## The heroes who came up with zero

‘Arabic’ numerals came to Europe from India — but ancient Sumerians invented them

Mark Ronan

Who invented zero? Certainly not the Greeks and Romans. When the Romans used a count-down they ended with 1, not 0. For example their calendar counted down to the Ides in the middle of the month, the previous day being “the day before the Ides”, and the day before that being “the third day before the Ides” — rather than the second day before. They had no concept of 0, and the New Testament of the Bible, written in Greek, tells us that after the crucifixion “he rose again on the third day”, meaning two days later. Church scholars had determined that the day of the crucifixion was a Friday, so “the third day” meant Sunday (two days later), hence the Christian choice of Sunday as the holy day, rather than the Jewish Sabbath. In the Classical world counting — up or down — started or ended with 1. There was no zero, and no year zero: 1BC is followed immediately by 1 AD.

European mathematics only adopted the concept of zero in the 13th century after the publication of Liber Abaci (book of computation) by Leonardo of Pisa, aka Fibonacci. He used the Arabic numeral system, learned from North Africa. The shape of the digits was rather different from the customary Western forms, but the point was that the order of the digits in a number determined its value. You did not simply add together the value of digits representing 1 up to 9, 10 up to 90, 100 up to 900, etc, as in the Greek system. For instance, 327 is not the same number as 273. This is the “positional” or “place-value” system, a fantastic advance on the previous methods of writing numbers used in Europe and the Classical world.

Thus in his 1911 book An Introduction to Mathematics the philosopher Alfred North Whitehead applauded our modern notation for numbers:

By relieving the brain of all unnecessary work, a good notation sets it free to concentrate on more advanced problems, and in effect increases the mental power of the race. Before the introduction of the Arabic notation, multiplication was difficult, and the division even of integers called into play the highest mathematical faculties.

His term “Arabic notation” refers to the positional system, which the Arabic world acquired from India where it had been in use since the 6th century BC. In modern parlance we talk of Hindu-Arabic or Indo-Arabic numerals, which use just ten digits including zero to represent all counting numbers. A brilliant system, it was later extended to decimal fractions, with a decimal point separating the fractional part of a number from the rest.

This is the accepted narrative, but neither the Arabs nor the Indians invented the positional system with its inevitable use of zero. That honour goes without question to the Sumerians, inhabitants of southern Mesopotamia since before the dawn of history. They invented writing on clay tablets, starting in the late 4th millennium BC in the city of Uruk, home to the legendary Gilgamesh, and origin of the modern name Iraq. In 3000 BC Uruk was the greatest city in the world but during the 21st century BC that role belonged to the city of Ur, where the “positional” system of writing numbers was invented.
The chief difference between 6th-century BC India and the Sumerians a millennium and a half earlier is that the Indians worked to base-10 as we do today, but the Sumerians to base-60. It was a fantastic system that became an essential component of mathematics and astronomy, later passed on to the Greek world which then used base-60 for fine divisions of angles: each degree splits into 60 minutes, each minute into 60 seconds, and so on. But for everyday use the Greeks already had the alphabet and a counting system that operated to base-10, so they made no use of cuneiform, and the sexagesimal system — base-60 — was only adopted for fractions in technical computations, not day-to-day matters.

When the cuneiform script eventually disappeared into the sands of time the ancient Sumerian system was lost, until its recovery in modern times. Their zero was represented by a space, rather than a specific symbol (that came later), and we have examples of multi-digit fractions — some having as many as eight digits — with zeros in the middle. The positional system makes multiplication easier, particularly when you use multiplication tables, as they did. But the real genius of their invention was for division. Dividing by a number is the same as multiplying by its reciprocal, and in base-60 this was very simple for such numbers as 2, 3, 4, 5, 6, 8, 9, 10, 12, etc: you simply looked up the reciprocal in one table and then used multiplication tables. The floating-point nature of the system facilitated this, because while the reciprocal of 2 is one-half, in floating-point base-60 it is written as 30 (think “point 30”), the reciprocal of 3 is written as 20, the reciprocal of 5 as 12, and so on. This takes modern minds a while to get used to, but for the Sumerians it became second nature.

As Whitehead wrote, “a good notation sets [us] free to concentrate on more advanced problems”, and this is precisely what the Sumerians, and their successors the Babylonians, did. They pursued metrical geometry 1,500 years before Book II of Euclid’s Elements, they computed the square root of 2 to what amounts to six places of decimals, and discovered Pythagoras’s theorem well over a thousand years before Pythagoras.

Any positional system of numbers necessarily uses zero, so the Sumerians certainly got there first, and only if you wish to invest the concept with additional philosophical attributes do you have to appeal to later mathematics. Today zero is associated with the null-set, and perhaps there are cultures whose creation myths start with the void, but not the Judaeo-Christian tradition. It begins with chaos. The Babylonian Enuma Elish starts with Tiamat, a watery monster whose name has Sumerian roots, and the Hebrew Bible refers to tehom (the deep), which derives from Tiamat.

As for the “positional” system for numbers in India, there was trade between Mesopotamia and the Indus Valley as early as the third millennium BC, but what is certain is that the Sumerians invented zero, and gave us the origins of our most fundamental myths before their spoken language sadly died out in about 2,000 BC.