The number 60 holds a special significance in mathematics, timekeeping, geography, and culture. Many systems of counting and measuring are based on the number 60 or divisibles of 60 because it has so many factors. This allows for easy division into fractions. In this article, we’ll explore the many unique properties of this number and how it came to dominate various aspects of civilization.
Why is 60 important in math?
The number 60 is a highly composite number, meaning it has more divisors than most other numbers. The factors of 60 are 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, and 60. This abundance of factors makes 60 a useful base for counting systems and calendars. Other significant qualities of 60 are:
- It is the smallest number divisible by the numbers 1 to 6.
- It is the smallest number divisible by all one-digit numbers from 1 to 9 except 7.
- It is a superior highly composite number, meaning no other number under 60 has as many divisors.
- The sum of its divisors (1 + 2 + 3 + 4 + 5 + 6 + 10 + 12 + 15 + 20 + 30 + 60) is equal to 120. Numbers with this property are known as perfect highly composite numbers.
The multitude of factors in 60 allows it to be halved, thirded, and quartered without leaving a fraction – a useful property for weights and measures. Its division by 2, 3, 4, 5, and 6 creates the common fractions 1⁄2, 1⁄3, 1⁄4, 1/5, and 1/6. These fractions enabled early civilizations to easily divide commodities like grain into portions for trading and taxation.
Use of 60 in ancient civilizations
The Babylonians were among the first to adopt 60 as a base for mathematics and timekeeping. Their number system was sexagesimal (base 60) rather than decimal (base 10). Some theories suggest this is because 60 is the lowest number divisible by the first six counting numbers (1 to 6), making it useful for commerce and calculations. The Babylonians used 60 to create fractions and preferred 60 over 10 because of the many divisors.
The Sumerians and Babylonians divided circles into 360 degrees. This may have been because 360 is approximately the number of days in a year, and 60 divides 360 evenly into 6 parts of 60 degrees each. Later Greek mathematicians used the Babylonian sexagesimal system and further divided the degree into 60 minutes and the minute into 60 seconds.
The ancient Chinese also used a sexagesimal cycle of 60 days to keep their calendar aligned with the seasons. China designated every 60th day as a festival day.
The Mayans used base 20 (vigesimal) for their Long Count calendar. But they also tracked a sacred year of 260 days (13 x 20). The multiples of 13 and 20 as cycle lengths allowed their calendar systems to regularly realign.
Origins of timekeeping in base 60
Civilizations settled on units for timekeeping that divide into many factors because this created calendars aligned with solar and lunar cycles. While the Egyptians and ancient Europeans used base 10 and counted in decades, the Mesopotamian culture settled on base 60 early on. Here are some theories as to why:
- The Mesopotamians observed 12 lunar cycles per solar year. Each cycle was roughly 30 days. 12 x 30 = 360. So their number system needed to be highly factorable into at least 12 parts.
- A sexagesimal system allows for easy quartering (dividing by 4) related to the phases of the moon.
- The number 360 has at least 12 factors (2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20). The next multiple of 12, 420, has only 11 factors.
- When counting on fingers, 60 is the first highly composite number. Counting 12 knuckles on one hand by 5 digits on the other hand gets you to 60.
By 1500 BCE, Babylonian astronomers used base 60 for calculations, which survived in astronomy and timekeeping thereafter. The key subdivisions of time – the minute divided into 60 seconds, and the hour into 60 minutes – stem from the sexagesimal system.
Use of base 60 units today
Remnants of the utility of base 60 systems are still evident today in both timekeeping and the measurement of spatial angles:
- There are 60 seconds in a minute and 60 minutes in an hour.
- There are 360 degrees in a circle, with each degree divided into 60 arcminutes and 1 arcminute divided into 60 arcseconds.
Interestingly, circles play a role in setting the divisions of time. The daily rotation of the Earth led to dividing the day into 24 hours. The monthly orbit of the moon led to divisions by 7 (days per week) and 28-31 (days per month). And the annual orbit around the sun divided seasons into quarters of 3 months each.
So the interplay of factors 2, 3, 4, 7, 12, and 28-31 from astronomical cycles is why we wind up with:
- 24 hours in a day
- 60 minutes per hour
- 7 days per week
- 28-31 days per month
- 4 seasons per year
- 12 months per year
The ancient astronomers and mathematicians settled on divisions that allowed the various cycles to synchronize over time. And base 60 proved optimal for dividing by many astronomical factors.
60 in geography and navigation
Base 60 was useful in geography and navigation because of the many ways it divides into fractions. The key examples are:
- Each degree of latitude is divided into 60 nautical miles.
- Each nautical mile contains 8000 feet, making it easy to work with distances in miles and feet measurements.
- Longitude lines are drawn at 6 degree spacings around the globe. So there are 60 longitude lines from the 0° line in Greenwich to the 180° line on the opposite side of Earth.
The French Cassini maps of France in the 17th and 18th centuries used base 60 heavily. These professional maps divided the realm into triangles with sides approximately 60 French miles. Surveyors found base 60 convenient for dividing these triangles into subunits for mapping terrain and settlements accurately.
Use of 60 in other cultures
The pervasive use of base 60 timekeeping and spatial divisions influenced many other cultures as these systems spread:
- In ancient India, Hindu texts described time units down to 60 atom-sized moments called truti. Traditional Hindi and Vedic calendars used 60-year cycles.
- Ancient Greek astronomers like Hipparchus and Ptolemy inherited the Babylonian sexagesimal system which influenced Greek mathematics and science.
- The 60-year traditional calendar cycle in China and the East Asian zodiac has its roots in the ancient use of base 60 counting.
- In Mesoamerica, the Aztecs and Mayans used interlocking calendar cycles based on days, 13-day “weeks”, 20-day months, and ritual years of 260 days – reflecting the widespread ritual significance given to base 60 time cycles.
Special properties of 60
Let’s recap some of the mathematical properties that make 60 extraordinary:
- It’s the smallest number divisible by 1 through 6.
- It has 12 factors, more than any number under 60.
- It’s a highly composite number, superior highly composite number, and perfect highly composite number.
- It has numerous divisors – 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, and 60 – useful for fractions.
- It’s the lowest number divisible by all one-digit numbers from 1 to 9 except 7.
- It has abundant fractions thanks to many divisors – halves, thirds, quarters, fifths, sixths, tenths, twelfths, etc.
In addition, some innovative civilizations discovered useful cycles that involved 60, such as:
- 60 seconds per minute and 60 minutes per hour.
- A circle has 360 degrees, with 60 arcminutes per degree.
- The Maya and Aztec ritual calendar cycles of 260 days (13 x 20) and 52 years (13 x 4).
- Ancient Hindu time cycles of 60 atom-sized moments and 60 year cycles.
These unique mathematical properties made base 60 privileged historically as a highly useful system for counting, calendars, timekeeping, geometry, navigation, mapping, and astronomy.
Why 60 was favored over other numbers
The fact that 60 has so many divisors made it better suited to early counting and calendaring needs compared to other small numbers. Consider these drawbacks of other potential base numbers:
- Base 10 has only four divisors (1, 2, 5, 10) – too few for calendaring cycles.
- Base 12 has six divisors (1, 2, 3, 4, 6, 12) – decent but less than 60.
- Base 20 has six divisors (1, 2, 4, 5, 10, 20) – limited divisibility.
- Base 30 has eight divisors (1, 2, 3, 5, 6, 10, 15, 30) – more than 12 or 20 but less than 60.
Hence, base 60 offered the optimal blend of small size relative to other highly composite numbers and significantly more divisors than the closest competitors like 10, 12, 20, and 30. That made all the difference in its adoption for key applications like ancient counting, calendaring, astronomy, timekeeping, and geometry.
Lasting influence of 60-based systems
The entrepreneurial Mesopotamian civilizations that first developed base 60 mathematics gifted this amazingly useful number system to the world. Though decimal counting became widespread over time, vestiges of the versatile utility of base 60 remain with us today:
- We still use 60 seconds and minutes for timekeeping.
- Geographic coordinates are based on the 360 degrees of a circle, with each degree divided into 60 minutes and 60 seconds.
- Nautical miles of 60 to a degree facilitate navigation and measurement.
- The traditional 60-year cycle persists in some Asian calendars.
So today’s clocks, maps, GPS, and schedules retain evidence of the brilliance of the ancient mathematicians and astronomers who pioneered the use of this handy number to tackle problems in their world. Base 60 continues to influence how we chart time and space in the modern world.
The number 60 enjoys a special status across many fields thanks to its highly composite nature and multitude of factors. It provides versatility for counting, dividing, multiplying, measuring angles, timekeeping, and calendaring. Sexagesimal or base 60 systems profoundly shaped history, originating with the Babylonians’ innovative mathematics and astronomical observations. The lasting remnants in our systems for time, geometry, navigation, and cartography are a testament to 60’s underappreciated but ongoing contributions to science and civilization.