Ancient Egyptian Mathematics
It’s 1650 BCE. A scribe named Ahmose sits on a reed mat, copying a long mathematical text onto a papyrus scroll that is nearly six meters long. He is not doing this for fame or theory. He is solving practical problems that help keep society running. For example, how do you fairly divide 6 loaves of bread among 10 workers? What should be the slope of a pyramid’s sides? How much grain fits in a cylindrical granary?
Ancient Egyptian mathematics was not created by philosophers. It developed from the needs of farming and construction. This mathematics has quietly influenced the way we understand math today.
A Number System Carved in Stone (and Reed)
The ancient Egyptians started developing their number system around 3200 BCE. The earliest signs of this system appeared on ivory labels in Tomb U-j at Abydos. By the time of the Narmer Macehead, which recorded 400,000 oxen and over 1.4 million goats, Egypt had a fully functional counting system that could handle large quantities.
Their numeral system was based on the number 10, similar to ours, but it was different because it was non-positional. This means that they used different hieroglyphic symbols for each power of ten. Here are some examples of their symbols:
– A simple stroke for 1
– A heel bone for 10
– A coiled rope for 100
– A lotus plant for 1,000
– A finger for 10,000
– A frog for 1,000,000
To write a number, they repeated the appropriate symbol as many times as needed. This system was visual and effective, though it could become complicated with larger numbers. The Egyptians wrote from right to left and did not have a concept of zero, which they managed to work around creatively.
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The Papyrus Scrolls That Changed History
Most of what we know about ancient Egyptian mathematics comes from a small number of surviving texts. This is impressive since papyrus is a delicate material. The dry climate of Egypt is what has allowed these documents to survive.
The most important of these documents is the Rhind Mathematical Papyrus, which dates back to around 1650 BCE. Scottish archaeologist Alexander Henry Rhind bought it from an antiquities dealer in Luxor in the early 1860s. Today, it is housed in the British Museum, with a small part in Brooklyn. A scribe named Ahmose wrote that it shows “the method of calculating, for understanding and knowing everything that exists.” The scroll contains over 84 solved problems in arithmetic and geometry, as well as fraction tables that scribes used daily.
Another key source is the Moscow Mathematical Papyrus, which is older and has fewer problems — just 25. However, it includes one of the most impressive achievements of ancient mathematics: a formula for calculating the volume of a truncated pyramid, also known as a frustum. This calculation is still impressive to modern mathematicians.
Other important sources include the Egyptian Mathematical Leather Roll, which has a fraction conversion table; the Berlin Papyrus 6619; the Lahun Mathematical Papyri; and ostraca, which are pottery shards used as informal notepads, from Deir el-Medina, the village where pyramid workers lived and kept detailed records of their quarrying activities.
Arithmetic, Algebra, and the Art of Doubling
Egyptian arithmetic was simple and clear. Addition and subtraction involved counting symbols, which made them easy. However, multiplication and division showed the cleverness of the Egyptians.
Instead of memorizing multiplication tables, Egyptian scribes used a method of doubling. For example, to multiply 13 by 17, a scribe would start with 1 and 17, then double: 2 and 34, 4 and 68, 8 and 136. Next, they would add the rows that summed to 13 (1 + 4 + 8), leading to 17 + 68 + 136 = 221. This method is similar to binary arithmetic and works every time.
They performed division by reversing multiplication, asking, “What must I double to reach this number?”
For algebra, Egyptians used a method called “false position.” In Problem 24 of the Rhind Papyrus, the question is: “A quantity and its 7th together make 19 — what is it?” The scribe would guess the quantity is 7, see that 7 + 1 = 8 (not 19), and then adjust proportionally. The correct answer — 16 + ½ + ⅛ — was found through careful arithmetic. While this isn’t symbolic algebra like we know today, it shows genuine algebraic thinking. The Berlin Papyrus even shows that ancient Egyptians could solve quadratic equations.
Egypt's Obsession with Fractions
One major difference between ancient Egyptian mathematics and modern math is how they used fractions. Egyptians mainly worked with unit fractions, which are fractions that have a numerator of 1, such as ½, ⅓, and ¼. To represent other fractions, they added distinct unit fractions together.
For example, instead of writing ⅗, an Egyptian scribe would write ½ + 1/10. This method may seem complicated, but it was a logical system. The Rhind Papyrus included a table that helped convert fractions of the form 2/n into sums of unit fractions, acting like a reference guide used throughout Egypt.
This system mattered in practice. Dividing bread and beer rations fairly among groups of workers of various ranks was a daily task. Even though the fraction system seemed cumbersome, it worked reliably for the bureaucrats in the Nile Delta.
Geometry: When Math Moves Mountains
Egyptian geometry focused on getting results rather than proving theorems, and the results were impressive.
Scribes could accurately measure the areas of rectangles, triangles, and circles. For circles, Problem 48 of the Rhind Papyrus gives an approximation of π as about 3.16. They achieved this by comparing a circle inscribed in a square, long before Archimedes did. The error is less than 1%.
One remarkable measure is the seked, which describes the angle of a pyramid’s slope by showing how far it moves horizontally for each unit of vertical rise. Problem 56 of the Rhind Papyrus details how to calculate the slope of a pyramid, showing that Egyptian builders used calculations rather than just instinct.
The evidence is in the stone. The base of the Great Pyramid of Giza is accurate to within less than 1/14,000 of its planned length. Its right angles are off by less than 1/27,000. These figures show careful calculation and applied mathematics at a level that amazes us even today.
Math in Everyday Egyptian Life
Egyptian mathematics was important in everyday life. Tax collectors used geometric formulas to measure land after each yearly Nile flood because the floods washed away boundary markers. They calculated grain storage volumes to manage the state’s large food supply. Trade relied on standardized weights and measures that needed consistent arithmetic.
The calendar, which had a 365-day year, required astronomical calculations. This is shown by the large stone structures at Nabta Playa, which aligned with the early rising of Sirius to signal the flooding season.
Egypt's Legacy: The Math That Traveled the World
The contributions of Egyptian mathematics to later civilizations are important and often overlooked. In the 5th century BCE, Herodotus noted that the Greeks learned geometry from Egypt. Notable figures like Thales, Pythagoras, and Plato studied in Egyptian temples. Euclid also worked in Alexandria. The Egyptian decimal system influenced Greek and Roman numerals and eventually led to the base-10 system we use today.
Egyptian mathematics focused on practicality. It aimed to solve real problems, such as feeding people, building monuments, collecting taxes, and organizing a large society. This practical approach is perhaps its most important lesson. The most lasting mathematics might not always be the most elegant. Sometimes, it’s the kind that helps divide bread, build pyramids, and manage floods.
Got a Question?
F.A.Qs
The Egyptian numeral system was a non-positional decimal system. They used specific hieroglyphs for powers of ten: a single stroke for 1, a heel bone for 10, a coiled rope for 100, and a lotus plant for 1,000. To write a number, they simply repeated these symbols as many times as necessary.
Dated to approximately 1650 BCE, the Rhind Mathematical Papyrus is the most comprehensive source of ancient Egyptian math. Copied by the scribe Ahmose, it contains 84 problems covering arithmetic, geometry, and fractions, serving as a practical manual for officials and builders.
They used a clever method of doubling (similar to modern binary arithmetic). To multiply two numbers, they would create two columns, starting with 1 and the multiplier, then doubling both until the numbers in the first column could be added to equal the multiplicand. They then summed the corresponding numbers in the second column.
Egyptians used a measurement called the seked. The seked defined the inclination of a pyramid’s face by measuring the horizontal “run” for every one cubit of vertical “rise.”
Yes, though not in the symbolic form we use today. They used a technique called “false position,” where they would make an educated guess at an unknown value (the “heap”) and then adjust that guess proportionally to find the correct answer.
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