History of Large Numbers - Part 2: Babylon and Ancient Egypt

In the first part of our exploration of the history of large numbers, we examined the earliest systems of numeration. Now, in part 2, we shift our focus to two ancient civilizations - Assyrian Babylon and Ancient Egypt - whose innovative use of large numbers played a pivotal role in shaping both their societies and the future of mathematics. From the Babylonian base-60 system to the Egyptian hieroglyphic numerals, these cultures set the stage for more complex systems of counting, calculation, and measurement. Let’s dive into the mathematical legacies they left behind.

3. Assyrian Babylon (1500 BC - 300 BC)

Babylonian Clay Tablet (1)

Babylonian Clay Tablet (2)

The Babylonian Numerical System

The Babylonians inherited and refined the sexagesimal (base-60) numerical system from their Sumerian predecessors. Unlike the decimal (base-10) systems familiar today, their system allowed for a more compact representation of large numbers using positional notation. Numbers were constructed by combining symbols for 1 and 10, with positions indicating multiples of 60, 60², 60³, and so on. For example, 3600 (60²) was a single positional value in their notation.
This innovation provided them with unmatched flexibility in expressing both small fractions and enormous quantities, making it ideal for a variety of mathematical and practical applications.

Mathematical Achievements and Astronomical Uses

The mathematical ingenuity of the Babylonians is vividly displayed in artifacts like the Plimpton 322 tablet. This artifact, containing Pythagorean triples, reveals their deep understanding of squares and ratios. Such advanced knowledge predates similar developments in Greek mathematics, proving their early expertise in dealing with complex calculations.
Babylonians also used large numbers extensively in astronomy. They tracked planetary movements, predicted eclipses, and developed the division of a circle into 360 degrees - a practice still used in modern geometry. By using their base-60 system, they could calculate astronomical cycles spanning years or even centuries, demonstrating both their mathematical skill and their dedication to observing the cosmos.

Practical Applications and Legacy

Large numbers were indispensable in the Babylonians' daily lives, particularly in land management and resource allocation. The fertile lands of Mesopotamia were measured and divided using their mathematical methods, ensuring equitable distribution among communities.
The Babylonian base-60 system influenced neighboring civilizations, including the Greeks, and left an enduring legacy. Today, we still use their numerical concepts in time-keeping (60 seconds in a minute, 60 minutes in an hour) and geometry, bridging ancient innovation with modern practice.

Babylonian cuneiform numerals

 Babylonian Cuneiform Numerals

Babylonian numerals were written in cuneiform, a wedge-shaped script impressed on clay tablets. Their system used two basic symbols: a wedge for 1 and a corner mark for 10, arranged within a base-60 framework. For example, 23 was written as two corner marks (20) followed by three wedges (3).
Although they initially lacked a symbol for zero, spacing was used to indicate missing positional values, with a placeholder added later for clarity. This versatile system allowed the Babylonians to perform advanced calculations, from measuring land to tracking celestial events, leaving an enduring legacy in mathematics.

4. Ancient Egypt (3000 BC - 300 BC)

Egyptian Numerals Hieroglyphs

The Egyptian Hieroglyphic Numerical System

The Ancient Egyptians developed a base-10 numerical system represented by hieroglyphs for powers of ten. A vertical staff symbolized one, a horseshoe mark represented ten, and so on, up to one million, represented by a kneeling figure with raised hands. Numbers were expressed additively, combining these symbols to create larger values.
For example, 123 was written with one “hundred” symbol, two “ten” symbols, and three “one” symbols. This clear and straightforward system allowed the Egyptians to document large quantities efficiently, whether for architectural calculations or administrative records.

Monumental Calculations

The Egyptian expertise in handling large numbers is evident in their monumental architecture. Structures like the Great Pyramid of Giza required detailed planning involving massive quantities of stone blocks, precise angles, and careful workforce management. Calculations for such projects often involved numbers in the tens or hundreds of thousands, a testament to their practical application of mathematics.
Their understanding of geometry extended to approximations of π, as seen in the Rhind Mathematical Papyrus. This knowledge was essential not only for architecture but also for tasks like calculating the areas of fields and storage spaces for grain.

Administration and Symbolism

Large numbers played a central role in Egyptian society beyond construction. Administrative tasks, such as managing taxes, recording census data, and distributing resources, relied on meticulous numerical documentation. Scribes, skilled in hieroglyphics and mathematics, ensured the accuracy of these records on papyrus scrolls and stone tablets.
Symbolically, numbers carried significant meaning in Egyptian culture. The hieroglyph for one million, depicting a kneeling man with raised arms, symbolized vastness and reverence, often appearing in religious texts to express the infinite. Their ability to represent large numbers reflected not just mathematical skill but also their philosophical and spiritual outlook.

Conclusion

The civilizations of Assyrian Babylon and Ancient Egypt laid the foundation for the sophisticated use of large numbers in mathematics, astronomy, and administration. The Babylonians, with their base-60 system, revolutionized the way large numbers were represented and used, influencing time-keeping and geometry in the modern world. Meanwhile, the Egyptians applied their hieroglyphic-based number system to monumental architecture and daily governance, demonstrating the vital role of mathematics in the development of complex societies.
These early innovations in numeration and calculation not only reflect the intellectual prowess of these ancient cultures but also show how the concept of large numbers helped shape the course of human history, from land management to celestial observation. As we move forward in time, the legacy of these early systems paved the way for even more advanced and nuanced ways of dealing with numbers.
In the next part of our journey through the history of large numbers, we will explore the Roman Numerals (500 BC) and Greek Numerals (600 BC - 0 AD). The Romans introduced a system that combined simplicity and practicality, while the Greeks advanced mathematical thought, using numbers in profound ways for both philosophy and science. Stay tuned as we continue to uncover how these ancient cultures refined the art of numeration and laid the groundwork for modern mathematics.

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