Sumerian Base-60 Numbers: Origins of Our 60-Minute Hour

The sexagesimal, or base-60, number system was developed by scribes in Sumer and Babylonia, in southern Mesopotamia, between about 3000 and 2000 BC. By the end of that period it had become the first known place-value system, in which the position of a digit shows its value, as in our own numbers. Passed on through Greek astronomy, it is the reason why an hour has 60 minutes, a minute 60 seconds and a circle 360 degrees.

Origins in early bookkeeping

The roots of base 60 lie in the accounts kept at Uruk and other Sumerian cities around 3300-3000 BC, at the time of the invention of cuneiform writing. The earliest tablets did not use a single number system. Scribes counted different kinds of goods with different sets of signs: one for discrete objects such as animals or people, others for grain capacity, land area and time.

The system for counting objects already grouped units by 10 and 60, with special signs for 1, 10, 60, 600 and 3,600. Over the third millennium BC this sexagesimal count became the main way of writing numbers, although measures of length, weight and volume kept their own traditional units.

The place-value breakthrough

The key step came at the end of the third millennium BC, probably under the Third Dynasty of Ur (c. 2112-2004 BC), when the large royal administration needed fast, reliable calculations. Scribes began to use a purely positional notation for working out sums, writing numbers with just two signs.

A vertical wedge stood for 1 and a corner-shaped wedge for 10. Groups of these signs wrote every number from 1 to 59. A group’s position then showed its scale: ones, sixties, 3,600s (60 times 60) and so on, or, going the other way, sixtieths and 3,600ths. Modern scholars write such numbers with commas and semicolons, so 1,24 means 1 x 60 + 24 = 84.

A larger example shows how compact the system was. The number written 2,10,5 means 2 x 3,600 + 10 x 60 + 5, or 7,805. Fractions worked the same way: 0;30 is thirty sixtieths, a half, and 0;20 is a third. Because so many common fractions come out exactly, scribes rarely had to deal with endless repeating digits of the kind that 1/3 produces in decimals.

By the Old Babylonian period, about 2000-1600 BC, this system was used throughout the scribal schools. It allowed multiplication and division to be done with standard tables, especially tables of reciprocals. To divide by a number, a scribe multiplied by its reciprocal, which was easy for numbers whose reciprocals ended neatly in base 60, such as 2, 3, 4, 5, 6, 8, 9, 10, 12 and 15.

Gaps, zero and the floating point

The Babylonian system had two important limits. First, for many centuries it had no sign for an empty place. A number such as 1,0,5 (3,605) could look almost the same as 1,5 (65), and scribes relied on spacing or context.

From about the 4th or 3rd century BC, in late Babylonian and Seleucid astronomical texts, a sign of two slanting wedges was used as a placeholder inside numbers. It worked like a zero digit in the middle of a number but was not treated as a number on its own.

Second, there was no equivalent of the decimal point. The same signs could mean 1, 60 or 1/60; the reader decided the scale from the problem. Historians compare this to floating-point arithmetic: it made calculation simple, but the size of the answer had to be worked out separately.

What scribes could do with it

Old Babylonian mathematical tablets show remarkable skill. The tablet known as YBC 7289, in the Yale Babylonian Collection, shows a square with its diagonals and gives the square root of 2 as 1;24,51,10. In decimal terms this is about 1.4142130, which differs from the true value by less than one millionth.

The tablet Plimpton 322, from around 1800 BC, lists sets of numbers related to right-angled triangles, more than a thousand years before Pythagoras. Other tablets solve problems equivalent to quadratic equations, calculate areas and volumes, and work out interest on loans.

Later Babylonian astronomers used the same system for highly accurate calculations of the movements of the Moon and planets. Observations such as those in the Venus Tablet of Ammisaduqa belong to a long tradition of recording the sky in numbers.

Why sixty?

No Mesopotamian text explains why base 60 was chosen. The most common modern explanation is practical: 60 has many divisors (1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30 and 60), so halves, thirds, quarters, fifths and sixths can all be written without awkward fractions.

Other theories suggest that 60 resulted from merging two older counting systems, perhaps based on 10 and 6 or on 12 and 5, or that it grew from counting on the hands. One popular idea holds that people counted the 12 finger joints of one hand with the thumb and kept track of dozens on the 5 fingers of the other. These theories are plausible but cannot be proved.

From Babylon to our clocks

Babylonian astronomers divided the day into 12 double hours, each of 30 smaller units, and by about the 5th century BC they divided the zodiac into 12 signs of 30 units each, giving the 360-unit circle. Greek astronomers learned from Babylonian records. Hipparchus in the 2nd century BC and Ptolemy in his Almagest, around AD 150, used 360 degrees and base-60 fractions in their tables, even though they wrote whole numbers in the Greek system.

Medieval Arabic, Persian and Latin astronomers kept these sexagesimal fractions. In Latin, the first division of a degree was the pars minuta prima, “first small part”, and the second the pars minuta secunda, “second small part”, from which come the English words minute and second. The 24-hour day itself came from Egypt, but its division into minutes and seconds of sixty followed the astronomical tradition.

When mechanical clocks gained minute hands, especially after Christiaan Huygens built the first pendulum clock in 1656 and made timekeeping far more accurate, they adopted these astronomical units. An attempt by revolutionary France in the 1790s to introduce decimal time, with 10-hour days, failed and was soon abandoned.

Today the sexagesimal system survives in everyday life in three main places: time, angles and geographic coordinates. A ship’s navigator or a GPS screen giving a position as 51 degrees, 30 minutes north is using a way of dividing quantities that Mesopotamian scribes first worked out more than 4,000 years ago.

Timeline

• c. 3300-3000 BC: Proto-cuneiform tablets at Uruk use several number systems, including a sexagesimal count.
• c. 2112-2004 BC: Under the Third Dynasty of Ur, scribes develop a positional base-60 notation for calculation.
• c. 2000-1600 BC: Old Babylonian schools teach place-value arithmetic; tablets such as YBC 7289 and Plimpton 322 are written.
• c. 5th century BC: Babylonian astronomers use a zodiac of 12 signs of 30 units, a 360-unit circle.
• c. 4th-3rd century BC: A placeholder sign for an empty position appears in astronomical texts.
• 2nd century BC: Hipparchus uses Babylonian data and base-60 fractions in Greek astronomy.
• c. AD 150: Ptolemy’s Almagest uses 360 degrees and sexagesimal fractions.

Why it matters

The base-60 system was one of the great intellectual achievements of the ancient world. It introduced the principle of place value that underlies all modern arithmetic, and it lives on every time we read a clock, measure an angle or give a map position in degrees, minutes and seconds.