Sexagesimal system

The sexagesimal system is a system of positional numeral sets that uses the number 60 as the base. It originated in ancient Mesopotamia, in the Sumerian civilization. The sexagesimal system is used to measure times (hours, minutes and seconds) and angles (degrees) mainly.

The sexagesimal system was used only formally in numerical calculations, since the names of the numbers in Sumerian, Akkadian and other numeral languages did not follow a proper sexagesimal system, but rather like the numerals of most languages of the world had names based on the decimal or vigesimal system.^{[citation needed]}

Introduction

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 it easier to calculate with fractions. Notice that 60 is the smallest number that is divisible by 1, 2, 3, 4, 5, and 6.

The use of the number sixty as a basis for the measurement of angles, coordinates and measures of time is linked to old astronomy and trigonometry. It was common to measure the angle of elevation of a star and trigonometry uses right triangles. In ancient times, what we now call positive integers—zero excluded—were the only "bona fide" numbers. The current rational numbers were considered ratios between integers, since the prevailing philosophy resorted to proportion and a fraction, in short, was a proportional comparison between two segments of integer values. All this linked to what we call least common multiple. All right triangles with integer sides have the property that the product of their three sides is always a multiple of sixty. If one of the legs is prime, the other is at least a multiple of twelve and it 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 a prime hypotenuse, but the compound leg is a multiple of four: (3, 4, 5), although the product is sixty. Other examples of triangles with a prime leg and hypotenuse are: (11, 60, 61) and (71, 2520, 2521).

There are vestiges of the sexagesimal system 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 a second are measured with the decimal system.

This method of calculating units is also used in typography, where the base 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 be at the origin of this way of calculating the size of printing types and other typographical composition elements such as columns, colonels or streets, the reason for its permanence is due to comfort with the that divisions in half, fourth and third can be done mentally with integer typographical points without resorting to decimals.

To express the numbers in the sexagesimal system, an agreement is followed that consists of using the numbers of the decimal system (from 0 to 59), separated two by two by commas. To indicate the decimal point, a sexagesimal semicolon would be used. For example, the number 1;07.30 corresponds to 1 + 07/60 + 30/60² = 1.125 in decimal. During the Umayyad Caliphate, the sexagesimal system was used by the Arabs both for counting time and for geometry and trigonometry that had evolved from 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 all the 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 cemented the use of the sexagesimal system as we know it today. And no matter how curious it may be, it still works perfectly.

Addition and subtraction of the sexagesimal system in mathematics

The sexagesimal system is a numbering system in which each unit is divided into 60 units of lower order, that is, it is its numbering system in base 60. It is currently applied to the measurement of time and to that of the width of the angles.

 1 h 60 min 60 s
1 60′ 60′′′′

Operations in the sexagesimal system

Sum

1st step: Hours are placed below hours (or degrees below degrees), minutes below minutes, and seconds below seconds; and they add up.

2nd step: If the seconds add up to more than 60, divide that number by 60; the rest will be the seconds and the quotient will be added to the minutes.

3rd step: The same is done for the minutes.

Subtraction

1st step: Hours are placed below hours (or degrees below degrees), minutes below minutes, and seconds below seconds.

2nd step: The seconds are subtracted. If this is not possible, we convert one minute of the minuend into 60 seconds and add it to the seconds of the minuend. Next we subtract the seconds.

3rd step: We do the same with the minutes.

Origin

As in the case of the decimal system, the origin goes back to a way of counting using the fingers of the hands. In ancient times, the inhabitants of the so-called Fertile Crescent counted by pointing with the thumb of their right hand, if they 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 numbers, each time they perform this operation, a finger of the free hand —the left hand— is raised until completing 60 units (12 x 5 = 60), so this number was considered a "round figure", becoming a common reference in transactions and measurements. The number counted on the right hand had a similar fate, 12, and some multiples such as 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 way of counting on the fingers (up to 12 and then up to 60) is still used by some Middle Easterners today.

The mathematician Sergey Fomin reviewed two other old explanations about the origin of the sexagesimal system, although he considered them not very credible and poorly argued. The first hypothesis was that the sexagesimal system arose from the political compromise between two tribes that confederated, combining their respective number systems (senary 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 numeration.

Astronomical use at the origin

In Babylon the circumference was divided into 360 equal arcs. Each of these parts received the name of grade and each of them was assigned a god. Twelve reappears in the zodiac, since that number of signs or "houses" has the system, covering an arc of 30 degrees and a set of the same number of gods. The religious system was very strict and dogmatic and required that the angles be constructed by means of a non-graduated rule with a single edge and an indefinite length, plus a compass with a fixed opening, while drawing a circle, but which closed when raised, with which it was not possible to use it to transport segments or measures (See: Ruler and compass). This geometric construction system was considered of divine origin and use; According to these beliefs, the universe had been created with this method of geometric construction.

What constitutes a mystery is knowing how this religious geometric system was fully developed, since Gauss' General Cyclotomy Theorem of 1801 demonstrates that it is impossible to construct for many angles an integer number of degrees: whatever is not multiple of 3°.

It is an open question whether the priests settled for approximations or non-sacred methods, such as making marks on the ruler. This would have destroyed an entire philosophy and if it had happened they would have had to be carefully hidden from non-clergy learned men.

Every 315 years the Sun and the Moon return to the same place in the sky, with an error of 7 or 8 minutes of arc. This is slightly more than twice the minimum separation that can be detected by the human eye without magnifying instruments. The small error should have a religious meaning unknown to our civilization, since the degree was occupied by a god and was divided into 60 minutes. But both the Sun and the Moon fell into the same "divine" domain. Four periods span 1260 years, which is equal to 3 + ½ times 360 years. Taking the set to its lowest integer expression we have the astronomical period of 2520 years, which is part of a right triangle with one leg and the hypotenuse prime: (71, 2520, 2521). These numbers, 1260 and 2520, are multiples of 12, 40 and 60 and can occupy any leg and hypotenuse of right triangles similar to the Egyptian sacred triangle (3, 4, 5) and, in general, of any right triangle with full sides, especially those with a prime leg.

Examples

• The length of the diagonal of a square side 1 is equal to the square root of 2. A very good approximation of this value is:

1,414212... = 30547/21600 = 1;24,51,10 (sexagesimal = 1 + 24/60 + 51/602 + 10/603),

a constant that was already used by the Babylonian mathematicians of the Old Babylonian Period (1900 a. C. – 1650 a. C.), and that is collected on the YBC 7289 clay tablet. A more accurate value of 2{displaystyle {sqrt {2}}} It's 1;24,51,10,07,46,06,04,44,...

• The duration of the tropical year in neo-babilonic astronomy (See: Hiparc):

365,24579... = 06,05;14,44,51 (365 + 14/60 + 44/602 + 51/603).

• The value of π used Ptolomeo

3,141666... = 377/120 = 3;08,30 (3 + 8/60 + 30/602).

Fractions

The number 60 has as prime divisors the first three prime numbers, that is, 2, 3 and 5. Any fraction whose denominator is of the form 2^a · 3^b 5^c will have an exact sexagesimal development, with a, b and c are integers equal to or greater than 0.

However, in cases where the development is not exact, the period will generally be long. Since both the number before and after 60 are prime (59 and 61, respectively), for the period to only be one or two digits, the denominator has to be 59, 61, the product of the two (3599) or whatever of the above by a number of the form 2^a 3^b 5 ^c. In any other case, the period will be longer.

1/2 = 0;30

1/3 = 0;20

1/4 = 0;15

1/5 = 0;12

1/6 = 0;10

1/7 = 0.08.34.17

1/8 = 0.07.30

1/9 = 0.06.40

1/10 = 0.06

1/11 = 0.05.27.16.21.49

1/12 = 0.05

1/13 = 0.04.36.55.23

1/14 = 0.04.17.08.34

1/15 = 0.04

1/16 = 0.03.45

1/17 = 0.03.31.45.52.56.28.14.07

1/18 = 0.03.20

1/19 = 0.03.09.28.25.15.47.22.06.18.56.50.31.34.44.12.37.53.41

1/20 = 0.03

1/24 = 0.02.30

1/25 = 0.02.24

1/27 = 0.02.13.20

1/30 = 0.02

1/32 = 0.01.52.30

1/36 = 0.01.40

1/40 = 0;01.30

1/45 = 0;01.20

1/48 = 0;01.15

1/50 = 0;01.12

1/54 = 0.01.06.40

1/59 = 0;01

1/1,00 = 0;01 (1/60 in decimal).

Time Tables

The multiplication tables in base-sixties are relatively difficult to memorize, since it involves memorizing 59×60/2 = 1770 different products. For comparison, in the decimal system you have to memorize 9×10/2 = 45 products. Example: 8×5 = 40 = 5×8

 10×9 = 90 = 9×10

Finding Prime Numbers

Prime numbers can end in the following digits: 01, 07, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 49, 53, or 59.

In other words, if we have a natural number whose last digit, in base-sixty, is a prime number (other than 02, 03 or 05), 01 or 49, then that number can be prime — and could be checked using some method of primality. If it doesn't end in one of those figures, then it has to be composited.

Bibliography

• Dantzig, Tobias (1971). Number. Language of Science. Buenos Aires: Editorial Hobbs Sudamericana. (Translated from the fourth edition in English). The book makes a history of the evolution of the concept of number for the non-mathematic cult reader. He had a praiseworthy comment from Albert Einstein and was a reference to educational reform in Argentina. It deals with numbering systems and the complete development of the concept of number, not deliberately explaining the concepts that are not assimilable by a non-mathematic reader such as ideals. Particularly in it you will find an incomplete demonstration of the theorem that says that the product of the sides of a rectangle triangle is always a multiple of sixty, located on page 295 of the above reference; this due to an omission as to the prime character of a cateto. It was edited at least four times in English in the United States of America and two in Spanish in Argentina. The reference corresponds to the second edition in Argentina and the first was made by another editorial in 1947.

Notes