Babylonian Sexagesimal Number Converter

Decode ancient Mesopotamian base-60 mathematics — the system behind our 60-second minutes and 360° circles

Convert Numbers

Enter a decimal (base-10) number

Enter any positive integer up to 999,999,999

Please enter a valid positive integer. Include fractional part?

No Yes (enter decimal number)

Sexagesimal (base-60) Representation

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Cuneiform Visual Approximation:

Place	Power of 60	Value of Place	Digit (0–59)	Contribution

Enter sexagesimal digits (separated by commas or spaces)

Enter each base-60 digit (0–59) separated by commas or spaces. E.g. "1 0 0" = 3,600. For fractional parts, use a semicolon: "1 0; 30" means 60 + 30/60.

Each digit must be between 0 and 59.

Decimal (base-10) Equivalent

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Position	Digit	× Power of 60	= Value

The Babylonians used two cuneiform wedge marks to write all numbers 1–59. The vertical wedge 𒁹 = 1, the horizontal corner wedge 𒐐 = 10.

Decimal	Cuneiform Symbols	How it works

Historical Context

The Babylonian sexagesimal (base-60) number system emerged in Mesopotamia around 2000–1800 BCE, building on earlier Sumerian traditions. Babylonian scribes pressed a reed stylus into soft clay tablets to create wedge-shaped marks — the origin of the term "cuneiform" (from Latin cuneus, wedge). These tablets, preserved by the very fires that burned ancient Babylon, reveal a sophisticated mathematical culture.

The system used just two symbols: a vertical wedge for units (1) and a corner wedge for tens (10). Grouped together, these formed numbers 1–59; then a new "place" began — just as 10 ones become a "tens" column in our system, 60 ones became a new column in Babylonian math. Famous tablets like Plimpton 322 (c. 1800 BCE, now at Columbia University) show Pythagorean triplets calculated in this system — nearly 1,000 years before Pythagoras was born.

🏺 Did You Know?

Our 60-minute hour, 60-second minute, and 360° circle all descend directly from the Babylonian base-60 system — making it the oldest mathematical framework still embedded in daily life.
The Babylonians had no zero symbol for centuries; context determined whether a blank space meant "empty place" or "none." A proper zero placeholder finally appeared around 300 BCE in later Babylonian texts.
The famous clay tablet YBC 7289 (Yale Babylonian Collection) shows √2 calculated in sexagesimal to 6 decimal-equivalent places of accuracy — a feat not surpassed in Europe until the Renaissance.

How to Use This Converter

Choose a conversion direction using the tabs above. In Decimal → Sexagesimal mode, enter any positive integer (or decimal fraction) and click Convert. The tool displays the sexagesimal notation in standard scholarly format (digits separated by commas, with a semicolon before fractional places), a visual cuneiform approximation using Unicode wedge characters, and a full positional breakdown table.

In Sexagesimal → Decimal mode, enter the base-60 digits separated by spaces or commas. Use a semicolon (;) to separate integer and fractional sexagesimal places. The Symbol Table tab shows how every number 1–59 is written in cuneiform.

Why This Matters

The Babylonian sexagesimal system is not merely a historical curiosity — it is the direct ancestor of how we measure time and angles today. Every time you read "3:45 PM" or set a compass to 270°, you are using a number system perfected in ancient Mesopotamia around 1800 BCE.

Why base 60? Historians believe 60 was chosen because it is highly composite — divisible by 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, and 60 — making fractions far simpler to express. One-third of 60 is a clean 20; one-quarter is 15. Try that cleanly in base-10! This made Babylonian merchants and astronomers unusually efficient at practical calculation.

Babylonian astronomers used sexagesimal math to predict lunar eclipses with remarkable accuracy. Their "saros cycle" of 223 lunar months (≈18 years, 11 days) was calculated in base-60 arithmetic. Greek astronomers inherited these methods, which passed through Islamic scholars like Al-Battani into Renaissance Europe. When Copernicus and Tycho Brahe recorded celestial angles in degrees, minutes, and seconds, they were using Babylonian notation.

How It's Calculated

Sexagesimal is a positional numeral system with base 60, exactly analogous to our decimal (base-10) system. Each position represents a successive power of 60:

Value = d₃×60³ + d₂×60² + d₁×60¹ + d₀×60⁰ = d₃×216,000 + d₂×3,600 + d₁×60 + d₀×1

For fractional parts (after the "sexagesimal point", denoted by semicolon):

Fractional Value = f₁/60 + f₂/3,600 + f₃/216,000 + ...

Converting decimal → sexagesimal: Repeatedly divide by 60 and record remainders (for integer part). For fractions, multiply by 60 and record the integer portion at each step.

Example: 3,661 ÷ 60 = 61 remainder 1. Then 61 ÷ 60 = 1 remainder 1. So 3,661 = 1,1,1 in sexagesimal (1×3,600 + 1×60 + 1×1).

Scholarly notation writes sexagesimal integers with commas between digits and a semicolon before the fractional part. For example: 2,24;30 = 2×3,600 + 24×60 + 30/60 = 8,640 + 1,440 + 0.5 = 10,080.5

Tips & Common Mistakes

Remember: digits go up to 59, not 9. In base-60, a single "digit" can be any value from 0 to 59. "1,0,0" in sexagesimal means 3,600 — not one hundred.
The semicolon is the sexagesimal point. Scholars use ";" (not ".") to separate integer and fractional places. So 1;30 = 1 + 30/60 = 1.5, not 1.3.
Beware the missing zero. Ancient Babylonian texts often left a space where we'd write 0. The number 1,0,1 (= 3,601) could look like 1,1 (= 61) without careful reading of the tablet layout.
Context was everything. Without an absolute magnitude marker, "1" could mean 1, 60, 3,600, or even 1/60. Babylonians relied on context (like the problem being about areas vs. lengths) to determine scale.
Multiplying is elegant in base-60. Try it: 0;30 × 0;30 = 0;15 (i.e., 0.5 × 0.5 = 0.25). Fractions that are ugly in base-10 are often clean in base-60.

Frequently Asked Questions

Why did the Babylonians choose base 60 instead of base 10?

The most widely accepted explanation is that 60 is the smallest number divisible by 1, 2, 3, 4, 5, and 6 — making it extremely convenient for fractions in trade, land measurement, and astronomy. Some historians also suggest it arose from combining a 5-finger counting system with a 12-segment system (counting finger segments on one hand). The Sumerians who preceded the Babylonians likely developed the base-60 foundation around 3000 BCE.

How do I read a cuneiform number on an actual clay tablet?

Cuneiform numbers are read right to left within each sexagesimal place, and each group is separated spatially. The vertical wedge 𒁹 represents 1 and the corner wedge 𒐐 represents 10. Three 𒐐 symbols plus two 𒁹 symbols = 32. Large numbers are stacked vertically in columns. The symbol table in this tool's third tab shows all values 1–59.

Is base-60 still used anywhere today?

Yes — directly and extensively. Hours are divided into 60 minutes, minutes into 60 seconds (both time and arc). A full circle is 360° (= 6 × 60). Navigation uses degrees, minutes, and seconds of latitude/longitude. Even in modern astronomy, right ascension is measured in hours (0–24) broken into minutes and seconds. The Babylonian legacy is embedded in every GPS coordinate you read.

What is the largest number this converter handles?

This tool converts integers up to 999,999,999 (about one billion), which in sexagesimal requires up to 6 places (since 60⁶ = 46,656,000,000). Babylonian tablets rarely exceeded 5–6 sexagesimal places in practice, as most calculations involved areas, commodity quantities, and astronomical periods well within that range. JavaScript's floating-point precision limits very large decimal fractions.