Sexagesimal

Proto-cuneiform sexagesimal type Sa

Babylonian 1

Babylonian 2

Babylonian 3

Babylonian 4

Babylonian 9

Sexagesimal (base-60) is a numeral system with sixty as its base. It originated with the ancient Sumerians in the 3rd millennium BC, was passed down to the ancient Babylonians, and is still used—in a modified form—for measuring time, angles, and geographic coordinates. The sexagesimal system is notable for its use in the division of the hour into 60 minutes and the minute into 60 seconds.

Origins and History

The sexagesimal system's origins can be traced back to the Sumerians who inhabited ancient Mesopotamia. It was later adopted and adapted by the Babylonians. The choice of 60 is likely due to its status as a highly composite number, having 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, and 60 as divisors. This property made calculations easier and allowed for the division of units into fractions.

Applications

Time and Astronomy

In modern times, the most visible use of the sexagesimal system is in the measurement of time, angles, and astronomy. An hour is divided into 60 minutes and a minute into 60 seconds. This division is a direct inheritance from Babylonian astronomy. For angles, a circle is divided into 360 degrees, where each degree is further divided into 60 minutes, and each minute into 60 seconds.

Geographic Coordinates

Geographic coordinates also use the sexagesimal system. Latitude and longitude are traditionally expressed in degrees, minutes, and seconds. For example, the Great Pyramid of Giza is located at approximately 29°58′45″N 31°08′03″E in sexagesimal notation.

Mathematical Notation

In sexagesimal notation, numbers are represented using the digits 0–59. Each digit after the first represents a power of 60. For example, the sexagesimal number 1,2,3 represents 1 × 60^2 + 2 × 60^1 + 3 × 60^0. In modern usage, a comma or space is often used to separate the positions in sexagesimal numbers to avoid confusion with decimal notation.

Conversion to Decimal

Conversion from sexagesimal to decimal and vice versa is an important aspect of working with these numbers, especially in fields like astronomy and navigation. The process involves multiplying each digit by its corresponding power of 60 and summing the results for conversion to decimal. The reverse process requires division and the calculation of remainders.

Modern Usage

While the sexagesimal system is largely historical, its influence persists in the way we measure time, angles, and geographic positions. Its use in these contexts is a testament to the system's utility and the enduring legacy of ancient Mesopotamian mathematics and astronomy.

See Also

• Numeral system
• Babylonian mathematics
• Time
• Angle
• Geographic coordinate system

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