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

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

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.

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.

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