2.2 Mesopotamian cuneiform and sexagesimal system

Ancient Mesopotamians revolutionized math with cuneiform writing on clay tablets. They used wedge-shaped marks to record numbers, problems, and solutions. This system evolved from simple pictographs to complex symbols, preserving mathematical knowledge for centuries.

The sexagesimal (base-60) system was a game-changer. It allowed for easy division and precise calculations, influencing modern timekeeping. Babylonians used two symbols for numbers, creating a place value system that could handle large and small numbers efficiently.

Cuneiform and Clay Tablets

Development of Cuneiform Writing

• Cuneiform emerged as one of the earliest writing systems in Mesopotamia around 3200 BCE
• Consisted of wedge-shaped marks impressed into soft clay using a reed stylus
• Evolved from pictographs to more abstract symbols representing syllables and concepts
• Used for recording various types of information (administrative records, literary texts, mathematical calculations)
• Adapted by multiple civilizations and languages over time (Sumerian, Akkadian, Babylonian)

Clay Tablets as Mathematical Records

• Clay tablets served as the primary medium for recording mathematical information
• Tablets were shaped into various forms (rectangular, circular, lens-shaped) depending on their purpose
• Mathematical tablets included problem sets, solutions, and tables of calculations
• Preserved mathematical knowledge through their durability, providing insights into ancient mathematical practices
• Notable examples include the Plimpton 322 tablet (contains Pythagorean triples) and the YBC 7289 tablet (shows approximation of square root of 2)

Sumerian and Babylonian Mathematical Contributions

• Sumerian mathematics laid the foundation for later Babylonian developments
• Developed a complex number system based on alternating base-10 and base-6 counting
• Created standardized units of measurement for length, area, and volume
• Babylonian mathematics built upon Sumerian knowledge and made significant advancements
• Introduced more sophisticated algebraic and geometric concepts
• Developed methods for solving quadratic equations and calculating square roots
• Created extensive mathematical tables for practical and theoretical purposes

Sexagesimal System

Fundamentals of Base-60 Numeration

• Sexagesimal system uses base-60 for numerical calculations and representations
• Originated in ancient Sumer and later adopted by Babylonians
• Combines elements of base-10 (decimal) and base-6 systems
• Allows for easy division by many factors (1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30)
• Modern timekeeping and angular measurement still use sexagesimal divisions (60 seconds in a minute, 360 degrees in a circle)

Babylonian Numerals and Place Value

• Babylonian numerals used two basic symbols: a vertical wedge (𒁹) for 1 and a corner wedge (𒂗) for 10
• Employed a place value system similar to modern decimal notation
• Each digit position represented a power of 60 (1, 60, 3600, etc.)
• Lacked a symbol for zero, initially using a blank space to indicate an empty place value
• Later introduced a placeholder symbol (two small wedges) to represent zero in intermediate positions
• Could represent very large and very small numbers efficiently

Fractions and Calculations in Sexagesimal

• Expressed fractions as reciprocals or as sexagesimal place values
• Developed tables of reciprocals to aid in division calculations
• Used sexagesimal fractions for precise astronomical calculations
• Created approximations for irrational numbers (√2 ≈ 1;24,51,10 in sexagesimal, equivalent to 1.414213...)
• Performed complex calculations using sexagesimal multiplication and division tables
• Solved problems involving ratios and proportions using sexagesimal arithmetic

Mathematical Tools and Applications

Computational Aids and Techniques

• Abacus served as an early calculating device in Mesopotamian mathematics
• Consisted of pebbles or beads arranged on lines or in grooves
• Facilitated addition, subtraction, and place value operations
• Developed computational algorithms for multiplication and division
• Created mathematical tables for common calculations (multiplication, reciprocals, squares, square roots)
• Employed geometric methods to solve algebraic problems

Astronomical and Practical Applications

• Applied mathematical knowledge to predict celestial events and create calendars
• Developed methods to calculate planetary positions and lunar phases
• Created the zodiac system, dividing the ecliptic into 12 equal parts
• Used mathematics for practical purposes in agriculture, commerce, and engineering
• Calculated areas and volumes for land measurement and taxation
• Developed techniques for surveying and constructing large-scale buildings and irrigation systems
• Applied mathematical concepts to create musical scales and harmonies