Every time you read a clock showing 60 minutes in an hour or measure an angle of 360 degrees, you are using an idea invented in ancient Mesopotamia. Around 4,000 years ago, scribes in the land between the Tigris and Euphrates rivers built a number system and a set of calculation methods so effective that some of them are still taught in schools today.
Who Were the Babylonians?
Mesopotamia, roughly modern Iraq, was home to a series of civilisations: the Sumerians, the Akkadians and later the Babylonians. Most of the mathematical tablets we have come from the Old Babylonian period, around 2000 to 1600 BCE, the era of King Hammurabi.
Scribes wrote by pressing a cut reed into wet clay, making wedge-shaped marks called cuneiform. When the clay was baked or simply dried in the sun, it became nearly indestructible. As a result, hundreds of thousands of tablets survive, including thousands of school exercises, multiplication tables and problem texts. We know more about how Babylonian students learned maths than about many later civilisations.
The Base-60 (Sexagesimal) System
The Babylonians counted in base 60. They needed only two symbols: a vertical wedge for 1 and a corner wedge for 10. Numbers from 1 to 59 were built by repeating these. Beyond 59, they used place value: the position of a group showed whether it meant units, sixties, three-thousand-six-hundreds, and so on.
Why 60? Nobody knows for certain, but 60 has many divisors: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30 and 60. This makes fractions like \( \tfrac{1}{3} \), \( \tfrac{1}{4} \) and \( \tfrac{1}{5} \) easy to write exactly, which was very useful for dividing grain, land and silver.
Modern historians write Babylonian numbers with commas between places and a semicolon to mark where the whole part ends. For example, 1,40 means \( 1 \times 60 + 40 = 100 \), and 0;30 means \( \tfrac{30}{60} = \tfrac{1}{2} \).
There was one weakness: the Old Babylonians had no true zero and no sexagesimal point. A scribe had to judge from context whether a number meant 1,30 or 1;30. A placeholder symbol for an empty place appeared much later, in the Seleucid period after about 300 BCE.
YBC 7289: The Square Root of 2
One of the most famous mathematical objects in the world is a small, round tablet catalogued as YBC 7289 in the Yale Babylonian Collection, dated to around 1800–1600 BCE. It shows a square with its two diagonals. Along the diagonal is written the number
$$ 1;24,51,10 $$
Worked Example 1: Converting the YBC 7289 value to decimal
Each place after the semicolon is worth a power of \( \tfrac{1}{60} \):
$$ 1;24,51,10 = 1 + \frac{24}{60} + \frac{51}{60^2} + \frac{10}{60^3} $$
Now calculate each term:
- \( \tfrac{24}{60} = 0.4 \)
- \( \tfrac{51}{3600} \approx 0.0141667 \)
- \( \tfrac{10}{216000} \approx 0.0000463 \)
Adding:
$$ 1 + 0.4 + 0.0141667 + 0.0000463 \approx 1.4142130 $$
The true value of \( \sqrt{2} \) is 1.4142136..., so the scribe's value is correct to about six decimal places. The tablet also multiplies this by the side length 30 to give the diagonal 42;25,35, showing a clear understanding of the relationship we now call Pythagoras' theorem, more than a thousand years before Pythagoras.
How might they have found it?
Historians believe the Babylonians used a method of averaging, now often called the Babylonian (or Heron's) method. Start with a guess \( x \). If \( x \) is too big, then \( \tfrac{2}{x} \) is too small, so their average is better:
$$ x_{\text{new}} = \frac{1}{2}\left(x + \frac{2}{x}\right) $$
Starting from \( x = 1.5 \):
$$ x_1 = \frac{1}{2}\left(1.5 + \frac{2}{1.5}\right) = \frac{1}{2}(1.5 + 1.3333) = 1.41667 $$
$$ x_2 = \frac{1}{2}\left(1.41667 + \frac{2}{1.41667}\right) = \frac{1}{2}(1.41667 + 1.41176) = 1.414216 $$
Just two steps give six correct digits. The same method is used inside calculators today.
Plimpton 322: A Table of Triples
Another celebrated tablet is Plimpton 322, held at Columbia University and dated to around 1800 BCE. It contains 15 rows of numbers that, when interpreted, are closely linked to Pythagorean triples, whole numbers satisfying \( a^2 + b^2 = c^2 \). One row corresponds to the triple 119, 120, 169, and another to 4601, 4800, 6649, which are far too large to find by trial and error. Scholars still debate the tablet's purpose: a trigonometric table, a teacher's aid for generating problems, or something else.
Early Algebra: Solving Quadratics
Babylonian scribes did not use symbols like \( x \). Instead they wrote problems in words, often about lengths and areas of fields, and solved them with a step-by-step recipe. In effect, they were completing the square.
Worked Example 2: A problem from tablet BM 13901
The tablet BM 13901 in the British Museum contains this problem (translated): "I added the area and the side of my square: 0;45."
In modern notation, if the side is \( x \):
$$ x^2 + x = \frac{3}{4} $$
The scribe's recipe, with our explanation:
- Take half of the coefficient 1: \( \tfrac{1}{2} \) (written 0;30).
- Square it: \( \left(\tfrac{1}{2}\right)^2 = \tfrac{1}{4} \) (0;15).
- Add this to 0;45: \( \tfrac{1}{4} + \tfrac{3}{4} = 1 \).
- Take the square root: \( \sqrt{1} = 1 \).
- Subtract the half from step 1: \( 1 - \tfrac{1}{2} = \tfrac{1}{2} \).
So the side is \( \tfrac{1}{2} \) (0;30). Check:
$$ \left(\tfrac{1}{2}\right)^2 + \tfrac{1}{2} = \tfrac{1}{4} + \tfrac{1}{2} = \tfrac{3}{4} \checkmark $$
Compare this with the modern derivation:
$$ x^2 + x + \tfrac{1}{4} = \tfrac{3}{4} + \tfrac{1}{4} \quad\Rightarrow\quad \left(x + \tfrac{1}{2}\right)^2 = 1 \quad\Rightarrow\quad x = \tfrac{1}{2} $$
The steps are identical. The scribe was literally adding a small square of area \( \tfrac{1}{4} \) to a rectangle to "complete" a larger square, which is where our phrase comes from.
Reciprocal Tables and Division
The Babylonians did not divide directly. To compute \( a \div b \), they multiplied \( a \) by the reciprocal \( \tfrac{1}{b} \), read from a standard table. Numbers like 2, 3, 4, 5, 6, 8, 9, 10, 12 and 15 have reciprocals that terminate in base 60 (for example, \( \tfrac{1}{8} = 0;7,30 \)). Numbers like 7, whose reciprocal does not terminate, were treated as awkward and often avoided in textbook problems.
Did You Know?
- The division of the hour into 60 minutes and the minute into 60 seconds is a direct inheritance from Babylonian base 60, passed on through Greek astronomers.
- Babylonian astronomers around 350–50 BCE used a geometric method to track Jupiter's motion that resembles finding an area under a velocity-time graph, ideas once thought to have appeared first in medieval Europe.
- Many mathematical tablets were school exercises, and some still show students' mistakes.
Key Takeaways
- Babylonian mathematics flourished around 2000–1600 BCE and survives on durable clay tablets.
- Their base-60 place-value system made fractions easy and still shapes how we measure time and angles.
- Tablet YBC 7289 gives \( \sqrt{2} \) accurate to about six decimal places.
- Plimpton 322 records values linked to large Pythagorean triples.
- Scribes solved quadratic problems with a recipe equivalent to completing the square.
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