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". The early shekel in particular was one-sixtieth of a mana, though the Greeks later coerced this relationship into the more base-10 compatible ratio of a shekel being one-fiftieth of a mina.Īpart 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. 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 ancient texts this shows up in the fact that sexagesimal is used most uniformly and consistently in mathematical tables of data. The most powerful driver for rigorous, fully self-consistent use of sexagesimal has always been its mathematical advantages for writing and calculating fractions. Įarly Proto-cuneiform (4th millennium BCE) and cuneiform signs for the sexagesimal system (60, 600, 3600, etc.) Their use has also always included (and continues to include) inconsistencies in where and how various bases are to represent numbers even within a single text. 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. However, the Babylonian sexagesimal system was based on six groups of ten, not five groups of 12.Īccording to Otto Neugebauer, the origins of sexagesimal are not as simple, consistent, or singular in time as they are often portrayed. The five fingers would count five sets of 12, or sixty. In this system, a person's other hand would count the number of times that 12 was reached on their first hand. A traditional counting system still in use in many regions of Asia works in this way, and could help to explain the occurrence of numeral systems based on 12 and 60 besides those based on 10, 20 and 5. Using the thumb, and pointing to each of the three finger bones on each finger in turn, it is possible for people to count on their fingers to 12 on a single hand. For example, 10 means the number ten and 60 means the number sixty. In this article, all sexagesimal digits are represented as decimal numbers, except where otherwise noted. 60 is the smallest number that is divisible by every number from 1 to 6 that is, it is the lowest common multiple of 1, 2, 3, 4, 5, and 6. For example, one hour can be divided evenly into sections of 30 minutes, 20 minutes, 15 minutes, 12 minutes, 10 minutes, 6 minutes, 5 minutes, 4 minutes, 3 minutes, 2 minutes, and 1 minute. With so many factors, many fractions involving sexagesimal numbers are simplified. The number 60, a superior highly composite number, has twelve factors, namely 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, and 60, of which 2, 3, and 5 are prime numbers. It originated with the ancient Sumerians in the 3rd millennium BC, was passed down to the ancient Babylonians, and is still used-in a modified form-for measuring time, angles, and geographic coordinates. Sexagesimal, also known as base 60 or sexagenary, is a numeral system with sixty as its base.
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