Mesopotamia & Egypt

1.1 The Babylonian Sexagesimal System

The story of mathematics begins in the river valleys of Mesopotamia, in the land between the Tigris and Euphrates — modern-day Iraq. Here, around 3400 BCE, the Sumerians began pressing wedge-shaped marks into soft clay tablets to keep accounts of grain, livestock, and labor. Over the following millennium, these primitive tallies evolved into one of the most sophisticated number systems of the ancient world: the sexagesimal (base-60) positional system of the Old Babylonian period (c. 2000–1600 BCE).

Why base 60? The choice likely arose from the merger of two earlier traditions — a base-10 system (counting on fingers) and a base-6 system (possibly counting the twelve finger-segments of one hand with the thumb as pointer, using five positions of the other hand). The number 60 has extraordinary divisibility: it is evenly divisible by 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, and 60 itself — a total of 12 factors. This made fraction arithmetic especially clean, since many common fractions terminate in base 60.

The system was positional, meaning the same symbol could represent different values depending on its position — just as the digit 3 means three hundred in 345 but three in 123. This was a revolutionary leap. The Egyptians, Greeks, and Romans all usedadditive systems where a symbol always meant the same value regardless of position. Positional notation enables compact representation of very large and very small numbers, and makes arithmetic algorithms (addition, subtraction, multiplication, division) far more systematic.

One critical limitation of the early Babylonian system was the absence of a zero as a placeholder. If a position was empty, scribes simply left a larger gap — an ambiguous convention that could lead to confusion between, say, 1,0,1 (= 3601) and 1,1 (= 61). It was not until the Seleucid period (c. 300 BCE) that a special placeholder symbol — two small slanted wedges — was introduced for internal empty positions, though even then it was never used at the end of a number.

The Babylonians compiled extensive tables of reciprocals to facilitate division. Since division by a number is equivalent to multiplication by its reciprocal, and since many reciprocals have finite sexagesimal representations (any number whose prime factors are only 2, 3, and 5 — the so-called "regular" numbers), this table-based approach was highly efficient. A standard reciprocal table included entries such as:

Numbers that were not regular (such as 7, 11, 13) have infinitely repeating sexagesimal expansions and were called igi nu ("no reciprocal") in Babylonian terminology. The scribes dealt with these through approximation tables or by avoiding such divisions when possible.

The legacy of base-60 persists to this day. We divide the hour into 60 minutes and the minute into 60 seconds. The circle is divided into 360 degrees (= 6 × 60), each degree into 60 arc-minutes, and each arc-minute into 60 arc-seconds. Even the convention of writing coordinates in degrees-minutes-seconds is a direct inheritance from Babylonian astronomy.

1.2 Babylonian Algebra

The Old Babylonian scribes were master algebraists — though they expressed their algebra not in symbols but in rhetorical (purely verbal) form. Their clay tablets contain hundreds of problems that, when translated into modern notation, correspond to linear equations, quadratic equations, systems of equations, and even some cubic equations. The problems are typically posed as word problems about lengths, widths, areas, and volumes, but the underlying algebraic structure is unmistakable.

The quintessential Babylonian algebraic technique is completing the square — the same method taught in algebra courses today, but discovered nearly four thousand years ago. The Babylonians did not have a general quadratic formula; instead, they had a procedure (an algorithm) that they applied to specific numerical problems.

The Babylonians also handled the related forms $x^2 - bx = c$ and $x^2 = bx + c$, with appropriate modifications to the procedure. They always sought positive solutions only — negative numbers had no physical meaning in their problem contexts.

A particularly important class of problems involved simultaneous equations of the form: given the sum and product of two unknowns, find the unknowns. If $x + y = s$ and $xy = p$, then the Babylonians recognized that $x$ and $y$ are the roots of the quadratic $t^2 - st + p = 0$. Their procedure was:

$$x = \frac{s}{2} + \sqrt{\left(\frac{s}{2}\right)^2 - p}, \qquad y = \frac{s}{2} - \sqrt{\left(\frac{s}{2}\right)^2 - p}$$

Remarkably, some tablets also present problems leading to cubic equations. The Babylonians compiled tables of cubes and cube roots (values of $n^3$ and $n^2(n+1)$) that allowed them to solve certain cubics by table lookup. The tablet BM 85200+ contains problems that reduce to cubics of the form $x^3 + x^2 = c$, solved by consulting a table of $n^2(n+1)$ values.

1.3 Plimpton 322

Among the most celebrated mathematical artifacts in the world is Plimpton 322, a small clay tablet (approximately 13 cm wide and 9 cm tall) now housed in the Rare Book and Manuscript Library at Columbia University. Dating to approximately 1800 BCE, this Old Babylonian tablet contains a table of 15 rows and 4 columns of numbers that encode Pythagorean triples — sets of integers $(a, b, c)$ satisfying $a^2 + b^2 = c^2$.

This means the Babylonians were systematically studying the Pythagorean relationship more than a thousand years before Pythagoras of Samos (c. 570–495 BCE) was born. The tablet does not merely list a few triples — it appears to present them in a mathematically organized way, generated by a deliberate algorithm.

The tablet's columns, when properly interpreted, correspond to the following for each row:

Here are several of the Pythagorean triples encoded in Plimpton 322, along with the generating parameters $p$ and $q$:

The interpretation of Plimpton 322 has been debated for decades. The three leading theories are:

• Trigonometric table: Neugebauer and Sachs (1945) proposed it is a table of secant-squared values, organized by decreasing angle from roughly 45 degrees to 31 degrees — effectively the world's first trigonometric table.
• Teacher's reference: Robson (2001) argued it is a pedagogical tool — a list of problems with known integer answers, designed for scribal training.
• Exact sexagesimal trigonometry: Mansfield and Wildberger (2017) proposed it is a "ratio-based trigonometric table" using exact rational arithmetic rather than the approximate angle-based approach of later Greek trigonometry.

Regardless of which interpretation is correct, Plimpton 322 demonstrates that Babylonian mathematicians had a deep and systematic understanding of the $a^2 + b^2 = c^2$ relationship, a sophisticated method for generating integer solutions, and the ability to organize mathematical knowledge into structured tables — all more than three millennia ago.

1.4 The Babylonian Square Root Algorithm

One of the most impressive achievements of Babylonian mathematics is their method for computing square roots to remarkable accuracy. The tablet YBC 7289, a small clay tablet from the Yale Babylonian Collection (c. 1800–1600 BCE), depicts a square with its diagonals drawn, and along the diagonal is inscribed the sexagesimal number 1;24,51,10 — the Babylonian approximation of $\sqrt{2}$.

The quadratic convergence of this method is striking. If the error at step $n$ is $\varepsilon_n = x_n - \sqrt{S}$, then the error at the next step satisfies:

$$\varepsilon_{n+1} = \frac{\varepsilon_n^2}{2x_n} \approx \frac{\varepsilon_n^2}{2\sqrt{S}}$$

This means the number of correct digits approximately doubles with each iteration. Starting with 1 correct digit, after three iterations we have approximately $2^3 = 8$ correct digits — more than enough for the five sexagesimal digits seen on YBC 7289.

The Babylonians also computed square roots of other numbers. Tablets contain approximations of $\sqrt{3}$, $\sqrt{5}$, and other values, all consistent with the application of this iterative method. It is one of the earliest known examples of an iterative numerical algorithm in mathematics.

1.5 The Egyptian Number System

Across the deserts and along the Nile, a parallel mathematical tradition developed independently. Egyptian mathematics, though less algebraically sophisticated than its Babylonian counterpart, possessed its own distinctive elegance and produced remarkable results in geometry and practical computation.

The Egyptians used a hieroglyphic decimal system that was fundamentally additive rather than positional. There were distinct symbols for each power of 10:

To write a number, the Egyptian simply repeated each symbol as many times as needed. For example, the number 4,523 would be written with 4 lotus flowers, 5 coiled ropes, 2 heel bones, and 3 strokes. The order of symbols did not matter (unlike a positional system), though they were conventionally arranged from largest to smallest.

This unit fraction system may seem cumbersome to modern eyes, but it had practical advantages. When dividing goods among workers, expressing shares as sums of distinct unit fractions made it clear how to physically partition a quantity. If 3 loaves must be divided among 5 workers, the Egyptian answer $\frac{1}{3} + \frac{1}{15}$ tells you to cut each loaf into thirds (giving 5 workers each one third-piece, with some adjustment for the remaining portion).

The 2/n table at the beginning of the Rhind Papyrus is a masterwork of computational ingenuity: it gives unit fraction decompositions of $\frac{2}{n}$ for every odd $n$ from 3 to 101. Some examples:

$$\frac{2}{3} = \frac{1}{2} + \frac{1}{6}, \qquad \frac{2}{5} = \frac{1}{3} + \frac{1}{15}, \qquad \frac{2}{7} = \frac{1}{4} + \frac{1}{28}$$

$$\frac{2}{9} = \frac{1}{6} + \frac{1}{18}, \qquad \frac{2}{11} = \frac{1}{6} + \frac{1}{66}, \qquad \frac{2}{13} = \frac{1}{8} + \frac{1}{52} + \frac{1}{104}$$

The choices of decomposition are not always the "simplest" — the scribes appear to have followed aesthetic and practical preferences (avoiding very large denominators, preferring even denominators for ease of halving). The algorithm underlying the table construction remains debated among historians.

Egyptian multiplication was performed by successive doubling (also called the "Russian peasant" method). To compute $a \times b$, the scribe would repeatedly double $a$ (which is trivial in any additive system) and then sum those doublings that correspond to the binary representation of $b$. For example, to compute $13 \times 12$:

1.6 The Rhind Mathematical Papyrus

The Rhind Mathematical Papyrus (also known as the Ahmes Papyrus) is the most extensive surviving document of ancient Egyptian mathematics. It was purchased by the Scottish antiquary Alexander Henry Rhind in Luxor in 1858 and is now held in the British Museum (with small fragments in the Brooklyn Museum). The papyrus is about 5.5 meters long and 33 cm wide, written in hieratic script by a scribe named Ahmes (c. 1650 BCE), who states that he is copying from an older document from the reign of Amenemhat III (c. 1860–1814 BCE).

Ahmes opens with the ambitious declaration that this is the "correct method of reckoning, for grasping the meaning of things and knowing everything that is, obscurities and all secrets." The papyrus contains 84 problems covering arithmetic (division of loaves, the 2/n table), algebra (aha problems solved by false position), geometry (areas and volumes), and practical applications.

The method of false position is mathematically equivalent to solving a linear equation by proportional reasoning. If $f(x) = x + \frac{x}{n}$ is linear, and we know that $f(x_0) = r$, then $f\left(\frac{m}{r} \cdot x_0\right) = m$. This works perfectly for linear problems and was later extended (by medieval Islamic and European mathematicians) to the double false position method for more complex equations.

The Egyptian Approximation of Pi

Problem 50 of the Rhind Papyrus asks: "A circular field has diameter 9 khet. What is its area?" The scribe's instruction is to subtract $\frac{1}{9}$ of the diameter and square the result:

How did the Egyptians arrive at this formula? One plausible explanation involves an octagonal approximation. If you inscribe a regular octagon inside a square of side 9, by cutting triangular corners from the square, the octagon has area approximately 63, which is close to 64. The scribes may have noticed empirically that $\left(\frac{8}{9}d\right)^2$ gives a very good approximation to the area of a circle inscribed in a square of side $d$.

1.7 Egyptian Volume Formulas

The Moscow Mathematical Papyrus (also known as the Golenischev Mathematical Papyrus), dating to approximately 1850 BCE and now in the Pushkin Museum in Moscow, is the second-most important mathematical papyrus from ancient Egypt. It contains 25 problems, and its crowning achievement is Problem 14 — the computation of the volume of a truncated pyramid (frustum).

The derivation of this formula is far from obvious. Unlike the simple formulas for rectangles and triangles, which can be discovered empirically, the frustum formula involves the cross-term $ab$, which suggests genuine mathematical reasoning. One possible approach the Egyptians might have used involves decomposing the frustum into simpler solids: a rectangular block of volume $hb^2$, four triangular prisms, and four corner pyramids, whose volumes sum to the formula above.

Note that when $b = 0$ (a complete pyramid), the formula reduces to:

$$V = \frac{h}{3}(a^2 + 0 + 0) = \frac{ha^2}{3}$$

which is the correct formula for the volume of a square pyramid. And when $b = a$(a rectangular prism), it reduces to:

$$V = \frac{h}{3}(a^2 + a^2 + a^2) = ha^2$$

which is the volume of a cube (when $h = a$) or a rectangular prism. These special cases give confidence in the formula's correctness.

The Moscow Papyrus also contains other geometrically interesting problems. Problem 10, for instance, involves the surface area of what may be a hemisphere or a half-cylinder — the interpretation is debated. If it is indeed a hemisphere of diameter $d = 4.5$, then the Egyptian answer of $32$ corresponds to $A = 2 \times (8d/9)^2$, using their circle-area approximation. This would imply knowledge of curved surface areas — an extraordinary achievement.

The Egyptians also knew the formula for the volume of a cylinder. Given a circular base of diameter $d$ and height $h$, they computed:

$$V = \left(\frac{8d}{9}\right)^2 \times h = \frac{64d^2 h}{81}$$

This is simply their circle-area formula multiplied by the height — conceptually identical to the modern $V = \pi r^2 h$, with the approximation $\pi \approx \frac{256}{81}$.

1.8 Legacy and Comparison

The mathematical traditions of Mesopotamia and Egypt, developing largely independently over the third and second millennia BCE, represent humanity's first sustained engagement with abstract mathematical thought. Though both civilizations developed mathematics primarily for practical purposes — taxation, surveying, construction, astronomy, and calendar-keeping — their achievements transcended mere utility.

Babylonian Strengths

• Positional notation: The sexagesimal place-value system enabled compact representation of numbers and efficient arithmetic algorithms — the most important innovation in the history of numeration until the Indian decimal system.
• Algebraic sophistication: Solutions of quadratic and some cubic equations, systematic generation of Pythagorean triples, and sophisticated number theory.
• Numerical methods: The square root algorithm demonstrates an understanding of iterative convergence that would not be formalized in the West until Newton.
• Astronomical computation: Late Babylonian astronomers (c. 400–100 BCE) developed arithmetic methods for predicting planetary positions that influenced Greek and Islamic astronomy profoundly.

Egyptian Strengths

• Geometric insight: The frustum formula and the circle-area approximation show genuine geometric reasoning beyond empirical measurement.
• Practical computational methods: The method of false position, multiplication by doubling, and unit fraction arithmetic were robust and reliable tools.
• Systematic pedagogy: The mathematical papyri are essentially textbooks, with problems arranged by type and difficulty, suggesting organized scribal education.
• Architectural mathematics: The precision of the pyramids (the Great Pyramid's base is level to within 2 cm across 230 meters) implies sophisticated practical mathematics.

Key Differences

The survival bias is important: clay tablets are nearly indestructible, while papyrus decays in anything but the driest conditions. We have hundreds of thousands of cuneiform mathematical tablets but only a handful of Egyptian mathematical papyri. The Egyptian mathematical tradition may have been far richer than the surviving evidence suggests.

Both traditions influenced the Greeks directly. Thales and Pythagoras reportedly traveled to both Egypt and Babylon. The Greek historian Herodotus credited the Egyptians with inventing geometry ("earth measurement"). Babylonian arithmetic and astronomical methods were adopted and extended by Hellenistic astronomers, most notably Hipparchus and Ptolemy. The sexagesimal system persists in astronomy and timekeeping to this day.

Perhaps the deepest legacy is conceptual: these civilizations established the idea that mathematical procedures should be systematic, teachable, and verifiable. The Babylonian problem texts and the Egyptian mathematical papyri are, in essence, the world's first mathematics textbooks — and the direct ancestors of every algebra, geometry, and arithmetic course taught today.

Video Lectures

These video lectures cover the key mathematical techniques of ancient Egypt and Mesopotamia, from numeral systems and arithmetic operations to geometry and algebra.