The Babylonians controlled much of modern-day Iraq, Iran and Syria, from 1800BC. Capital of Syria Damascus used to be the most important point on the trade routes, linking old Mesopotamia with Egypt. To manage their empire, they developed a sophisticated numerical system that laid the groundwork for many mathematical concepts still in use today.

Textbooks and Math Problems, 4,000 Years Ago

The keepers of this knowledge were the scribes: literates trained in specialized schools from as early as 2500 BC. These young scholars honed their skills on clay tablets, many of which have survived. Fascinatingly, some of the most enlightening discoveries about Babylonian mathematics come not from official records—but from children's homework found in the clay tablets.

One tablet from around the 18th century BC reads like an ancient geometry workbook. It presents problems visually: "I drew a square, 60 units long, and inside it, I drew four circles. What are their areas?" This shows us that Babylonian culture deeply engaged in applied and theoretical mathematics.

Like the Egyptians, Babylonians tackled practical problems, but their approach was different. Instead of theoretical abstractions, they followed step-by-step recipes—algorithms in their purest form. Scribe would simply follow and record a set of instructions to get a result.

Mathematical Equations Without Algebraic Expression

By manipulating weights and quantities, the Babylonians deduced the unknown weight without symbolic algebra. Let’s say I have a small clay pot, and I don’t know how much it weighs. Instead of weighing it directly, I do this:

1. I place 4 identical clay pots on one side of the scale — pretend I have four of the same kind.
2. Then I add 4 stones that weights 2 gin (ancient Babylonian measure of weight) to the same side, so in total it weights 4 pots + (4×2=8 gin)
3. Now I balance out the scale by putting 10 stones (10 × 2 = 20 gin) on the other side of the scale.

That gives us:

4 pots + 8 gin = 20 gin

After removing 4 stones from each side of the scale, it still balances out and I can tell that that 4 pots weights 12 gin and each pot is about 3 gins without writing this algebraic equation above.

Mathematical Equations Without Algebraic Symbols

One of the most fascinating things about Babylonian math is how they solved equations—without using any algebraic symbols at all.

Let’s imagine a simplified example based on their problem-solving style:

Suppose I have a clay pot, but I don’t know how much it weighs. Here's what I do:

• I place 4 identical pots on one side of the scale.
• Then I add 4 stones, each weighing 2 gin (the ancient Babylonian unit of weight), for a total of 8 gin.
• On the other side of the scale, I place 10 stones, each also 2 gin, totaling 20 gin.

Now the scale is balanced:

4 pots + 8 gin = 20 gin

To solve this, I remove the same weight—4 stones (8 gin)—from both sides. What’s left?

4 pots = 12 gin → 1 pot = 3 gin

No symbols, no variables—just logic and balance.

Base-60, Place Value, and the Birth of Zero

Unlike the Egyptians, who used a base-10 system like ours, the Babylonians used a base-60 system. Why 60? They likely got this from using 12 knuckles on one hand and 5 fingers on the other: 12 × 5 = 60. But the key reason was that 60 is incredibly divisible—you can split it into halves, thirds, fourths, fifths, sixths... That made calculations with fractions much easier.

The legacy of base-60 is still with us today:

• 60 seconds in a minute
• 60 minutes in an hour
• 360 degrees in a circle

Another brilliant feature of their system was place value—the idea that a digit’s position affects its value. Just like in our number system (where 101 means 100 + 1), the Babylonians used positions to represent increasing powers of 60.

For example, the number 3,661 would be written as 1 – 1 – 1, meaning:

• 1 × 60² (3,600)
• 1 × 60¹ (60)
• 1 × 60⁰ (1)

Total = 3,600 + 60 + 1 = 3,661

To make this possible, they needed a way to indicate an empty place—the Babylonian version of zero. At first, they just left a blank space. Later, they introduced a placeholder symbol, like a punctuation mark, to clearly show when a digit was missing in a position. This was the first recorded use of zero—though it would take another thousand years before zero was used as a full number.

The Babylonians didn’t develop this system just for fun—it was essential for their work in astronomy. They tracked lunar cycles and eclipses with astounding accuracy. From around 800 BC, they began keeping full records of lunar eclipses, month by month, year by year. Their system of angular measurement—360 degrees in a circle, with each degree divided into 60 minutes, and each minute into 60 seconds—was in perfect harmony with their number system and ideal for astronomical observations and calculations.

Mathematics in the Fields: Land, Irrigation, and Quadratics

Once the Babylonians had developed a sophisticated number system, they put it to work—not just for astronomy, but also for agriculture and engineering. Living in an often dry and harsh environment, Babylonian engineers and surveyors used mathematics to tame the landscape. They devised clever irrigation systems to bring water from rivers to their crops, an essential skill in regions like the Orontes Valley in Syria, which still benefits from these ancient methods today. Mathematics wasn’t just theoretical—it had practical, life-sustaining value.

Many Babylonian tablets focus on land measurement, and it's in this context that we see one of their most impressive innovations: the use of quadratic equations. These appear when solving problems involving areas of fields where the unknown value is squared (multiplied by itself).

🧮 A Geometric Puzzle with a Quadratic Twist

Here's an example drawn straight from Babylonian methods:

A field has an area of 55 square units. One side is 6 units longer than the other. How long is the shorter side?

Rather than using modern algebra, Babylonian scribes visualized the problem geometrically. They imagined reshaping the rectangle into a square:

• Cut 3 units from one end and move it to the side

• That leaves a missing 3×3 = 9 square

• Add that back in → total area becomes 64
• The square root of 64 is 8, so the original short side must have been 5

No formulas, just clever thinking.

What’s remarkable is that the Babylonians could solve quadratic equations like this around 4,000 years ago—long before algebraic symbols were invented.

Geometry and the Mystery of Right-Angled Triangles

Babylonian fascination with numbers also led them to explore geometry—especially the properties of right-angled triangles.

You may have heard of the 3-4-5 triangle—a classic example of the Pythagorean theorem, which states that the square on the diagonal (hypotenuse) is equal to the sum of the squares on the other two sides.

While we credit this to the Greek philosopher Pythagoras, evidence shows the Babylonians understood it a thousand years earlier.

📜 Plimpton 322: A Tablet Ahead of Its Time

One of the most famous mathematical artifacts from Babylon is Plimpton 322, a clay tablet over 3,700 years old. It lists sets of whole numbers that satisfy the Pythagorean relationship.

These are not just random numbers—they are 15 perfect right-angled triangles, arranged in steadily decreasing angles. Many historians believe this proves Babylonian mathematicians had a working knowledge of trigonometry, long before the Greeks formalized it.

There’s debate about whether this was a list of known relationships or a teaching tool used by a clever math instructor. Either way, it shows deep mathematical insight.

Square Roots and Irrational Numbers

Another incredible example is a school tablet that calculates a remarkably accurate approximation of the square root of two—around 1.41421, which is accurate to four decimal places.

Why is this significant?

Because √2 is an irrational number—it cannot be written as a simple fraction, and its digits go on forever without repeating. The fact that Babylonian students were working with irrational numbers nearly 4,000 years ago is a testament to their curiosity, precision, and mathematical brilliance.

References:

1. Houston Museum of Natural Science
2. A History of Mathematics 3rd Edition by Carb B. Boyer
3. https://www.youtube.com/watch?v=2WcbPcGrQZU&t=1188s
4. https://projectlovelace.net/problems/babylonian-square-roots/
5. https://www.scienceabc.com/eyeopeners/why-are-there-24-hours-in-a-day-and-60-minutes-in-an-hour.html