ABSTRACT
For any finite set of integers except for the set consisting of just 0, the set is not closed under any ordinary addition, subtraction, multiplication, and division. People in ancient civilizations devised clock arithmetic on a finite set of integers, which is closed under addition, subtraction, and multiplication. It is an arithmetic system for the set of integers {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}, where numbers wrap around after they reach 12. This kind of arithmetic is called modular arithmetic. The introduction of the first clock began with ancient astronomers noticing the phenomenon of the rising and setting of the sun. The clock arithmetic became known about 5000 years ago when Middle East and North Africa civilizations made the earliest clock to enhance their calendars. Historically, units of time in many civilizations are duodecimal (a positional notation number system of base-12). There are 12 months in a year, and the Babylonians had 12 hours in a day (at some point this was changed to 24 hours) [4].
