagesimal (proposed)

175 - centepentaseptagesimal (proposed)

176 - centehexaseptagesimal (proposed)

177 - centeheptaseptagesimal (proposed)

178 - centeoctoseptagesimal (proposed)

179 - centeenneaseptagesimal (proposed)

180 - centeoctogesimal (proposed)

181 - centeunioctogesimal (proposed)

182 - centeduoctogesimal (proposed)

183 - centetrioctogesimal (proposed)

184 - centetetroctogesimal (proposed)

185 - centepentoctogesimal (proposed)

186 - centehexoctogesimal (proposed)

187 - centeheptoctogesimal (proposed)

188 - centeoctooctogesimal (proposed)

189 - centeenneoctogesimal (proposed)

190 - centenonagesimal (proposed)

191 - centeuninonagesimal (proposed)

192 - centeduononagesimal (proposed)

193 - centetrinonagesimal (proposed)

194 - centetetranonagesimal (proposed)

195 - centepentanonagesimal (proposed)

196 - centehexanonagesimal (proposed) (related to base 14) (14 x 14)

197 - centeheptanonagesimal (proposed)

198 - centeoctononagesimal (proposed)

199 - centeenneanonagesimal (proposed)

For century particles 200s to 900s, just replace the cente with those below.

We begin the decentesimal century group of bases here.

200 - ducentesimal / duocentesimal / ducentimal / duocentimal

(...repeat naming pattern...)

225 - ducentepentavigesimal (proposed) (related to base 15) (15 x 15)

(...repeat naming pattern...)

240 - ducentequadragesimal (proposed) / duocentequadragesimal / duocentoquadragesimal

(...repeat naming pattern...)

256 - ducentehexaquinquagesimal / duocentehexaquinquagesimal (related to base 16) (16 x 16, 28)

(...repeat naming pattern...)

300 - trecentesimal / tercentesimal / tricentesimal / tercentimal

(...repeat naming pattern...)

360 - trecentesexagesimal / trecentosexagesimal(degrees for an angle)

(...repeat naming pattern...)

400 - quadringentesimal / quattrocentimal / quadricentesimal

(...repeat naming pattern...)

500 - quingentesimal / quincentimal / pentacentesimal

(...repeat naming pattern...)

600 - sescentesimal / hexacentimal / hexacentesimal

(...repeat naming pattern...)

700 - septingentesimal / heptacentimal / heptacentesimal

(...repeat naming pattern...)

800 - octingentesimal / octacentimal / octocentimal / octacentesimal / octocentesimal

(...repeat naming pattern...)

900 - noningentesimal / novacentimal / novacentesimal

(...repeat naming pattern...)

999 - noningenteenneanonagesimal (proposed)

Now we get to the thousand bases. Note that since we're at the millenium mark, there is nothing in the century column, so we just leave out the century part of the name and just use the thousand position and the decade/unit pairing as the base of naming. We color the thousand prefix green for easier reading.

Since it's one thousand, I used just the mille particle. I will use it only when the hundred-thousands and ten-thousands places are both zero. Alternatively, you can use the unimille particle to represent the same number.

When we get to the millions where there are all zeros in the thousands, ten-thousands and hundred-thousands positions of the numbers, I'll just leave out the mille particle. No sense of taking up more space and making words harder to pronounce.

1,000 - millesimal (10 x 10 x 10)

This one is difficult. In case hundreds of years from now somebody has a use for base 1001, here's my solution to the naming of that base. Note that since the hundreds (century) column is zero, we skip the particle related to its position and name it like this:

1,001 - milleunarsimal (proposed)

(...repeat naming pattern...)

For 1,010 and 1,011, they go like this:

1,010 - milledecimal (proposed)

1,011 - milleunidecimal (proposed)

(...repeat naming pattern...)

We get to the 1100 mark, and this time, we use the century part of the name since there is at least a non-zero digit in the hundreds position.

1,100 - millecentesimal (proposed)

The next one after that looks like this:

1,101 - millecenteunarsimal (proposed)

(...repeat naming pattern...)

We're at 1110 where there is a non-zero number in the base. It looks like this, using the red pairing seen in the base numbers ending in "10"

1,110 - millecentedecimal (proposed)

And for the base after that, this one is easy to figure out.

1,111 - millecenteunidecimal (proposed)

(...repeat naming pattern...)

1,200 - milleducentesimal (proposed)

(...repeat naming pattern...)

2,000 - bimillesimal (proposed)

(...repeat naming pattern...)

3,000 - trimillesimal (proposed)

(...repeat naming pattern...)

4,000 - quaternmillesimal (proposed)

(...repeat naming pattern...)

5,000 - quinarmillesimal (proposed)

(...repeat naming pattern...)

6,000 - senarmillesimal (proposed)

(...repeat naming pattern...)

7,000 - septenmillesimal (proposed)

(...repeat naming pattern...)

8,000 - octamillesimal (proposed)

(...repeat naming pattern...)

9,000 - nonamillesimal (proposed)

(...repeat naming pattern...)

10,000 - decimillesimal (10 x 10 x 10 x 10) (100 x 100)

(...repeat naming pattern...)

11,000 - unidecimillesimal (proposed)

(...repeat naming pattern...)

20,000 - vigesmillesimal (proposed)

(...repeat naming pattern...)

65,000 - pentasexagemillesimal (proposed)

(...repeat naming pattern...)

65,536 - pentasexagemillequingentehexatrigesimal (proposed) (216, 2562, used for 16-bit words or two 8-bit bytes)

(...repeat naming pattern...)

REVISION: Beginning with the ten-thousands, I will use the hundred particles in darker green. In this case where the thousands and ten-thousands are both zeroes, and the fact that there is a non-zero in the hundred-thousands column, I will use the mille particle after the hundred particle to describe the thousands portion of the name of the base number. (ex: 100, 200, 300...900)

In the case where there's a non-zero in the hundred-thouesands, a zero in the ten-thousands, and a one in the thousands, I'll use the unimille particle. (ex: 101, 201, 301,...901.)

So my rule for using the mille- particle grouping goes like this: I'll use it only if at least one of the three positions, thousands, ten-thousands and hundred-thousands, are non-zero. Else, it's left out of the name altogether.

Furthermore, for describing the nonzero hundred-thousands positions of the base number name, you use at least one of the hundred particles along with the other particles for thousands and ten-thousands.

So with that, here goes...

100,000 - centesmillesimal (proposed) (10 x 10 x 10 x 10 x 10 x 10)

(...repeat naming pattern...)

101,000 - centesunimillesimal (proposed)

(...repeat naming pattern...)

110,000 - centesdecimillesimal (proposed)

(...repeat naming pattern...)

111,000 - centesunidecimillesimal (proposed)

(...repeat naming pattern...)

999,999 - noningentesenneanonagesmillenoningenteenneanonagesimal (proposed)

106 - micresimal

(...repeat naming pattern if you really care...)

109 - nanesimal

1012 - pikesimal

1015 - femtesimal

1018 - attesimal

1021 - zeptesimal

1024 - yoctesimal

Personally, I don't yet see why anyone would need bases in the milions and beyond but we never know what use it would provide to mathematicians and scientists hundreds and thousands of years from since I wrote this in the year 2020.

Particles

Bases 1-9

1 - unary
2 - binary
3 - ternary
4 - quaternary
5 - quinary
6 - senary
7 - septenary
8 - octal
9 - nonary

Unit

1 - un-
2 - duo- or bi-
3 - tri-
4 - tetra-
5 - penta-
6 - hexa-
7 - hepta-
8 - octo-
9 - ennea-

Decade

10 - dec-
20 - viges-
30 - triges-
40 - quadrages-
50 - quinquages-
60 - sexages-
70 - septuages-
80 - octoges-
90 - nonages-

Hundred

100 - centes-
200 - ducentes-
300 - trecentes-
400 - quadringentes-
500 - quingentes-
600 - sescentes-
700 - septingentes-
800 - octingentes-
900 - nongentes-

Sources

Seximal: Names of Other Bases

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