# Sexagesimal system

The sexagesimal system (also hexagesimal system or sixties system ) is a place value system based on base 60 ( Latin sexagesimus 'the sixtieth' ).

It is still used today to indicate angles and geographical longitudes and latitudes . A degree has 60 arc minutes and a minute has 60 seconds . It has also survived in the field of timekeeping . An hour has 60 minutes and a minute has 60 seconds . In the late Middle Ages, some mathematicians further subdivided the seconds into tertians for their calculations . However, this has not caught on.

## origin

The first evidence of a written sexagesimal calculation system, which, however, was still an addition system, goes back to the Sumerian period around 3300 BC. BC back. In the further course of the Babylonian mathematics from about 2000 BC. A sexagesimal place system used. The main sources on mathematics date from 1900 BC. BC to 1600 BC BC, but the oldest table texts are from the neo-Sumerian period. The post-Alexandrian period shows increasing Greek influences under the Seleucids , which entered into a synergy with the Babylonian knowledge in order to later fully export the experiences of the Sumerians, Akkadians, Assyrians and Babylonians to Greece. Arab astronomers used the spelling of the famous Greek astronomer Ptolemy in their star maps and tables , which was based on sexagesimal fractions. Early European mathematicians like Fibonacci also used fractions like this when they couldn't work with whole numbers.

Many historians see a motive for the introduction of a sexagesimal system in astronomy , since the Babylonian years comprised twelve months of 30 days, but there was also an additional 13th leap month about every three years  . Further information can be found in the early counting of the lunar months, which dates back to 35,000 BC. Can be proven (calendar stick). In the Czech Republic , the spoke bone of a young wolf was found from around 30,000 BC. Discovered, which has a series of 55 notches in total, the 9th, 30th and 31st notches are around twice as long from above as the other notches. Because the mean period of the moon phases is 29.53 days, the markers could be related to the moon phases .

Other scientists see the reason for choosing the number 60 as the basis of the computing system to be able to simply express or calculate as many of the parts that occur in practical counting and measuring (trade) as possible. An indication of this is that the 60 with 12 factors is one of the highly compounded numbers (No. 9 in series A002182 in OEIS ).

## One- and two-handed counting with phalanges and fingers

In the usual decimal system (ten system) you count with the ten fingers (two times five) of both hands. In some areas of the world, however, there was counting with the help of the phalanx , which led to the number twelve ( duodecimal ) with one hand , but the number 60 with two hands.

### One-handed counting to 12

Counting is done with the thumb as a pointer and the phalanges of the same hand as the counting object.

• One-handed counting begins by touching the tip, i.e. the upper phalanx, of the little finger of the same hand for the first object.
• For the second object, the middle phalanx of the little finger is touched with the thumb; so you count on with your thumb by limb and finger.
• Three → lower link of the little finger
• Four → upper link of the ring finger
• Five → middle link of the ring finger
• Six → lower link of the ring finger
• Seven → upper phalanx of the middle finger
• Eight → middle link of the middle finger
• Nine → lower link of the middle finger
• Ten → upper part of the index finger
• Eleven → middle link of the index finger
• Twelve → lower link of the index finger

### Two-handed counting up to 60

After the first dozen has been counted using the thumb as a pointer with the three phalanges of the remaining four fingers of the same hand (4 × 3 = 12), the counting capacity of one hand is initially exhausted.

• The other hand is clenched in a fist. In order to remember that a dozen has been counted, one now extends a finger, e.g. B. the thumb.
• Now you continue to count by starting again at one with your first hand . At twelve , the second dozen is full.
• In order to remember that two dozen have been counted, one now extends the next finger of the other hand, e.g. B. after the thumb out the index finger.
• With the five fingers of the first hand you can count five times a dozen, so 5 × 12 = 60.
• Now you can count the next dozen again with the first hand, i.e. count up to 72 with two hands (12 on the first plus 60 on the other hand).

This finger counting system still exists in parts of Turkey , Iraq , India and Indochina .

You can also count up to 12 × 12 = 144 (a large ) or 156 (13 × 12) by using the second hand to count with phalanxes.

When counting a large amount, an aid can be used, such as sticks, stones, lines or the ten fingers of a helper. Five dozen at a time, i.e. 60, are noted with one of the aids. With the ten fingers of a human helper you can count up to 10 × 60 = 600, with the other aids even further.

## Sumerians

Among the Sumerians, the 60 was called gesch .

• 120: gesch-min (60 × 2)
• 180: gesch-esch (60 × 3)
• 240: gesch-limmu (60 × 4)
• 300: gesch-iá (60 × 5)
• 360: gesch-asch (60 × 6)
• 420: gesch-imin (60 × 7)
• 480: shot (60 × 8)
• 540: gesch-ilummu (60 × 9)
• 600: gesch-u (60 × 10)
• Now the Sumerians did not continue to count in steps of 60 ( gesch- steps), but in 600-steps ( gesch-u -steps), namely six times 600, i.e. up to 3600, which was called schàr .
• The 3600 were then increased ten times to schàr-u (3600 × 10) 36,000.
• The 36,000 were counted six times to 216,000 schàr-gal , literally the large 3600 ( i.e. 60 × 60 × 60).
• The 216,000 was counted ten times to 2,160,000 schàr-gal-u (= (60 × 60 × 60) × 10)
• The schàr-gal-u was initially multiplied five times. The sixth multiple 12,960,000, i.e. 60 × 60 × 60 × 60, was given its own name again, namely schàr-gal-shu-nu-tag (the great schàr superordinate unit).

The numbers 10 to 60 have a decimal (30 = uschu = esch-u = 3 × 10), and sometimes even a vigesimal structure (40 = nischmin = nisch-min = 2 × 20).

## The sexagesimal system in Babylonian usage

The Sumerians used before the cuneiform signs for the numbers 1 to 60 each of different size half ellipses and the numbers 10 and 3600 = 60² each different sized circles , with cylindrical pencils were pressed into clay tablets. From these symbols the symbols for 600 = 10 · 60 and 36000 = 10 · 60² were combined accordingly. There was also another system with decimal levels of 1, 10 and 100, as well as a third system in Akkadian time. Until the late Sumerian period, the individual characters changed their shape, but retained their individual character and formed an addition system similar to the Roman numerals . Only with the later Babylonian sexagesimal system was there a real system of place values ​​with only two individual characters: for 1 and for 10. With these the numbers 1 to 59 could be formed additively, which in turn got their actual value like the digits in the decimal system through their position.

### The numerals

Reasons for using the sexagesimal system lie in the effective calculation method and the very limited number of individual number characters from which the numbers were formed. Some examples of the Babylonian cuneiform script:

 1 2 3 4th 5 6th 7th 8th 9 10 11 12 13 14th 15th 16 17th 18th 19th 20th 30th 40 50

Further numerical examples:

= 62, = 122 and = 129.

The numerals consist of only two individual numerals. In this respect, the number of actual numerals was not limited, although reference was only made to two individual numerals, the sizes of which were changed as required. Nevertheless, there are always problems with the reading, because the digits of a number, which mostly resulted from the context, were not clear: e.g. B. could mean 30, 30x60 or 30/60 and so on. Likewise, there was no zero, so that occasionally a digit was missing - but this was very rare - and different numbers were written in the same way. Later, a gap was sometimes left at a missing point, from the 6th century BC onwards. A space with the value zero appeared as an additional number. However, this space was not used for direct calculations and it did not appear as a separate number symbol, so it did not have the meaning of the number zero . The Indians first gave their space as a symbol for the number zero .

Sexagesimal numbers are represented by Arabic numerals by writing a comma between two individual sexagesimal places. The whole sexagesimal places, on the other hand, are separated from the broken ones by a semicolon and if there are missing places or spaces, a “0” is written (this is then an interpretation): B. 30.0 = 30 * 60 and 0; 30 = 30/60.

### The computing technology

As with our decimal system , the place value system allowed the preceding digit to be extended or reduced by 1. The shape of the wedges made the sexagesimal system easier because only the wedges had to be put together. The technical terms used for addition and subtraction were “increase” and “move away” (the mathematical symbols + and - were first introduced by Johannes Widmann in the 15th century AD). A negative difference between two numbers is expressed with "Subtrahend goes beyond". Adding and subtracting works just as it does today in the decimal system.

${\ displaystyle {\ begin {array} {rr} 59 & \\ + \ 11 & \\ + \ 20 & \\\ hline 1 {,} 30 & (= 90), \ end {array}}}$
in notation of the sexagesimal system. The 1 in front of the decimal point indicates the value 1 · 60, to which the number 30 after the decimal point is added.

Example of a subtraction:

${\ displaystyle {\ begin {array} {rl} 4 {,} 40 & (= 280) \\ - \ 1 {,} 40 & (= 100) \\ - \ 1 {,} 50 & (= 110) \\\ hline 1 {,} 10 & (= 70), \ end {array}}}$
in notation of the sexagesimal system. The 4 and 1 in front of the decimal point indicate the values ​​4 · 60 and 1 · 60, to which the numbers 40, 50 and 10 respectively are added after the decimal point.

#### Multiply

The same procedure as in the decimal system was used for multiplication. But while in the decimal system one has to have the multiplication table from 1 · 1 to 9 · 9 in mind, the Babylonians should have been able to memorize the multiplication table from 1 · 1 to 59 · 59. To make things easier, multiplication tables were used, from which the required products could be read: Each line of a multiplication table began with the same head number, e.g. B. 2, followed by the expression “times” and the multiplier, e.g. B. 1, and finally the result, e.g. B. 2. The multipliers went from 1 to 20 and then came 30, 40 and 50.

Because in the sexagesimal system 60 was graded in steps of 10 (see above under numerals) and in general, everyday life decimal numbers were much in use, head numbers such as e.g. B. 1.40 = 100 and 16.40 = 1000 multiplication tables created. Another reason is the interaction with the values ​​from reciprocal tables (see below under division). If other values ​​were required, the numbers were put together.

 1.15 1.2 1.3 1.4 2 2.13.20 2.15 2.24 2.30 3 3.2 3.45 4th 4.30 5 6th 6.40 7th 7.12 7.3 8th 8.2 9 10 12 12.3 15th 16 16.4 18th 20th 22.3 24 25th 30th 36 40 44.26.40 45 48 50

Example of a multiplication:

${\ displaystyle 29 \ cdot 1 {,} 12 = 29 \ cdot 1 {,} 0 + 20 \ cdot 0 {,} 12 + 9 \ cdot 0 {,} 12 = 29 {,} 00 + 4 {,} 00 +1 {,} 48 = 34 {,} 48}$.

#### To divide

The Babylonians divided a number by a number in which they with the reciprocal of multiplied:${\ displaystyle a}$${\ displaystyle n}$${\ displaystyle a}$ ${\ displaystyle n}$

${\ displaystyle a: n = a \ cdot {\ frac {1} {n}}}$.

The reciprocal of a number could be found in a multiplication table with the head number if a power of 60 divided. Because it was there as a result , i.e. H. a power of 60, then the corresponding multiplier was the reciprocal value you were looking for ( and have the same representation in the Babylonian sexagesimal system): ${\ displaystyle n}$${\ displaystyle n}$${\ displaystyle n}$${\ displaystyle m}$${\ displaystyle m}$${\ displaystyle {\ frac {m} {60 ^ {l}}}}$

${\ displaystyle n \ cdot m = 60 ^ {l}}$, so .${\ displaystyle {\ frac {m} {60 ^ {l}}} = {\ frac {1} {n}}}$

The reciprocal values ​​of natural numbers were put together again in reciprocal tables to make things easier . In such tables, for values ​​that had no reciprocal in a multiplication table, "is not" was written instead of the reciprocal. For these irregular numbers, which have prime factors ≥ 7, approximate values were used as for irrational numbers .

The reciprocal table mainly used contains the following pairs of numbers:

n 1 / n n 1 / n n 1 / n n 1 / n n 1 / n n 1 / n n 1 / n n 1 / n n 1 / n n 1 / n
2 30th 3 20th 4th 15th 5 12 6th 10 8th 7.30 9 6.40 10 6th 12 5 15th 4th
16 3.45 18th 3.20 20th 3 24 2.30 25th 2.24 27 2.13.20 30th 2 32 1.52.30 36 1.40 40 1.30
45 1.20 48 1.15 50 1.12 54 1,640 60 1 1.4 56.15 1.12 50 1.15 48 1.20 45 1.21 44.26.40

A lot can be read from a reciprocal table, u. a. or or , but also the other way around etc. ${\ displaystyle {\ frac {1} {3}} = 0; 20}$${\ displaystyle {\ frac {1} {3 {,} 0}} = {\ frac {1} {180}} = 0; 0 {,} 20}$${\ displaystyle 1: 0; 3 = 60: 3 = 20}$${\ displaystyle {\ frac {1} {20}} = 0; 3}$

Examples of divisions:

${\ displaystyle 4: 3 = 4 \ cdot {\ frac {1} {3}} = 4 \ cdot 0; 20 = 1; 20}$.
${\ displaystyle 0; 12: 25 = 0; 12 \ cdot {\ frac {1} {25}} = 0; 12 \ cdot 0; 2.24 = 0; 0.28.48}$.

#### Root calculation

The ancient Greek mathematician and engineer Heron of Alexandria used the method already known in the ancient Babylonian empire in his Metrica for calculating the roots

${\ displaystyle {\ sqrt {a ^ {2} \ pm b}} \ approx a \ pm {\ frac {b} {2a}}}$.

${\ displaystyle a}$was taken from a table of squares. For the (irrational) square root of 2 we get:

${\ displaystyle {\ sqrt {2}} = {\ sqrt {1; 20 ^ {2} +0; 13.20}} \ approx 1; 20 + 0; 5 = 1; 25}$,

d. H.

${\ displaystyle {\ sqrt {2}} = {\ sqrt {\ left ({\ frac {4} {3}} \ right) ^ {2} + {\ frac {2} {9}}}} \ approx {\ frac {4} {3}} + {\ frac {1} {12}} = {\ frac {17} {12}} \ approx 1 {,} 41666667}$.

On a Babylonian clay tablet (Yale Babylonian Collection 7289), however, there is also a better approximation on the diagonal of a square:

${\ displaystyle {\ sqrt {2}} \ approx 1; 24,51,10 \ left (= {\ frac {305470} {216000}} \ approx 1 {,} 41421296 \ right)}$.

Because of

${\ displaystyle 1; 25> {\ sqrt {2}} = {\ frac {2} {\ sqrt {2}}}> {\ frac {2} {1; 25}} \ approx 1; 24.42, 21 (\ approx 1 {,} 41176389)}$,

lies between 1; 25 and 1; 24,42,21 their arithmetic mean

${\ displaystyle (1; 25 + 1; 24,42,21) \ cdot 0; 30 \ approx 1; 24,51,10}$

closer to

${\ displaystyle {\ sqrt {2}} \ approx 1 {,} 41421356}$.

Now the side length of the square on the clay tablet is given as 30 and the length of the diagonals as 42, 25, 35, which can be interpreted as the following calculation:

${\ displaystyle 0; 30 \ cdot 1; 24,51,10 = 0; 42,25,35}$.

The example shows that the Babylonians had algebraic and geometric knowledge (here the “ Pythagorean theorem ” could have been used).

## Further information

A direct relative of the sexagesimal system is the duodecimal system with base 12.

## literature

• Robert Kaplan: The History of Zero. Hardcover: Campus Verlag, Frankfurt am Main 2000, ISBN 3-593-36427-1 . Paperback edition: Piper Verlag, 2003, ISBN 3-492-23918-8 .
• Richard Mankiewicz: Time Travel of Mathematics - From the Origin of Numbers to Chaos Theory. VGS Verlagsgesellschaft, Cologne 2000, ISBN 3-8025-1440-8 .
• Kurt Vogel : Pre-Greek Mathematics. Part II: The Mathematics of the Babylonians. Schroedel, Hanover and Schöningh, Paderborn 1959.