Editing Positional notation (section)

=== Other bases in human language ===
Base-12 systems ([[duodecimal]] or dozenal) have been popular because multiplication and division are easier than in base-10, with addition and subtraction being just as easy. Twelve is a useful base because it has many [[divisor|factors]]. It is the smallest common multiple of one, two, three, four and six. There is still a special word for "dozen" in English, and by analogy with the word for 10<sup>2</sup>, ''hundred'', commerce developed a word for 12<sup>2</sup>, ''gross''. The standard 12-hour clock and common use of 12 in English units emphasize the utility of the base. In addition, prior to its conversion to decimal, the old British currency [[Pound Sterling]] (GBP) ''partially'' used base-12; there were 12 pence (d) in a shilling (s), 20 shillings in a pound (£), and therefore 240 pence in a pound. Hence the term LSD or, more properly, [[£sd]].

The [[Maya numerals|Maya civilization]] and other civilizations of [[pre-Columbian]] [[Mesoamerica]] used base-20 ([[vigesimal]]), as did several North American tribes (two being in southern California). Evidence of base-20 counting systems is also found in the languages of central and western [[Africa]].

Remnants of a [[Gaulish language|Gaulish]] base-20 system also exist in French, as seen today in the names of the numbers from 60 through 99. For example, sixty-five is ''soixante-cinq'' (literally, "sixty [and] five"), while seventy-five is ''soixante-quinze'' (literally, "sixty [and] fifteen"). Furthermore, for any number between 80 and 99, the "tens-column" number is expressed as a multiple of twenty. For example, eighty-two is ''quatre-vingt-deux'' (literally, four twenty[s] [and] two), while ninety-two is ''quatre-vingt-douze'' (literally, four twenty[s] [and] twelve). In Old French, forty was expressed as two twenties and sixty was three twenties, so that fifty-three was expressed as two twenties [and] thirteen, and so on.

In English the same base-20 counting appears in the use of "[[20 (number)|scores]]".  Although mostly historical, it is occasionally used colloquially.  Verse 10 of Psalm 90 in the King James Version of the Bible starts: "The days of our years are threescore years and ten; and if by reason of strength they be fourscore years, yet is their strength labour and sorrow".  The Gettysburg Address starts: "Four score and seven years ago".

The [[Irish language]] also used base-20 in the past, twenty being ''fichid'', forty ''dhá fhichid'', sixty ''trí fhichid'' and eighty ''ceithre fhichid''. A remnant of this system may be seen in the modern word for 40, ''daoichead''.

The [[Welsh language]] continues to use a [[vigesimal|base-20]] [[Welsh language#Counting system|counting system]], particularly for the age of people, dates and in common phrases. 15 is also important, with 16–19 being "one on 15", "two on 15" etc. 18 is normally "two nines". A decimal system is commonly used.

The [[Inuit languages]] use a [[base-20]] counting system. Students from [[Kaktovik, Alaska]] invented a [[Kaktovik numerals|base-20 numeral system]] in 1994<ref name="kakt">{{cite journal |last=Bartley |first=Wm. Clark |date=January–February 1997 |title=Making the Old Way Count |url=http://www.ankn.uaf.edu/sop/SOPv2i1.pdf |journal=Sharing Our Pathways |volume=2 |issue=1 |pages=12–13 |access-date=27 February 2017 |archive-url=https://web.archive.org/web/20130625225547/http://ankn.uaf.edu/SOP/SOPv2i1.pdf |archive-date=25 June 2013 |url-status=live }}</ref>

[[Danish language#Numerals|Danish numerals]] display a similar [[vigesimal|base-20]] structure.

The [[Māori language]] of New Zealand also has evidence of an underlying base-20 system as seen in the terms ''Te Hokowhitu a Tu'' referring to a war party (literally "the seven 20s of Tu") and ''Tama-hokotahi'', referring to a great warrior ("the one man equal to 20").

[[Binary numeral system|The binary system]] was used in the Egyptian Old Kingdom, 3000&nbsp;BC to 2050&nbsp;BC. It was cursive by rounding off rational numbers smaller than 1 to {{nowrap|1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64}}, with a 1/64 term thrown away (the system was called the [[Eye of Horus#Mathematics|Eye of Horus]]).

A number of [[Australian Aboriginal languages]] employ binary or binary-like counting systems. For example, in [[Kala Lagaw Ya]], the numbers one through six are ''urapon'', ''ukasar'', ''ukasar-urapon'', ''ukasar-ukasar'', ''ukasar-ukasar-urapon'', ''ukasar-ukasar-ukasar''.

North and Central American natives used base-4 ([[Quaternary numeral system|quaternary]]) to represent the four cardinal directions. Mesoamericans tended to add a second base-5 system to create a modified base-20 system.

A base-5 system ([[quinary]]) has been used in many cultures for counting. Plainly it is based on the number of digits on a human hand. It may also be regarded as a sub-base of other bases, such as base-10, base-20, and base-60.

A base-8 system ([[octal]]) was devised by the [[Yuki tribe]] of Northern California, who used the spaces between the fingers to count, corresponding to the digits one through eight.<ref>{{citation|page=38|title=Pi in the sky: counting, thinking, and being|first=John D.|last=Barrow|publisher=Clarendon Press|year=1992|isbn=9780198539568|url-access=registration|url=https://archive.org/details/piinskycounting00barr}}.</ref> There is also linguistic evidence which suggests that the Bronze Age [[Proto-Indo European]]s (from whom most European and Indic languages descend) might have replaced a base-8 system (or a system which could only count up to 8) with a base-10 system. The evidence is that the word for 9, ''newm'', is suggested by some to derive from the word for "new", ''newo-'', suggesting that the number 9 had been recently invented and called the "new number".<ref>(Mallory & Adams 1997) [[Encyclopedia of Indo-European Culture]]</ref>

Many ancient counting systems use five as a primary base, almost surely coming from the number of fingers on a person's hand. Often these systems are supplemented with a secondary base, sometimes ten, sometimes twenty. In some [[African languages]] the word for five is the same as "hand" or "fist" ([[Dyola language]] of [[Guinea-Bissau]], [[Banda languages|Banda language]] of [[Central Africa]]). Counting continues by adding 1, 2, 3, or 4 to combinations of 5, until the secondary base is reached. In the case of twenty, this word often means "man complete". This system is referred to as ''quinquavigesimal''. It is found in many languages of the [[Sudan]] region.

The [[Telefol language]], spoken in [[Papua New Guinea]], is notable for possessing a base-27 numeral system.