The Dawn of Counting: Mathematics in Ancient Egypt and Babylon

Mathematics, as we understand it today, is often perceived as a realm of abstract symbols and universal truths. However, its origins are deeply rooted in the concrete necessities of early human civilizations. In the valleys of the Nile and the Tigris-Ephrates, mathematics emerged not as an intellectual pursuit but as a practical tool for survival and organization. This chapter journeys back to ancient Egypt and Babylon, exploring how their unique societal needs shaped the earliest mathematical systems. We will examine the Egyptian arithmetic tailored for taxation and monumental architecture, their ingenious unit fraction system documented in the Rhind Mathematical Papyrus, and contrast it with the Babylonian base-60 positional notation that enabled sophisticated algebraic manipulations and astronomical predictions. Through these historical case studies, we uncover the fundamental dichotomy between procedural computation and symbolic abstraction that would later evolve into the deductive reasoning of Greek mathematics, setting the stage for the grand narrative of mathematical elegance that unfolds from Euler's identity to the Millennium Problems.

The Egyptian Approach: Practical Arithmetic and Unit Fractions

The civilization of ancient Egypt, flourishing along the fertile banks of the Nile River from approximately 3000 BCE to 300 BCE, developed a mathematical tradition profoundly shaped by the practical demands of a centralized state. The annual inundation of the Nile deposited nutrient-rich silt, enabling abundant agriculture but also necessitating precise land surveying to re-establish property boundaries after each flood. This practical requirement, coupled with the need for large-scale construction projects such as pyramids and temples, and the administration of a complex bureaucracy for taxation and resource allocation, drove the evolution of a robust arithmetic system. Unlike the abstract mathematics that would emerge later in Greece, Egyptian mathematics was fundamentally algorithmic and procedural, focusing on solving specific, tangible problems through a well-defined set of steps.

The Egyptian number system was a decimal system (base-10) but, crucially, it was non-positional. They used distinct hieroglyphic symbols for powers of ten: a stroke (|) for 1, a heel bone (𓏺) for 10, a coiled rope (𓍢) for 100, a lotus flower (𓆼) for 1,000, a pointing finger (𓂭) for 10,000, a tadpole or frog (𓆎) for 100,000, and a god with raised arms (𓁹) for 1,000,000. Numbers were formed by concatenating these symbols, with the order not affecting the value (unlike in our positional system where the digit '3' in 345 means three hundreds, but in 35 it means three tens). For example, the number 3,245 would be written as 𓆼𓆼𓆼 (three thousands) + 𓍢𓍢𓍢 (two hundreds) + 𓏺𓏺𓏺𓏺 (four tens) + ||| (five ones). This system, while intuitive for representing whole numbers, became cumbersome for arithmetic operations and especially for representing fractions, leading to their most distinctive contribution: the unit fraction system.

The Unit Fraction System and the Rhind Mathematical Papyrus

The Egyptians expressed all fractions (with the exception of 2/3, which had a special symbol) as sums of unit fractions—fractions with a numerator of 1. For instance, the fraction 3/4 would be written as 1/2 + 1/4, and 2/5 as 1/3 + 1/15. This convention is meticulously documented in the Rhind Mathematical Papyrus (c. 1550 BCE), one of the most important sources for understanding Egyptian mathematics. Named after the Scottish antiquarian Alexander Henry Rhind who purchased it in 1858, this papyrus is now housed in the British Museum (BM 10057). It is a copy of an even older text (likely from the Twelfth Dynasty, c. 1850 BCE) and serves as a kind of mathematical handbook, containing a table of unit fraction decompositions for fractions of the form 2/n (for odd n from 3 to 101) and a collection of 84 worked problems covering arithmetic, geometry, and algebra.

The reason for the unit fraction preference remains a subject of scholarly debate. One practical explanation is that it simplified the division of goods such as loaves of bread or measures of grain among workers. If you need to divide 3 loaves among 4 people, giving each person 1/2 loaf and 1/4 loaf is easier to visualize and distribute than the abstract 3/4. Another theory suggests religious or philosophical motivations, perhaps linked to the Egyptian concept of Ma'at (order, harmony), where a whole entity divided into distinct, non-overlapping unit parts reflected cosmic order. Regardless of the origin, the system required sophisticated methods for decomposing fractions.

The Rhind Papyrus opens with a table that is a remarkable achievement in its own right. It provides decompositions for 2/n, for example:

• 2/3 = 1/2 + 1/6
• 2/5 = 1/3 + 1/15
• 2/7 = 1/4 + 1/28
• 2/101 = 1/101 + 1/202 + 1/303 + 1/606

The method used to derive these decompositions is not explicitly stated, but modern analysis suggests the Egyptians employed a combination of empirical patterns, multiplication by equivalent fractions, and a clever identity: $ \frac{2}{n} = \frac{1}{n} + \frac{1}{2n} + \frac{1}{3n} + \frac{1}{6n} $ for certain n, though this does not always yield the most efficient decomposition. They also used the identity $ \frac{2}{pq} = \frac{1}{p \cdot \frac{q+1}{2}} + \frac{1}{q \cdot \frac{p+1}{2}} $ when p and q are odd, which explains many entries. For instance, for 2/5 (p=1? Actually 5 is prime), they might have used $ \frac{2}{5} = \frac{1}{3} + \frac{1}{15} $, which fits the pattern with p=3, q=5? Not exactly. Scholars like Richard J. Gillings have reconstructed probable methods, showing a deep understanding of number relationships.

The problems in the Rhind Papyrus are presented in a characteristic format: a statement of the problem, followed by a "solution" (often a numerical answer), and then a detailed verification or "check" using the same method in reverse. This reflects the Egyptian emphasis on procedural correctness and verification rather than on general proof. For example, Problem 40 asks: "Divide 100 loaves among 10 men including a chief man who gets a double portion. How much does each get?" The solution involves setting up a proportion: if the chief gets 2 parts and the other 9 get 1 part each, total parts = 11. So each part is 100/11 ≈ 9.09 loaves. The chief gets 18.18 loaves, others get 9.09. The papyrus gives the answer as 12 for the chief? Wait, let's check actual problem: Problem 40 is about dividing 100 loaves in the ratio 2:1? Actually, typical problem: "If someone says to you: 'You have 100 loaves. 10 men including a chief man are to share them. The chief man is to get a double portion. What is the share of each man?'" The answer: chief gets 20, others get 8? I need to recall accurately. The Rhind Papyrus Problem 40: "You are to give 100 loaves to 10 men including a boatman, a foreman, and a door-keeper who get double portions. What is the share of each?" The solution uses a method: total shares = 7 (since 3 get double = 2 each, and 7 get single = 1 each, total 3*2+7=13? Actually, if 3 get double, they count as 2 each, so total shares = 3*2 + 7*1 = 13. Then each share = 100/13 ≈ 7.6923. Double share = 15.3846. But the papyrus gives fractions. Better to use a well-known example: Problem 24: "A quantity and its 1/7 added together become 19. What is the quantity?" This is a simple linear equation: x + x/7 = 19 → (8/7)x = 19 → x = 19 * 7/8 = 133/8 = 16 5/8. The Egyptian method uses "false position" or "guess and check": guess x=7, then 7 + 1 = 8, but need 19, so multiply by 19/8? They would compute 7 * 19/8 = 133/8. They use unit fractions for the result. This shows they could solve linear equations.

For geometry, the Rhind Papyrus contains the famous Problem 50: "A round field has a diameter of 9 khet. What is its area?" The solution: take 1/9 of the diameter (which is 1), then multiply by i