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BY THE RIVERS OF BABYLON

Between the Tigris and Euphrates rivers is a stretch of fertile land that has been home to many great ancient civilizations. The rivers have their sources separately in what is now Turkey, meandering through modern-day Iraq, Syria and Iran before flowing into the Persian Gulf. Together they form a natural border for the area that was once known as Mesopotamia.

By 3000 BCE, the Sumerian civilization was thriving here. The Sumerians built complex cities with vast irrigation systems. They also had one of the earliest legal systems, complete with courts, jails and government records. They had developed the earliest known writing system, cuneiform—needed for those records—and a counting system, to boot. They even created a postal service.

Over the next thousand years, the Akkadians became the dominant force in the region. They brought with them their own technology, including the abacus—a tool they had invented. (It worked slightly differently to later versions, such as the Chinese one.) Eventually, their empire would fall, leaving behind two distinct Akkadian-speaking groups: Assyrians in the north and Babylonians in the south. Each produced a vast civilization of its own, but it was in the south that mathematics really accelerated.

The city of Babylon, roughly 100 kilometers south of modern-day Baghdad, was the capital of the empire of Babylonia. Under the direction of King Hammurabi, who ruled from around 1792 to 1750 BCE, Babylonia became a force to be reckoned with. He controlled several city-states in the region, making Babylonia extremely rich and powerful. This provided the stability and resources needed for a mathematical community to develop and thrive.

An extensive collection of clay tablets that still survive today record many details about Babylonia in this era. Scribes would scratch what they wanted to record with a sharpened stick on wet clay and then leave it to harden in the sun. These tablets, to Babylonians, were what paper and spreadsheets are to us today—crucial tools for record-keeping. They recorded Hammurabi's legal system, known as the Code of Hammurabi, which consisted of 282 written laws and contained one of the earliest examples of being innocent until proven guilty—though how guilty you were depended on whether you were a person with property, free or a slave. They also recorded transactions and told stories, including myths about creation, and relayed news.

One tablet has survived that is essentially a bad review. Written around 1750 BCE, it is from an unsatisfied customer called Nanni, who had agreed to buy copper ingots from a merchant called Ea-nasir. However, when the ingots arrived, they were not to Nanni's liking. In his complaint, he wrote that he was unhappy with the copper and that the seller had been rude to his servant when completing the transaction. Scraping and baking a review into a form that would last for thousands of years displays consumer power at its finest.

The Babylonians used mathematics for many practical purposes, including splitting plots of land and calculating tax. Some clay-tablet writers recorded revenues and budgets, and so familiarized themselves with numbers. Unfortunately, they did not sign their names, so we know almost nothing about individual mathematicians from this time. But some certainly studied mathematics systematically, taking in topics such as algebra and uncovering that famous theorem about triangles often named after Pythagoras (who lived much later). They also approximated the square root of two correct to six decimal digits.

The counting system of the day came from the Sumerians and was sexagesimal—based around the number 60. Our preferences for dividing circles into 360 degrees and hours into 60 minutes stems from this system. Below are the cuneiform symbols they used to represent the numbers 1 to 59 (Not Shown)

The Babylonian number system was a positional system like ours, meaning that the order the numbers are written in tells you something about the amounts they represent. For example, when we write the number 271, it is with the implicit understanding that the number furthest to the right represents one unit, then, moving toward the left, there are seven tens and two hundreds. Or, in numbers (Not Shown)

Similarly, the Babylonians used positions to represent powers of 60, so 271 could be expressed as (Not Shown)

Or, in cuneiform (Not Shown)