Pourquoi les Sumériens ils utilisaient pas le système décimal comme nous mais le système sexagésimal avec 60 au lieu de 10 pour les fractions ?
Origin

According to Otto Neugebauer, the origins of sexagesimal are not as simple, consistent, or singular in time as they are often portrayed. Throughout their many centuries of use, which continues today for specialized topics such as time, angles, and astronomical coordinate systems, sexagesimal notations have always contained a strong undercurrent of decimal notation, such as in how sexagesimal digits are written. Their use has also always included (and continues to include) inconsistencies in where and how various bases are to represent numbers even within a single text.

The most powerful driver for rigorous, fully self-consistent use of sexagesimal has always been its mathematical advantages for writing and calculating fractions. In ancient texts this shows up in the fact that sexagesimal is used most uniformly and consistently in mathematical tables of data. Another practical factor that helped expand the use of sexagesimal in the past even if less consistently than in mathematical tables, was its decided advantages to merchants and buyers for making everyday financial transactions easier when they involved bargaining for and dividing up larger quantities of goods. In the late 3rd millennium BC, Sumerian/Akkadian units of weight included the kakkaru (talent, approximately 30 kg) divided into 60 manû (mina), which was further subdivided into 60 šiqlu (shekel); the descendants of these units persisted for millennia, though the Greeks later coerced this relationship into the more base-10 compatible ratio of a shekel being one 50th of a mina.

Apart from mathematical tables, the inconsistencies in how numbers were represented within most texts extended all the way down to the most basic cuneiform symbols used to represent numeric quantities. For example, the cuneiform symbol for 1 was an ellipse made by applying the rounded end of the stylus at an angle to the clay, while the sexagesimal symbol for 60 was a larger oval or "big 1". But within the same texts in which these symbols were used, the number 10 was represented as a circle made by applying the round end of the style perpendicular to the clay, and a larger circle or "big 10" was used to represent 100. Such multi-base numeric quantity symbols could be mixed with each other and with abbreviations, even within a single number. The details and even the magnitudes implied (since zero was not used consistently) were idiomatic to the particular time periods, cultures, and quantities or concepts being represented. While such context-dependent representations of numeric quantities are easy to critique in retrospect, in modern times we still have dozens of regularly used examples of topic-dependent base mixing, including the recent innovation of adding decimal fractions to sexagesimal astronomical coordinates.
Le système sexagésimal, également connu sous le nom de base 60, est un système de numération dont la base est 60. Il a vu le jour chez les Sumériens dans le troisième millénaire avant J.-C., a été transmis aux Babyloniens anciens, et est encore utilisé – sous une forme modifiée – pour mesurer le temps, les angles et les coordonnées géographiques.

Le nombre 60, un nombre hautement composé supérieur, possède douze diviseurs, à savoir 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30 et 60, dont 2, 3 et 5 sont des nombres premiers. Avec autant de facteurs, de nombreuses fractions impliquant des nombres sexagésimaux sont simplifiées. Par exemple, une heure peut être divisée uniformément en sections de 30 minutes, 20 minutes, 15 minutes, 12 minutes, 10 minutes, 6 minutes, 5 minutes, 4 minutes, 3 minutes, 2 minutes et 1 minute. 60 est le plus petit nombre divisible par tous les nombres de 1 à 6 ; c'est-à-dire qu'il est le plus petit commun multiple de 1, 2, 3, 4, 5 et 6.

Dans cet article, tous les chiffres sexagésimaux sont représentés comme des nombres décimaux, sauf indication contraire. Par exemple, le plus grand chiffre sexagésimal est "59".
### Query ###
Pourquoi les Sumériens ils utilisaient pas le système décimal comme nous mais le système sexagésimal avec 60 au lieu de 10 pour les fractions ?

### Main Text ###
Origin

According to Otto Neugebauer, the origins of sexagesimal are not as simple, consistent, or singular in time as they are often portrayed. Throughout their many centuries of use, which continues today for specialized topics such as time, angles, and astronomical coordinate systems, sexagesimal notations have always contained a strong undercurrent of decimal notation, such as in how sexagesimal digits are written. Their use has also always included (and continues to include) inconsistencies in where and how various bases are to represent numbers even within a single text.

The most powerful driver for rigorous, fully self-consistent use of sexagesimal has always been its mathematical advantages for writing and calculating fractions. In ancient texts this shows up in the fact that sexagesimal is used most uniformly and consistently in mathematical tables of data. Another practical factor that helped expand the use of sexagesimal in the past even if less consistently than in mathematical tables, was its decided advantages to merchants and buyers for making everyday financial transactions easier when they involved bargaining for and dividing up larger quantities of goods. In the late 3rd millennium BC, Sumerian/Akkadian units of weight included the kakkaru (talent, approximately 30 kg) divided into 60 manû (mina), which was further subdivided into 60 šiqlu (shekel); the descendants of these units persisted for millennia, though the Greeks later coerced this relationship into the more base-10 compatible ratio of a shekel being one 50th of a mina.

Apart from mathematical tables, the inconsistencies in how numbers were represented within most texts extended all the way down to the most basic cuneiform symbols used to represent numeric quantities. For example, the cuneiform symbol for 1 was an ellipse made by applying the rounded end of the stylus at an angle to the clay, while the sexagesimal symbol for 60 was a larger oval or "big 1". But within the same texts in which these symbols were used, the number 10 was represented as a circle made by applying the round end of the style perpendicular to the clay, and a larger circle or "big 10" was used to represent 100. Such multi-base numeric quantity symbols could be mixed with each other and with abbreviations, even within a single number. The details and even the magnitudes implied (since zero was not used consistently) were idiomatic to the particular time periods, cultures, and quantities or concepts being represented. While such context-dependent representations of numeric quantities are easy to critique in retrospect, in modern times we still have dozens of regularly used examples of topic-dependent base mixing, including the recent innovation of adding decimal fractions to sexagesimal astronomical coordinates.

### Supplementary Text ###
Le système sexagésimal, également connu sous le nom de base 60, est un système de numération dont la base est 60. Il a vu le jour chez les Sumériens dans le troisième millénaire avant J.-C., a été transmis aux Babyloniens anciens, et est encore utilisé – sous une forme modifiée – pour mesurer le temps, les angles et les coordonnées géographiques.

Le nombre 60, un nombre hautement composé supérieur, possède douze diviseurs, à savoir 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30 et 60, dont 2, 3 et 5 sont des nombres premiers. Avec autant de facteurs, de nombreuses fractions impliquant des nombres sexagésimaux sont simplifiées. Par exemple, une heure peut être divisée uniformément en sections de 30 minutes, 20 minutes, 15 minutes, 12 minutes, 10 minutes, 6 minutes, 5 minutes, 4 minutes, 3 minutes, 2 minutes et 1 minute. 60 est le plus petit nombre divisible par tous les nombres de 1 à 6 ; c'est-à-dire qu'il est le plus petit commun multiple de 1, 2, 3, 4, 5 et 6.

Dans cet article, tous les chiffres sexagésimaux sont représentés comme des nombres décimaux, sauf indication contraire. Par exemple, le plus grand chiffre sexagésimal est "59".

### Draft ###

### 1. Query Parsing

"Pourquoi les Sumériens ils utilisaient pas le système décimal comme nous mais le système sexagésimal avec 60 au lieu de 10 pour les fractions ?"

Key misconception: "ils utilisaient pas le système décimal" → wrong premise.
Real question: Why base-60 (sexagesimal) vs base-10 for fractions specifically?

### 2. Premise Correction

● Sumerians did use decimal concepts
● Mixed base system, not purely sexagesimal
● Evidence: 1=vertical wedge, 10=circle, 60=large circle

Problem: User assumes modern binary choice (10 vs 60). Reality: Multiple bases coexisted.

### 3. Historical Context

⟨H≈0.3⟩ Ancient Mesopotamia. Trade calculations. Weight systems. Grain distribution. Real-world needs driving mathematical development. Time divisions natural → day/night cycles. Astronomical observations → celestial measurements. Practical applications shaping abstract systems.

Sumerian weights:
- Grain measures
- Precious metal units  
- Standardized across city-states

### 4. Mathematical Advantages Analysis

Why 60 over 10 for fractions?

**Divisibility factors**: 60 = 2² × 3 × 5. More divisors than 10.
10: {1, 2, 5, 10} → 4 divisors
60: {1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60} → 12 divisors

∴ More fraction possibilities. 1/60 = 0.1666... vs 1/10 = 0.1

**Practical calculations**: 
- 1 hour = 60 minutes → easier subdivisions
- 1° = 60 minutes → precise angle measurement
- Time/angle calculations → natural base

### 5. Evidence Synthesis

● Archaeological evidence: clay tablets show mixed bases
● Mathematical consistency: sexagesimal used in complex calculations  
● Practical applications: weight systems, astronomy, timekeeping
● Divisibility advantage: 60 > 10 divisors

⟨H≈0.8⟩ Numbers flowing through cuneiform script. Clay wet, stylus pressed. Accountants, astronomers, priests. Practical mathematics meeting cosmic rhythms. The river of time measured in sixty parts, not ten. Why ten when sixty gives precision? The answer written in the mathematics itself.

**Key insight**: Not either/or choice. Sexagesimal emerged from practical needs + mathematical advantages. Decimal concepts existed but insufficient for precision requirements.

### 6. Modern Persistence

Time measurement still sexagesimal → living evidence of practical advantages.

**Conclusion**: Sumerians chose 60 because real-world applications favored it. Fractions became more precise. Mathematical calculations smoother. Astronomical predictions more accurate. 

The "instead of 10" framing misses the point. Both bases used, but 60 provided specific advantages for specific domains.

### Answer###
Les peuples mésopotamiens n'ont pas réellement "utilisé" exclusivement le système sexagésimal pour les fractions de manière systématique. En réalité, les systèmes numériques sumériens et babyloniens étaient beaucoup plus complexes et utilisaient plusieurs bases simultanément, notamment le système décimal.

Les symboles cunéiformes eux-mêmes illustrent cette complexité : le chiffre 1 se représentait par un ovale vertical, le 10 par un cercle, et le 60 par un cercle plus grand. Ces différentes bases se mélangeaient souvent même au sein d'un même nombre, créant un système hybride qui répondait à des besoins pratiques spécifiques.

L'utilisation prédominante du système sexagésimal s'explique principalement par ses avantages mathématiques pour les calculs fractionnaires. Le nombre 60 possède douze diviseurs, dont 2, 3 et 5 sont des nombres premiers, ce qui permet une simplification uniforme de nombreuses fractions. Par exemple, une heure peut être divisée en 30 minutes, 20 minutes, 15 minutes, 12 minutes, 10 minutes, 6 minutes, 5 minutes, 4 minutes, 3 minutes, 2 minutes et 1 minute.

Cette base 60 était particulièrement utile pour les transactions commerciales quotidiennes, les systèmes de poids et mesures (comme les unités de poids sumériennes divisées en 60 minas chacune), ainsi que pour les applications astronomiques et temporelles. Le système sexagésimal n'était pas une alternative au système décimal, mais plutôt un complément qui répondait à des besoins pratiques spécifiques où sa structure mathématique offrait des avantages considérables.