Egyptian Mathematics

Origins in the Middle Kingdom

While the roots of Egyptian mathematics extend back to the Old Kingdom — where arithmetic was already essential for monumental architecture and the redistribution of harvests — it is during the Middle Kingdom (c. 2055–1650 BCE) that Egyptian mathematics truly flourished as a systematic discipline. This era, sometimes called the classical age of Egyptian civilisation, saw the production of the great mathematical papyri that would define our understanding of ancient Egyptian thought for millennia.

The Rhind Mathematical Papyrus (also called the Ahmes Papyrus, c. 1650 BCE) is a copy of an older Middle Kingdom text, written by the scribe Ahmes. It contains 84 problems covering arithmetic, algebra, geometry, and practical applications. Its opening line declares that it provides “accurate reckoning — the entrance into the knowledge of all existing things and all obscure secrets”, revealing that the Egyptians saw mathematics not merely as a tool but as a gateway to deeper understanding.

The Moscow Mathematical Papyrus (c. 1850 BCE) is even older and contains 25 problems. Its Problem 14 is celebrated as one of the greatest achievements of ancient mathematics: the correct formula for the volume of a truncated pyramid (frustum),

where a and b are the sides of the top and bottom squares, and h is the height. This result, derived without any formal concept of integration, demonstrates a level of geometric insight that would not be surpassed until Greek mathematics centuries later.

Several other texts survive from this period, including the Egyptian Mathematical Leather Roll (a table of unit fraction decompositions), the Lahun Mathematical Papyri, and fragments from the Berlin Papyrus 6619, which remarkably contains a problem equivalent to solving two simultaneous equations.

Hieratic Script & Mathematical Notation

While the monumental hieroglyphic script adorned temple walls and tombs, the everyday mathematics of Egypt was conducted in hieratic — a cursive script written with reed pens on papyrus. Hieratic was faster and more practical than hieroglyphs, and it was the standard writing system for all administrative, literary, and scientific texts throughout the Middle and New Kingdoms.

In hieratic, numbers had distinct symbols for units (1–9), tens (10–90), hundreds (100–900), and thousands. Unlike the additive repetition of hieroglyphs (where, for instance, the number 7 required seven vertical strokes), hieratic used a single symbol for each value — making it far more compact and efficient for computation.

Fractions were indicated by writing the denominator with a dot or a special stroke above it. The unit fraction system — where every fraction was expressed as a sum of distinct unit fractions (fractions with numerator 1) — was deeply embedded in both the notation and the mathematical thinking of the scribes.

The Eye of Horus & Sacred Fractions

One of the most striking intersections of Egyptian mathematics and mythology is the Eye of Horus (Wadjet or Oudjat), the sacred eye that was a powerful symbol of protection, royal power, and good health. According to myth, when Horus fought his uncle Seth to avenge his father Osiris, Seth tore out Horus’s left eye and shattered it into six pieces. The god Thoth — patron of scribes, wisdom, and mathematics — magically restored the eye, and each of its six parts came to represent a specific fraction.

The sum falls short of unity by exactly 1/64. According to tradition, this missing fraction was supplied by Thoth’s magic when he restored the eye — a beautiful metaphor in which divine intervention completes what arithmetic alone cannot. Some scholars have interpreted this as a deliberate mathematical statement about the limits of finite binary subdivision, while others see it as a symbolic reminder that perfection belongs to the gods.

In practical terms, the Horus Eye fractions formed a system for measuring grain. The basic unit was the hekat (approximately 4.8 litres), and each fraction of the Eye represented a specific subdivision of this measure. A scribe could write any quantity of grain from 1/64 to 1 hekat using combinations of these six symbols — essentially a binary notation system predating modern binary arithmetic by over three millennia.

For quantities smaller than 1/64 of a hekat, the Egyptians used a separate unit called the ro (1/320 of a hekat), ensuring precise measurement even for small amounts of grain, medicine, or pigment.

The Unit Fraction System

Perhaps the most distinctive feature of Egyptian mathematics is its exclusive use of unit fractions — fractions of the form 1/n. With the sole exception of 2/3 (which had its own special symbol), every fraction was expressed as a sum of distinct unit fractions. For example:

The Rhind Papyrus opens with a 2/n table — a systematic decomposition of fractions of the form 2/n for all odd values of nfrom 3 to 101. This table is a masterpiece of computational ingenuity. The scribes had to find decompositions that were not only correct but also practical — favouring small denominators and short sums for ease of further calculation.

Why did the Egyptians insist on unit fractions? The answer may lie in the concept of fair division. When dividing 2 loaves among 5 workers, giving each person “1/3 + 1/15” of a loaf is a concrete instruction a scribe can execute: cut each loaf into thirds, then subdivide the remaining pieces. The unit fraction system turns abstract arithmetic into a sequence of physical operations.

Egyptian Arithmetic: Doubling & Halving

Egyptian multiplication was based on a remarkably efficient algorithm of successive doubling. To multiply two numbers, the scribe would create two columns: one beginning with 1 and doubling at each step, the other beginning with the multiplicand and likewise doubling. The scribe then selected rows whose left column summed to the multiplier, and added the corresponding entries from the right column.

This method is mathematically equivalent to binary decomposition — the same principle at the heart of modern computer arithmetic. The algorithm requires only the ability to double, halve, and add — operations that are easy to perform on an abacus or counting board.

Geometry & the Seked of the Pyramids

The construction of the pyramids required sophisticated geometric knowledge. Egyptian scribes used the seked — the horizontal displacement per cubit of vertical rise — to specify the slope of a pyramid face. The seked is essentially the reciprocal of the modern concept of slope (or, more precisely, the cotangent of the angle of inclination).

The Great Pyramid of Giza has a seked of 5 palms and 2 fingers per cubit (where 1 cubit = 7 palms = 28 fingers), giving a rise-to-run ratio that produces the famous 51.84° angle of its faces. Problems 56–60 of the Rhind Papyrus are devoted to seked calculations, showing that this was a standard part of scribal training.

For computing areas, the Egyptians used correct formulas for rectangles, triangles, and trapezoids. Their approximation of the area of a circle — taking the diameter, subtracting 1/9 of it, and squaring the result — yields:

This is within 0.6% of the true value of π — a remarkable achievement for a civilisation working without algebra or the concept of irrational numbers.

Video: The Mathematics of the Great Pyramid

N.J. Wildberger explores the remarkable mathematical relationships embedded in the Great Pyramid of Giza — from its seked ratios and geometric proportions to the deep connections between Egyptian metrology and the emergence of mathematical thought.

The Aha Problems & Proto-Algebra

Some of the most mathematically interesting problems in the Rhind Papyrus are the aha problems (from the Egyptian word aha, meaning “heap” or “quantity” — essentially the unknown). These problems ask: “A quantity, its 1/7 part added to it, becomes 19. What is the quantity?”

In modern notation, this is the equation x + x/7 = 19. The Egyptian scribe solved it by the method of false position: assume the answer is 7 (a convenient choice because 7 + 7/7 = 8), then scale: since 8 must become 19, the true answer is 7 × 19/8 = 16 + 5/8. The scribe then verified by substitution.

These problems represent the earliest known examples of algebraic thinking — solving for an unknown quantity using a systematic method. While the Egyptians never developed symbolic notation, their algorithmic approach to equations laid the conceptual groundwork for the algebra that would later flourish in Mesopotamia, Greece, and the Islamic world.

The Berlin Papyrus & Simultaneous Equations

The Berlin Papyrus 6619 (c. 1800 BCE) contains one of the most mathematically sophisticated problems from ancient Egypt. Problem 1 asks for two numbers whose squares sum to 100, where one number is three-quarters of the other.

This problem is remarkable because it involves a system of two equations with two unknowns, a quadratic relationship, and the implicit extraction of a square root. The 3-4-5 right triangle lurking within the solution suggests the Egyptians were aware of what we now call Pythagorean triples long before Pythagoras.

The 2/n Table: A Masterpiece of Computation

The opening section of the Rhind Papyrus contains the famous 2/n table, which decomposes fractions of the form $2/n$ into sums of distinct unit fractions for every odd $n$ from 3 to 101. This table was not arbitrary — the scribes followed specific principles in choosing their decompositions.

A general identity that the scribes may have used for prime $p$:

The question of why the scribes chose particular decompositions over others has fascinated historians of mathematics for over a century. Multiple algorithms have been proposed, but no single rule accounts for all entries. The table likely represents centuries of accumulated computational experience, refined and transmitted through the scribal schools.

Division by Successive Doubling

Egyptian division was performed as the inverse of multiplication. To divide $a$ by $b$, the scribe asked: “What must I multiply$b$ by to obtain $a$?” The answer was built up by doubling the divisor and selecting the appropriate combinations.

Volumes & Advanced Geometry

Beyond the famous frustum formula, the Egyptians computed volumes of several solid shapes. The formula for the volume of a cylinder (used for calculating grain storage in cylindrical granaries) appears in multiple papyri:

Rhind Papyrus Problem 42: Volume of a Cylindrical Granary

Problem 42 of the Rhind Mathematical Papyrus (RMP plate 14, Peet reference P253 / Problem Register 149) is a masterful demonstration of how Egyptian scribes computed the volume of a circular granary. The problem asks for the volume of a granary with diameter 10 cubits and height 10 cubits, with the result expressed in khar (the standard unit for grain volume).

What makes Problem 42 particularly valuable is that the original hieratic text on the papyrus preserves the complete working — every intermediate step, every doubling table entry, every unit fraction sum. This allows modern scholars to reconstruct exactly how the scribe thought, computed, and verified the answer. The color-coded analysis of the hieratic sections reveals four distinct computational blocks on the papyrus: the doubling table for the squaring (shown in the main body), the intermediate products, the final summation, and the unit conversion — each carefully laid out in the scribe's own hand, as a model of clarity and method.

Video Lectures: The Rhind Papyrus

The following video lectures walk through several problems from the Rhind Mathematical Papyrus, demonstrating how the Egyptian scribes computed areas and volumes of circles, cylinders, cubes, and parallelepipeds.

Legacy & Transmission

Egyptian mathematics directly influenced Greek mathematical thought. Greek historians, from Herodotus to Proclus, attributed the origins of geometry to Egyptian land surveyors. Thales is said to have studied in Egypt, and Pythagoras reportedly spent over 20 years there. While the Greeks would transform mathematics with the introduction of deductive proof, the computational and geometric foundations they built upon were unmistakably Egyptian.

The unit fraction tradition persisted remarkably. Greek mathematicians continued to use Egyptian-style unit fractions (which they called “parts”), and the system survived in European commerce well into the Renaissance. The Fibonacci sequence itself was introduced in a problem about Egyptian-style fraction decompositions.

Today, the study of Egyptian mathematics remains an active field of research. The translation and reinterpretation of hieratic mathematical texts continues to reveal new insights — reminding us that along the banks of the Nile, four thousand years ago, human beings were already engaged in the timeless pursuit of understanding the world through numbers.