# Babylonian Mathematics and the Base 60 System

Babylonian mathematics used a sexagesimal (base 60) system that was so functional it remains in effect, albeit with some tweaks, in the 21st century. Whenever people tell time or make reference to the degrees of a circle, they rely on the base 60 system.

## Base 10 or Base 60

The system surfaced circa 3100 BCE, according to The New York Times. “The number of seconds in a minute — and minutes in an hour — comes from the base-60 numeral system of ancient Mesopotamia,” the paper noted.

Although the system has stood the test of time, it is not the dominant numeral system used today. Instead, most of the world relies on the base 10 system of Hindu-Arabic origin.

The number of factors distinguishes the base 60 system from its base 10 counterpart, which likely developed from people counting on both hands. The former system uses 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, and 60 for base 60, while the latter uses 1, 2, 5, and 10 for base 10. The Babylonian mathematics system may not be as popular as it once was, but it has advantages over the base 10 system because the number 60 “has more divisors than any smaller positive integer,” the Times pointed out.

Instead of using times tables, the Babylonians multiplied using a formula that depended on knowing just the squares. With only their table of squares (albeit going up to a monstrous 59 squared), they could compute the product of two integers, a and b, using a formula similar to:

ab = [(a + b)2 - (a - b)2]/4. The Babylonians even knew the formula that’s today known as the Pythagorean theorem.

## History

Babylonian math has roots in the numeric system started by the Sumerians, a culture that began about 4000 BCE in Mesopotamia, or southern Iraq, according to ​USA Today.

“The most commonly accepted theory holds that two earlier peoples merged and formed the Sumerians,” USA Today reported. “Supposedly, one group based their number system on 5 and the other on 12. When the two groups traded together, they evolved a system based on 60 so both could understand it.”

That’s because five multiplied by 12 equals 60. The base 5 system likely originated from ancient peoples using the digits on one hand to count. The base 12 system likely originated from other groups using their thumb as a pointer and counting by using the three parts on four fingers, as three multiplied by four equals 12.

The main fault of the Babylonian system was the absence of a zero. But the ancient Maya’s vigesimal (base 20) system had a zero, drawn as a shell. Other numerals were lines and dots, similar to what is used today to tally.

## Measuring Time

Because of their mathematics, the Babylonians and Maya had elaborate and fairly accurate measurements of time and the calendar. Today, with the most advanced technology ever, societies still must make temporal adjustments — almost 25 times per century to the calendar and a few seconds every few years to the atomic clock.

There’s nothing inferior about modern math, but Babylonian mathematics may make a useful alternative to children who experience difficulty learning their times tables.

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