What Is Base-60? — The Sexagesimal System Behind Every Clock
Base-60 (sexagesimal) is a number system that groups quantities in powers of 60 instead of powers of 10. Originated by the ancient Sumerians around 3000 BCE, it is the reason every clock divides hours into 60 minutes and minutes into 60 seconds. Base-60 also underpins the 360-degree circle and geographic coordinates used in navigation worldwide.
What Does Base-60 Mean?
Base-60, also known as the sexagesimal system, is a positional number system where each digit position represents a power of 60 — the same principle that makes 60 seconds equal 1 minute and 60 minutes equal 1 hour on every clock.
Base-60 (also called the sexagesimal system) is a positional numeral system that uses sixty as its fundamental grouping number. Just as the familiar decimal system groups quantities in powers of 10 (ones, tens, hundreds), the sexagesimal system groups quantities in powers of 60 (ones, sixties, thirty-six-hundreds).
Every clock on Earth uses base-60 arithmetic: 60 seconds make a minute, and 60 minutes make an hour.
Base-60 is not a relic of the past — it is the active, living number system behind every stopwatch, navigation chart, and geographic coordinate on the planet. Whenever you read "2 hours, 45 minutes, 30 seconds," you are reading a three-digit base-60 number.
Also called: Sexagesimal system. Origin: Ancient Sumerians, ~3000 BCE, southern Mesopotamia. Refined by: Babylonians, ~2000 BCE (positional notation). Why 60: Has 12 divisors (1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60) — more than any smaller number. Used today for: Timekeeping (60 s/min, 60 min/hr), angles (360°), and geographic coordinates (DMS). International standard: ISO 8601 (HH:MM:SS format).
In base-10 (decimal), each position is worth ten times the position to its right. The number 125 means (1 × 100) + (2 × 10) + (5 × 1). In base-60 (sexagesimal), each position is worth sixty times the position to its right. So the time 2:05:30 means (2 × 3,600) + (5 × 60) + (30 × 1) = 7,530 seconds.
Modern notation separates base-60 digits with colons. You already read this notation instinctively when you glance at a clock — 1:30:00 is one sixty-group-of-minutes, thirty single-minutes, and zero seconds. That colon is the base-60 equivalent of the gap between digits in a decimal number.
What Are the Powers of 60?
| Position | Power | Decimal Value | Time Equivalent |
|---|---|---|---|
| 1st (rightmost) | 60⁰ | 1 | 1 second |
| 2nd | 60¹ | 60 | 1 minute |
| 3rd | 60² | 3,600 | 1 hour |
| 4th | 60³ | 216,000 | 60 hours |
Why Did the Sumerians Use Base-60?
The Sumerians chose 60 because it has 12 divisors — more than any smaller number — making it exceptionally easy to divide quantities into halves, thirds, quarters, fifths, and sixths without producing awkward fractions.
The Sumerians, who built the world's earliest known civilization in southern Mesopotamia (modern-day Iraq) during the 3rd millennium BCE, are the originators of the base-60 system. But why choose sixty over a seemingly simpler base like ten?
Why Does 60 Have So Many Divisors?
The number 60 is a superior highly composite number — it has more divisors relative to its size than any smaller positive integer. Its twelve factors are 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, and 60. By contrast, the number 10 has only four factors: 1, 2, 5, and 10.
This rich divisibility means common fractions terminate neatly in base-60. One-half is simply 30/60, one-third is 20/60, one-quarter is 15/60, one-fifth is 12/60, and one-sixth is 10/60. In base-10, one-third becomes the repeating decimal 0.333…, which is awkward for practical trade and measurement. For a civilization dividing harvests, water allotments, and land parcels, a base that makes equal-sharing easy was far more practical than a base that creates recurring fractions.
| Property | 60 (Base-60) | 10 (Base-10) |
|---|---|---|
| Total divisors | 12 | 4 |
| Divisors list | 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60 | 1, 2, 5, 10 |
| 1/3 representation | 0;20 (terminates) | 0.333… (repeats) |
| 1/6 representation | 0;10 (terminates) | 0.1666… (repeats) |
| Smallest divisible by 1–6 | Yes | No (not by 3, 4, or 6) |
How Did Ancient People Count to 60 on Their Hands?
One widely cited hypothesis for why sixty was chosen involves a counting technique still used in parts of the Middle East today. Using the thumb as a pointer, you can count the twelve knuckle-segments (three per finger) on the four non-thumb fingers of one hand. Multiplying those twelve segments by the five fingers of the other hand — used to tally completed groups — yields 5 × 12 = 60. This theory, while not definitively proven, elegantly explains how an illiterate population could have settled on such a large base without written arithmetic.
Sixty is the smallest number that is evenly divisible by every integer from 1 through 6 — mathematically, it is the least common multiple of 1, 2, 3, 4, 5, and 6. Separately, 60 is also a "highly composite" number, meaning it has more divisors than any smaller positive integer. Together, these properties made it extraordinarily useful for ancient commerce, where goods regularly needed to be split into halves, thirds, quarters, fifths, and sixths.
How Did Base-60 Evolve From Sumerians to Babylonians to Today?
Base-60 traveled from Sumerian scribes (~3000 BCE) to Babylonian mathematicians (~2000 BCE), who invented positional notation, then to Greek astronomers, medieval scholars, and finally into the ISO 8601 standard that governs digital timekeeping worldwide.
The base-60 system has traveled a remarkable path across more than four millennia, passing through several civilizations before embedding itself permanently in modern timekeeping and geometry.
In southern Mesopotamia, Sumerian scribes develop a numeral system built on groupings of 60 for trade, taxation, and astronomical record-keeping. Early tablets from Uruk and Ur show sexagesimal accounting records.
After inheriting Sumerian culture, the Babylonians develop a true positional number system — the first in history — where a digit's place determines its value, just like our modern place-value system. This made multiplication, division, and astronomical prediction far more efficient.
Hipparchus and later Claudius Ptolemy adopt Babylonian base-60 notation for astronomical tables. Ptolemy's Almagest (c. 150 CE) uses sexagesimal fractions extensively, transmitting the system into Western and Islamic scientific traditions.
Islamic and later European scholars continue to use sexagesimal fractions in astronomy and navigation. The Latin terms pars minuta prima (first small part) and pars minuta secunda (second small part) give us the modern words "minute" and "second."
By the time Pope Gregory XIII reforms the calendar, the 60-second minute and 60-minute hour are already universally accepted across Europe. Mechanical clocks solidify the convention, and the base-60 structure of time becomes permanent.
The international standard ISO 8601 formalizes time representation as HH:MM:SS — a direct descendant of Sumerian base-60 arithmetic. Every smartphone, GPS unit, and stock exchange timestamp uses this format.
Revolutionary France tried. In 1793 the French Republic introduced a 10-hour day with 100 minutes per hour and 100 seconds per minute. It lasted only 17 months. The experiment failed because base-60 time was already too deeply woven into navigation, astronomy, and daily life — and because 60's superior divisibility made it genuinely more practical for dividing hours into useful fractions.
How Is Base-60 Used in Time Calculation?
Base-60 governs all time arithmetic: when adding or subtracting hours, minutes, and seconds, you must carry or borrow at 60 (not 10), which is why 75 minutes becomes 1 hour and 15 minutes rather than staying as "75."
Every time you add, subtract, or convert time values, you perform base-60 arithmetic. The rules are simple but differ from the decimal math most people are comfortable with, and overlooking the carry-over rules is the single most common source of time-calculation errors.
How Does the Base-60 Carry-Over Rule Work?
In base-10, you carry when a column exceeds 9. In base-60 time arithmetic, you carry when seconds or minutes exceed 59. The process works identically to decimal carrying, just with a different threshold:
Notice that 75 seconds becomes 1 minute and 15 seconds (75 − 60 = 15, carry 1), and 76 minutes becomes 1 hour and 16 minutes (76 − 60 = 16, carry 1). If you mistakenly treat time columns like decimal columns and write "75 seconds" as the answer, you have introduced an error. Use the Time Addition Calculator to practice carry-over calculations hands-on, or review the detailed time addition formulas for the underlying mathematics.
Why Do Decimal Shortcuts Fail With Time?
A common payroll mistake is entering "1.5 hours" to mean "1 hour 30 minutes" and then adding it as a decimal. While 1.5 decimal hours does equal 90 minutes, the relationship breaks down with less tidy values: 1 hour 45 minutes is 1.75 in decimal hours, not 1.45. Confusing the two formats leads to underpayments or overpayments. For guidance on avoiding this pitfall in work contexts, see our guide to calculating work hours.
Treating minutes as decimal fractions of an hour. Remember: 0.50 hours = 30 minutes, but 0:50 on a clock = 50 minutes. Always confirm whether a value is in decimal-hours or hours-and-minutes format before calculating.
How Is Base-60 Used in Angles and Coordinates?
Base-60 defines how angles are measured (360 degrees = 6 × 60, each degree split into 60 arcminutes and 60 arcseconds) and how geographic coordinates express latitude and longitude in degrees-minutes-seconds (DMS) format.
Time is not the only domain where base-60 thrives. Angular measurement and geographic navigation both depend on the same sexagesimal structure, inherited from the same Sumerian and Babylonian tradition.
Why Does a Circle Have 360 Degrees?
A full circle contains 360 degrees — and 360 = 6 × 60. This is not a coincidence. Babylonian astronomers, who tracked celestial movements using their base-60 system, observed that the sun appeared to move roughly one degree per day across the sky (the year being approximately 360 days). Choosing 360 for a full circle tied angular measurement neatly to their existing numeral system and to the calendar.
Like the number 60, 360 is remarkably divisible: it has 24 factors, allowing a circle to be cleanly split into halves (180°), thirds (120°), quarters (90°), fifths (72°), sixths (60°), eighths (45°), tenths (36°), and twelfths (30°). This flexibility is why 360 degrees has survived every attempt to replace it.
What Are Degrees, Arcminutes, and Arcseconds?
Each degree is subdivided into 60 arcminutes (′), and each arcminute into 60 arcseconds (″) — a direct application of base-60 subdivision. When a surveyor records a bearing as 47° 23′ 15″, they are writing a three-place base-60 number, just as a clock showing 2:23:15 does for time.
How Does Base-60 Appear in GPS Coordinates?
Latitude and longitude use the same degrees-minutes-seconds (DMS) format. For example, the NIST headquarters in Boulder, Colorado, sits at approximately 39° 59′ 45″ N, 105° 15′ 45″ W. Modern GPS devices often display decimal degrees instead, but the underlying grid and many aviation, maritime, and military applications still use DMS — sexagesimal notation in action.
Earth rotates 360° in 24 hours, which means it turns 15° per hour (360 ÷ 24 = 15). This relationship — linking base-60 time to base-60 angular measurement — is the mathematical foundation of time zones.
Base-60 vs Base-10 vs Base-12: How Do They Compare?
Base-10 has 4 divisors and dominates everyday math; base-12 has 6 divisors and survives in dozens and inches; base-60 has 12 divisors — more than either — which is why it endures wherever precise, even subdivision matters most: time, angles, and navigation.
Humanity has experimented with many number bases throughout history. Base-10, base-12, and base-60 are the three that left the deepest marks on modern life. Each has distinct strengths rooted in its mathematical properties.
| Feature | Base-10 (Decimal) | Base-12 (Duodecimal) | Base-60 (Sexagesimal) |
|---|---|---|---|
| Origin | Finger counting (10 digits) | Knuckle counting (12 per hand) | Sumerians, 3rd millennium BCE |
| Number of divisors | 4 (1, 2, 5, 10) | 6 (1, 2, 3, 4, 6, 12) | 12 (1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60) |
| 1/3 representation | 0.333… (repeating) | 0.4 (terminates) | 0;20 (terminates) |
| 1/7 representation | 0.142857… (repeating) | 0.186A35… (repeating) | 0;8,34,17… (repeating) |
| Modern use | General mathematics, currency, science | Dozens, inches per foot, months per year | Time, angles, geographic coordinates |
| Unique symbols needed | 10 (0–9) | 12 (0–9 + two extra) | 60 (historically written using sub-groups of 10) |
| Ease of learning | Very easy (familiar) | Moderate | Harder to memorize, but positional notation keeps it manageable |
| Divisibility advantage | Low | Medium | High — best for equal-part division |
Base-10 dominates general-purpose mathematics because it aligns with human finger counting and is entrenched in global education and commerce. Base-12 survives in units like dozens, feet-and-inches, and the 12-month year, valued for its better divisibility than ten. Base-60, with its unmatched factor count, persists wherever precise subdivision matters most — time, navigation, and astronomy — because no other common base splits as evenly into as many useful parts.
For deeper exploration of how these bases affect practical time arithmetic, see our Elapsed Time Calculator, which handles conversion between all three representation styles.
Try It: Base-10 to Base-60 Converter
Frequently Asked Questions About Base-60
Base-60 (also called sexagesimal) is a number system that uses 60 as its base, rather than the 10 we normally count with. Every time you read a clock — 60 seconds in a minute, 60 minutes in an hour — you are using base-60 arithmetic that dates back over 4,000 years to ancient Sumer.
Time uses base-60 because the ancient Sumerians chose 60 for its exceptional divisibility. The number 60 can be evenly divided by 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, and 60 — a total of 12 factors. This makes splitting time into halves, thirds, quarters, fifths, and sixths easy without messy fractions, which was critical for early astronomy and trade.
The base-60 system originated with the ancient Sumerians in the 3rd millennium BCE in southern Mesopotamia (modern-day Iraq). The Babylonians later inherited and refined it around 2000 BCE by developing a true positional notation, making complex calculations more practical. For further reading, see the Wikipedia article on the Sexagesimal system or Britannica's overview.
A circle has 360 degrees because 360 equals 6 × 60, tying directly to the Sumerian base-60 system. This number was also convenient because ancient astronomers estimated roughly 360 days in a year. Like 60, the number 360 has many divisors (24 in total), making it easy to divide a circle into halves, thirds, quarters, fifths, sixths, eighths, tenths, and twelfths.
In base-10 (decimal), each digit position represents a power of 10 (1, 10, 100, 1000…), and you carry over at 10. In base-60 (sexagesimal), each position represents a power of 60 (1, 60, 3,600…), and you carry over at 60. Base-60 has far more divisors, making it superior for dividing quantities evenly, which is why it endures in timekeeping and angular measurement.
Yes, base-60 is still used every day worldwide. Timekeeping (60 seconds per minute, 60 minutes per hour), angular measurement (360 degrees, with each degree divided into 60 arcminutes and each arcminute into 60 arcseconds), and geographic coordinates (latitude and longitude in degrees-minutes-seconds) all rely on base-60.
You count from 0 to 59 in the first position, just like counting 0 to 9 in base-10. When you reach 60, you reset the first position to 0 and increment the next position to 1. So 61 in base-10 is written as 1:01 in base-60, and 3,661 in base-10 is 1:01:01 in base-60 (1 × 3,600 + 1 × 60 + 1). This is exactly how clock time works: 1 hour, 1 minute, 1 second.
Sixty is a superior highly composite number, meaning it has more divisors relative to its size than any smaller number. With 12 factors (1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60), it is the smallest number divisible by every integer from 1 through 6. This property makes base-60 uniquely practical for dividing quantities into equal parts — a key reason the Sumerians adopted it and it persists in modern timekeeping.
Base-60 means you cannot simply add or subtract time values as regular decimal numbers. When seconds exceed 59, you must carry 1 into the minutes column. When minutes exceed 59, you carry 1 into the hours column. For example, 45 minutes + 30 minutes = 75 minutes, but in base-60 you write this as 1 hour and 15 minutes. Forgetting this carry-over is the most common time-math mistake. Try it yourself with our Time Addition Calculator.
Before adopting base-60, early Sumerian communities used a mix of counting systems, including base-10 (for everyday counting) and base-12 (possibly from counting finger knuckles). Scholars believe base-60 may have emerged from merging these two systems, since 5 × 12 = 60. Simpler tally systems and token-based counting preceded formal numeral systems entirely.
There are 60 seconds in a minute because medieval scholars subdivided the hour using the Babylonian base-60 system. The Latin term pars minuta prima ("first small part") gave us the word "minute," and pars minuta secunda ("second small part") gave us "second." Each level divides by 60, preserving the sexagesimal tradition that began with the Sumerians over 4,000 years ago.
Sexagesimal means "relating to or based on the number 60." It comes from the Latin sexagesimus, meaning "sixtieth." In mathematics, a sexagesimal system is any numeral system that uses 60 as its base. The most familiar modern example is timekeeping, where 60 seconds make a minute and 60 minutes make an hour.
The 24-hour day comes from ancient Egypt, where the night was divided into 12 periods (marked by star risings) and the day into 12 periods. The number 12 connects to base-60 because 60 ÷ 5 = 12, and both 12 and 60 share the same divisibility advantages. The Babylonians later combined the Egyptian 24-hour framework with their own base-60 subdivisions, giving us 24 hours of 60 minutes each.
To convert a base-60 value like 2:15:30 (hours:minutes:seconds) to base-10 seconds, multiply each position by its power of 60 and add them: (2 × 3,600) + (15 × 60) + (30 × 1) = 7,200 + 900 + 30 = 8,130 seconds. You can try this instantly with the converter above or our Time Addition Calculator.
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This article was researched and written by the editorial team at TimeAdditionCalculator.com. All historical claims are cross-referenced against peer-reviewed academic sources, including entries in Encyclopædia Britannica and Wikipedia's Sexagesimal article (with primary citations verified). Mathematical examples are programmatically tested. Time-calculation references comply with ISO 8601. If you find an error, please contact us.
Last updated: February 8, 2026