History of Large Numbers - Part 1: Notched Bones and Sumerian
The history of large numbers traces the evolution of human understanding and use of numbers over millennia. In this first part of the series, we’ll explore the origins of large numbers through early forms of counting, starting with the ancient practice of marking numbers on notched bones and moving into the development of early numeration systems in Sumer. These foundations laid the groundwork for our modern concepts of large numbers.
1. Notched Bones (44,200 BC - 3400 BC)
The earliest evidence of humans attempting to quantify and track large numbers dates back to the Upper Paleolithic era, around 44,200 BC. Early humans used objects like bones to make notches, which are believed to have served as counting tools.
The Lebombo Bones
One of the earliest known artifacts related to counting is the Lebombo bone, a baboon's fibula (leg bone) discovered in the Lebombo Mountains, which straddle the border between South Africa and Eswatini (formerly Swaziland). The bone is between 43,000 and 44,200 years old, according to 24 radiocarbon datings. This is far older than the Ishango bone with which it is sometimes confused. Other notched bones are 80,000 years old but it is unclear if the notches are merely decorative or if they bear a functional meaning.
(*) Significance of the Notches:
The notches carved into the Lebombo bone are arranged in a way that suggests a deliberate pattern, indicating that it was used for some sort of mathematical purpose. Though the exact meaning of the notches remains unclear, scholars have proposed several theories:
- Lunar Calendar: Some researchers believe the notches represent the phases of the moon, with 29 being a close approximation to the lunar cycle. This would imply that early humans were aware of the cyclical nature of time, which could have helped with tracking seasons, tides, and other natural phenomena essential for survival.
- Counting System: Others suggest that the notches could have been a simple tally system, used to track quantities, whether for counting days, animals, or other resources. The consistent and evenly spaced notches hint at a structured approach to counting, suggesting that early humans were beginning to conceptualize numbers.
- Symbolic Representation: Another possibility is that the notches held symbolic or ritualistic meaning. The number 29 might have held significance in a broader spiritual or cultural context, which could be tied to early beliefs about the natural world and the relationship between numbers and the cosmos.
The Ishango Bones
The notches on the Ishango bone seem to represent a system of tallying numbers, though the exact purpose is still debated. Some believe that the patterns might correspond to lunar calendars, while others propose that the bone may have been used for recording the reproductive cycles of animals or even as a primitive form of multiplication and division.
2. Sumer (3400 BC - 1500 BC)
Sumerian Numeral System
The Sumerians are credited with the invention of the sexagesimal system, a base-60 numeral system that would eventually influence both timekeeping (60 seconds in a minute, 60 minutes in an hour) and the way we measure angles. Their system used a combination of symbols to represent numbers, with a distinct emphasis on the number 60. Large numbers were represented by combining these symbols in ways that allowed them to handle increasingly larger values.
Sumerian numerals were initially simple, consisting of just a few symbols. For example, a vertical line represented a unit, and a symbol shaped like a corner represented ten. As the need for larger numbers grew, the system became more complex, introducing symbols for powers of ten, and eventually creating methods to combine symbols to represent very large numbers.
Mathematical Achievements
In Sumer, large numbers were used not just for counting but for complex administrative and scientific tasks. Sumerian scribes kept records of everything from grain inventories and livestock to astronomical observations. The Sumerians understood that numbers could be used to calculate large quantities, and they began to apply them to both their economy and their understanding of the cosmos.
One notable example is the Plimpton 322 tablet, a clay tablet dating from around 1900 BC, which is thought to be a record of a Pythagorean theorem-like relationship. Though the Sumerians did not have the formalized geometry we recognize today, they were already capable of using numbers in sophisticated ways to solve practical problems.
Large Numbers in Trade and Astronomy
Large numbers were essential for the burgeoning trade networks and astronomical observations of the Sumerians. The Sumerians had a clear understanding of cyclical time and used their advanced numeration system to predict astronomical events such as the movement of stars and planets. These astronomical records were not just of scientific interest - they also had practical applications for agriculture, as they could predict the best times to plant and harvest crops.
In trade, large numbers were used to represent the quantities of goods being exchanged, often in the form of barley, silver, or livestock. The Sumerian system made it possible to manage these exchanges over vast distances, tracking large inventories and ensuring the accuracy of transactions.
Conclusion
The early history of large numbers reveals the remarkable ways ancient humans began to quantify and make sense of the world around them. From the notched bones of prehistoric times to the structured numeration of the Sumerians, these early counting methods were crucial steps toward the complex systems that would evolve over time. The use of numbers for tracking cycles, quantities, and time set the stage for the advanced mathematical and administrative practices of later civilizations.
In the next installment of our series on the history of large numbers, we will explore the rise of numerical systems in ancient civilizations like Assyria and Babylonia (1500 BC - 300 BC) and Ancient Egypt (3000 BC - 300 BC). These cultures built upon the foundations laid by early numeracy, creating systems of mathematics and astronomy that still influence us today. Stay tuned to learn how large numbers played a key role in shaping ancient empires!
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