Editing Sexagesimal (section)

==Origin==

According to [[Otto Neugebauer]], the origins of sexagesimal are not as simple, consistent, or singular in time as they are often portrayed. Throughout their many centuries of use, which continues today for specialized topics such as time, angles, and astronomical coordinate systems, sexagesimal notations have always contained a strong undercurrent of decimal notation, such as in how sexagesimal digits are written. Their use has also always included (and continues to include) inconsistencies in where and how various bases are used to represent numbers even within a single text.<ref name=Neugebauer>{{citation|title=The Exact Sciences In Antiquity|last=Neugebauer|first=O.|journal=Acta Historica Scientiarum Naturalium et Medicinalium|volume=9|publisher=Dover|year=1969|isbn=0-486-22332-9|pages=17–19|pmid=14884919}}</ref>

[[File:Proto-cuneiform sexagesimal type Sa.svg|thumb|upright=1.2|Early [[Proto-cuneiform]] (4th millennium BCE) and [[cuneiform]] signs for the sexa&shy;gesimal system (60, 600, 3600, etc.)]]
The most powerful driver for rigorous, fully self-consistent use of sexagesimal has always been its mathematical advantages for writing and calculating fractions. In ancient texts this shows up in the fact that sexagesimal is used most uniformly and consistently in mathematical tables of data.<ref name=Neugebauer/> Another practical factor that helped expand the use of sexagesimal in the past, even if less consistently than in mathematical tables, was its decided advantages to merchants and buyers for making everyday financial transactions easier when they involved bargaining for and dividing up larger quantities of goods. In the late 3rd millennium BC, Sumerian/Akkadian units of weight included the ''kakkaru'' ([[Talent (measurement)|talent]], approximately 30 kg) divided into 60 ''manû'' ([[Mina (unit)|mina]]), which was further subdivided into 60 ''šiqlu'' ([[shekel]]); the descendants of these units persisted for millennia, though the Greeks later coerced this relationship into the more base-10–compatible ratio of a ''shekel'' being one 50th of a ''mina''.

Apart from mathematical tables, the inconsistencies in how numbers were represented within most texts extended all the way down to the most basic [[cuneiform]] symbols used to represent numeric quantities.<ref name=Neugebauer/> For example, the cuneiform symbol for 1 was an ellipse made by applying the rounded end of the stylus at an angle to the clay, while the sexagesimal symbol for 60 was a larger oval or "big 1". But within the same texts in which these symbols were used, the number 10 was represented as a circle made by applying the round end of the style perpendicular to the clay, and a larger circle or "big 10" was used to represent 100. Such multi-base numeric quantity symbols could be mixed with each other and with abbreviations, even within a single number. The details and even the magnitudes implied (since [[0#History|zero was not used consistently]]) were idiomatic to the particular time periods, cultures, and quantities or concepts being represented. In modern times there is the recent innovation of adding decimal fractions to sexagesimal astronomical coordinates.<ref name=Neugebauer/>