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 the largest sexagesimal digit is "59". 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. This makes for a confusing system, the details of which we will skip.Sexagesimal, also known as base 60 or sexagenary, is a numeral system with sixty as its base. The days of the year were thought to be gods, so the formal symbols for the days were decorated heads, like the sample to the left Since the basic calendar was based on 360 days, the priestly numeral system used a mixed base system employing multiples of 20 and 360. For the priests, the number system was governed by ritual. Not only did these two systems use different symbols, they also used different base systems. There were two numeral systems developed by the Mayans-one for the common people and one for the priests. Another important source of information on the Mayans is the writings of Father Diego de Landa, who went to Mexico as a missionary in 1549. In fact, much of what we know about this culture comes from their calendar records and astronomy data. The calendar, and calculations related to it, were thus very important to the ritual life of the priestly class, and hence the Mayan people. This class of priests developed a philosophy with time as divine and eternal. The Mayans had a sophisticated ritual system that was overseen by a priestly class. The Yucatan Peninsula (see figure 16 ) in Mexico was the scene for the development of one of the most advanced civilizations of the ancient world. The Mayan civilization is generally dated from 1500 BCE to 1700 CE. In this chapter, we wrap up with a specific example of a civilization that actually used a base system other than 10. The Babylonians used a base-sixty (sexigesimal) system. For example, the Natives of Queensland used a base-two system, counting as follows: “one, two, two and one, two two’s, much.” Some Modern South American Tribes have a base-five system counting in this way: “one, two, three, four, hand, hand and one, hand and two,” and so on. However, other civilizations have had a variety of bases other than ten.
Our own base-ten system probably arose from the fact that we have 10 fingers (including thumbs) on two hands. Use two different methods for converting numbers between basesĪs you might imagine, the development of a base system is an important step in making the counting process more efficient.Identify bases that have been used in number systems historically.Become familiar with the history of positional number systems.
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