The pyramids of Giza are so precisely built that their sides line up with the compass directions to within a fraction of a degree. Behind that precision lay a practical, carefully taught system of mathematics. Thanks to a few surviving papyrus scrolls, we can still follow the exact methods Egyptian scribes used nearly 4,000 years ago.
The Sources: Papyri and Scribes
In ancient Egypt, mathematics belonged to the scribes, the trained officials who measured fields after the Nile flood, calculated taxes, rationed bread and beer for workers and planned building projects. Papyrus is fragile, so only a handful of mathematical texts survive.
- The Rhind Mathematical Papyrus is the most important. It was copied by a scribe named Ahmes around 1550 BCE, and he tells us he was copying an older text from around 1850 BCE. The Scottish antiquarian Alexander Henry Rhind bought it in Luxor in 1858, and it is now in the British Museum. It is over 5 metres long and contains 84 problems.
- The Moscow Mathematical Papyrus, from around 1850 BCE, has 25 problems, including the famous calculation of the volume of a truncated pyramid.
Ahmes opens his scroll with a promise: it offers "accurate reckoning for inquiring into things, and the knowledge of all things". It was, in effect, a textbook.
Egyptian Numerals
Egyptians used a base-10 additive system. Hieroglyphs included a single stroke for 1, a heel bone for 10, a coiled rope for 100, a lotus flower for 1,000, a finger for 10,000, a tadpole for 100,000 and a kneeling god with raised arms for 1,000,000. To write 352 you drew three ropes, five heel bones and two strokes. There was no place value and no zero symbol, but for addition and subtraction this system worked well.
Multiplication by Doubling
Egyptians did not memorise multiplication tables. Instead, they multiplied using only doubling and adding, a method that works because every whole number can be written as a sum of powers of 2.
Worked Example 1: Calculate 13 × 24 the Egyptian way
Step 1: Make two columns. Start the left with 1 and the right with 24, and keep doubling both.
- 1 → 24
- 2 → 48
- 4 → 96
- 8 → 192
Stop at 8, because the next doubling (16) is bigger than 13.
Step 2: Pick left-hand numbers that add to 13:
$$ 13 = 8 + 4 + 1 $$
Step 3: Add the matching right-hand numbers:
$$ 192 + 96 + 24 = 312 $$
So \( 13 \times 24 = 312 \). Check: \( 13 \times 24 = 13 \times 20 + 13 \times 4 = 260 + 52 = 312 \). This is essentially binary multiplication, the same principle computers use today.
Unit Fractions
The most distinctive feature of Egyptian maths is the way it handled fractions. Apart from special symbols for \( \tfrac{2}{3} \) (and sometimes \( \tfrac{3}{4} \)), Egyptians wrote every fraction as a sum of different unit fractions, fractions with numerator 1. For example, they did not write \( \tfrac{3}{4} \) but \( \tfrac{1}{2} + \tfrac{1}{4} \).
This might seem clumsy, but it is very practical for sharing. Suppose you must divide 3 loaves among 4 workers. Cut two loaves into halves and one loaf into quarters: each worker gets \( \tfrac{1}{2} + \tfrac{1}{4} \) of a loaf, and everyone can see they received identical pieces.
The 2/n table
Because doubling was the basis of multiplication, scribes constantly needed to double unit fractions. Doubling \( \tfrac{1}{5} \) gives \( \tfrac{2}{5} \), which must be rewritten. So the Rhind Papyrus begins with a table expressing \( \tfrac{2}{n} \) for every odd \( n \) from 3 to 101 as a sum of unit fractions.
Worked Example 2: Writing 2/5 as unit fractions
Step 1: Find the largest unit fraction less than \( \tfrac{2}{5} \). Since \( \tfrac{2}{5} = 0.4 \) and \( \tfrac{1}{3} \approx 0.333 \), the largest is \( \tfrac{1}{3} \).
Step 2: Subtract:
$$ \frac{2}{5} - \frac{1}{3} = \frac{6}{15} - \frac{5}{15} = \frac{1}{15} $$
Step 3: The remainder is already a unit fraction, so
$$ \frac{2}{5} = \frac{1}{3} + \frac{1}{15} $$
This is exactly the entry in Ahmes' table. Similarly, the table gives
$$ \frac{2}{7} = \frac{1}{4} + \frac{1}{28} $$
which you can verify: \( \tfrac{7}{28} + \tfrac{1}{28} = \tfrac{8}{28} = \tfrac{2}{7} \).
Geometry: Circles and Pyramids
The area of a circle
Problem 50 of the Rhind Papyrus finds the area of a round field with diameter 9 khet. The rule was: subtract \( \tfrac{1}{9} \) of the diameter and square the result.
$$ 9 - \tfrac{1}{9}(9) = 8, \qquad 8^2 = 64 $$
Comparing with the modern formula \( A = \pi r^2 = \pi (4.5)^2 = 20.25\pi \), the Egyptian rule implies
$$ \pi \approx \frac{64}{20.25} = \frac{256}{81} \approx 3.1605 $$
That is less than 1% away from the true value 3.1416, a remarkable achievement around 1850 BCE.
Worked Example 3: The seked of a pyramid
Egyptians measured the steepness of a pyramid face using the seked: the horizontal distance the face moves inward for a vertical rise of one cubit, measured in palms. One royal cubit equals 7 palms.
Problem 56 asks: a pyramid has a base side of 360 cubits and a height of 250 cubits. What is its seked?
Step 1: The horizontal run from the edge to the centre is half the base:
$$ \frac{360}{2} = 180 \text{ cubits} $$
Step 2: The run per cubit of rise is
$$ \frac{180}{250} = 0.72 \text{ cubit} $$
Step 3: Convert to palms:
$$ 0.72 \times 7 = 5.04 \text{ palms} = 5\tfrac{1}{25} \text{ palms} $$
Ahmes gives exactly \( 5\tfrac{1}{25} \) palms. The Great Pyramid of Khufu has a seked of about \( 5\tfrac{1}{2} \) palms, which corresponds to a slope angle of roughly \( 51.8^\circ \). In modern terms, the seked is simply the cotangent of the slope angle, multiplied by 7.
The truncated pyramid
Problem 14 of the Moscow Papyrus finds the volume of a truncated square pyramid with height 6, bottom side 4 and top side 2. The scribe's steps match the correct formula
$$ V = \frac{h}{3}\left(a^2 + ab + b^2\right) = \frac{6}{3}(16 + 8 + 4) = 2 \times 28 = 56 $$
How the Egyptians discovered this formula remains one of the mysteries of ancient mathematics.
Did You Know?
- The Rhind Papyrus ends with a playful puzzle: 7 houses, each with 7 cats, each cat eating 7 mice, each mouse eating 7 ears of grain, each ear producing 7 measures. The total is \( 7 + 49 + 343 + 2401 + 16807 = 19607 \).
- Egyptian unit fractions were still being used in European business arithmetic in the Middle Ages, including in Fibonacci's Liber Abaci (1202).
- Every fraction can be written as a sum of distinct unit fractions, a fact later proved using exactly the "greedy" method in Worked Example 2.
Key Takeaways
- The Rhind Papyrus, copied by Ahmes around 1550 BCE, is our richest source on Egyptian mathematics.
- Egyptians multiplied by repeated doubling, an early form of binary arithmetic.
- Fractions were written as sums of distinct unit fractions, supported by a \( \tfrac{2}{n} \) table.
- Their circle rule implies \( \pi \approx 3.16 \), and the seked measured pyramid slopes precisely.
- The Moscow Papyrus contains the correct formula for the volume of a truncated pyramid.
If fractions, volumes or trigonometry are giving you trouble, Amathify's subject experts and PhD scholars can provide clear step-by-step solutions and assignment help, so contact us on WhatsApp or through the Ask a Question form.
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