Sexagesimal system. It is a positional numbering system that uses the number 60 as the arithmetic base.
Summary
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- 1 Introduction
- 2 Operations in the sexagesimal system
- 1 Sum
- 2 Subtraction
- 3 Multiplication
- 4 Ratio
- 3 History
- 4 Use of the sexagesimal system
- 5 Sources
Introduction
The Sexagesimal System is a numbering system in which each unit is divided into 60 units of a lower order, that is, it is a base 60 numbering system. It is currently applied to the measurement of time and the extent of the angles .
The number 60 has the advantage of having many divisors such as: (1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30 and 60), which makes calculating with fractions easier . Note that 60 is the smallest number that is divisible by 1, 2, 3, 4, 5, and 6.
The use of the number 60 as a basis for the measurement of angles, coordinates and time measurements is linked to old astronomy and trigonometry . It was common to measure the elevation angle of a star and trigonometry uses right triangles . In antiquity, what we now call positive integers excluding zero were the only bona fide numbers . The current rational numbers were considered ratios between whole numbers, because the philosophyprevailing resorted to the proportion and a fraction, in short, was a proportional comparison between two segments of integer values. All this linked to what we call the least common multiple. All integer-sided right triangles have the property that the product of their three sides is always a multiple of 60. If one of the legsis prime, the other is at least a multiple of twelve and is a multiple of sixty if the hypotenuse is also prime. If there is no prime leg, one leg is divisible by three and the other by four; any of the three sides is a multiple of five. This penultimate statement has the exception of the Egyptian sacred triangle, which has a prime leg and the hypotenuse prime, but the compound leg is a multiple of four: (3, 4, 5), although the product is sixty. Other examples of triangles with prime and hypotenuse primes are: (11, 60, 61) and (71, 2520, 2521).
Remains of the sexagesimal system remain in the measurement of time. There are 24 hours in a day, 60 minutes in an hour, and 60 seconds in a minute. Units less than one second are measured with the decimal system.
This method of calculating units is also used in typography, where the basic units, Cicero or Pica, are divided into twelve points, which in turn are divided into tenths of a point . In addition to the historical reasons that may exist at the origin of this way of calculating the size of the printing press and other elements of typographic composition such as columns, columns or streets , the reason for their permanence is due to the comfort with that divisions in means, quarters and thirds can be made mentally with whole typographic points without resorting to decimals.
During the Umayyad Caliphate, the sexagesimal system was used by the Arabs both for counting time and for the geometry and trigonometry that had evolved from the Babylonian ancestors, passing through ancient Egypt and many other cultures. It was precisely the Arabs who established the use of the sexagesimal system in modern culture, since for almost 500 years they held the full scientific potential without discussion. Just as at the time the Babylonians drew the first lines for the Arabs to use their system years later, they founded the use of the sexagesimal system as we know it today. And curious as it may be, it still works perfectly.
Operations in the sexagesimal system
Sum
- Hours are placed below hours (or degrees below degrees), minutes below minutes, and seconds below seconds; and they add up.
- If the seconds add up to more than 60, this number is divided by 60; the rest will be the seconds and the quotient will be added to the minutes.
- The same is done for the minutes.
Example
Subtraction
- Hours are placed below hours (or degrees below degrees), minutes below minutes, and seconds below seconds.
- The seconds are subtracted. If this is not possible, we convert one minute of the minuend to 60 seconds and add it to the seconds of the minuend. Then we subtract the seconds.
- We do the same with the minutes. And then we subtract the hours.
Example
Multiplication
- We multiply the seconds, minutes, and hours (or degrees) by the number.
- If the seconds exceed 60, this number is divided by 60; the rest will be the seconds and the quotient will be added to the minutes.
- The same is done for the minutes.
Example 1
Example 2
Quotient
- Hours (or degrees) are divided by the number.
- The quotient is the degrees and the rest, multiplying by 60, the minutes.
- These minutes are added to those that we have and the same process is repeated with the minutes.
- These seconds are added to those we have and the seconds are divided.
Example
History
The origin goes back to a way of enumerating using the fingers of the hands. In antiquity they counted pointing with the thumb of the handright, if you were right-handed, each of the 3 phalanges of the remaining fingers of the same hand, starting with the little finger. With this method you can count up to 12. And to continue with higher figures, each time they performed this operation, a finger is lifted from the free hand to the left until completing 60 units (12 x 5 = 60), so this number was considered a round figure, becoming a regular reference in transactions and measurements. Similar luck ran the number counted in the right hand, 12, and some multiples like 24, 180 (12 x 15, or 60 x 3) and 360 (12 x 30, or 60 x 6). For this reason, the sexagesimal system is related in its historical roots to the duodecimal system.
This form of counting on the fingers (up to 12 and then up to 60) is still used today by some inhabitants of the Middle East.
The mathematician Sergei Fomin outlined two other ancient explanations for the origin of the sexagesimal system, although he considered them unbelievable and poorly argued. The first hypothesis was that the sexagesimal system arose from the political compromise between two tribes that were confederated, combining their respective numerical systems (senario and decimal). The second presented the system as a derivative of astronomical observation and not a result of everyday use. In such a way that, since the Mesopotamians established their year in 360 days, they would have concluded the use of a divisor of that number (60) as the base figure of all their numbering.
Use of the sexagesimal system
The sexagesimal system is a base 60 numbering system. Strictly speaking, such a system should assign different names to the digits 1, 2, 3, …, 59, which is clearly impossible. Therefore, notation based on the name of the decimal digits has been used in all sexagesimal systems used throughout history .
In the world everyday persist two applications common to the sexagesimal system:
- The measure of angles in degrees, minutes and seconds (for example 23º15m47s). In the International System of Units, the sexagesimal degree has been removed as a standard measure to be replaced by the radian.
- The subdivision of time: one hour is divided into 60 minutes and one minute into 60 seconds. This time system is combined with the duodecimal system, base 12, which is used to measure the number of hours in the day (in two twelve-hour blocks). Again, these subdivisions have value only in the everyday world; in the scientific field, the second is used as the base unit of time and with a decimal numbering system (tenths of a second, hundredths,
