Mesopotamian Mathematics: The Ancient Roots of Base 60

Mathematics for a Working World

More than 4,000 years ago, the Sumerians and Babylonians were developing mathematical techniques in ancient Mesopotamia. Their calculations addressed practical needs: measuring farmland, organising trade, tracking loans and studying the sky.

This was mathematics built around real tasks. Numerical methods also helped planners estimate construction materials, labour requirements and the capacity of canals.

Why Count in Sixties?

Mesopotamian mathematics used a base-60 system, also called sexagesimal. Unlike our familiar base-10 system, its place values increased by powers of 60. This tradition helped shape the divisions we still use for time and angles: 60 minutes in an hour, 60 seconds in a minute and 360 degrees in a circle.

One proposed explanation for the choice of 60 involves finger counting. A thumb can count 12 finger segments on the other four fingers of the same hand. The five fingers of the opposite hand can then track five groups of 12, producing 60. This is a possible explanation, not a proven origin.

Clay Tablets and Place Value

Scribes pressed wedge-shaped marks into moist clay with reed styluses. This writing system is called cuneiform. Tablets could then be dried in the sun; paper was not their recording medium.

Mesopotamian scribes developed one of history’s earliest place-value systems. A numerical sign’s position helped determine its value, allowing relatively few signs to represent large quantities.

For long periods, however, the notation lacked a true numerical zero. Empty positions could be indicated by gaps, leaving some readings dependent on context. Later placeholder signs helped distinguish empty positions, but a placeholder was not the same as zero treated as a number.

Plimpton 322 and Right Triangles

Plimpton 322 is a Babylonian clay tablet approximately 3,700 years old. It records numerical relationships associated with Pythagorean triples: sets of three positive whole numbers in which the squares of the two smaller numbers add up to the square of the largest.

These relationships describe the sides of right triangles. The tablet shows that Babylonian mathematicians knew such relationships long before Pythagoras lived, although its exact purpose remains debated. It does not, by itself, establish that they produced a formal proof of the theorem.

Tools for Complex Calculations

• Reference tables supplied multiplication results, reciprocals, square roots and cube roots. Reciprocals were especially useful for division.
• Babylonian methods could solve problems we would express as quadratic equations, as well as certain cubic problems.
• Geometric procedures covered triangles, trapezoids and circles. Some Babylonian calculations approximated pi as 3.125.
• Mathematical planning supported agriculture, lending, canal works and monumental construction, including stepped temple structures called ziggurats.

Together, these techniques reveal a sophisticated mathematical culture whose influence extends far beyond the surviving clay tablets.

Key facts

• Mesopotamian mathematics used a base-60, or sexagesimal, number system.
• Scribes recorded calculations in cuneiform on clay tablets.
• Plimpton 322 contains numerical relationships associated with Pythagorean triples.
• Babylonian place-value notation developed without a true numerical zero.
• Some Babylonian calculations used 3.125 as an approximation of pi.