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Babylonian Numbers โ€” The Sexagesimal Legacy

Convert between decimal and base-60 notation. Discover why we have 60 seconds in a minute and 360 degrees in a circle โ€” gifts from ancient Mesopotamia.

Concept Fundamentals
60
Base
2
Basic Symbols
~2000 BCE
Origin
360ยฐ
Circle

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60 has 12 divisors, making fractions easier than base-10. Plimpton 322 tablet (~1800 BCE) contains Pythagorean triples in sexagesimal. Our 60 sec/min and 360ยฐ circle come directly from Babylonian mathematics.

Key quantities
60
Base
Key relation
2
Basic Symbols
Key relation
~2000 BCE
Origin
Key relation
360ยฐ
Circle
Key relation

Ready to run the numbers?

Why: Babylonian numerals were used in ancient Mesopotamia for astronomy, commerce, and mathematics. Their base-60 system survives today in time and angles.

How: Divide by 60 repeatedly for decimalโ†’Babylonian; multiply each digit by its place value (60^position) and sum for Babylonianโ†’decimal.

60 has 12 divisors, making fractions easier than base-10.Plimpton 322 tablet (~1800 BCE) contains Pythagorean triples in sexagesimal.
Sources:Britannica โ€” Babylonian NumeralsWolfram MathWorld โ€” Sexagesimal

Run the calculator when you are ready.

Start ConvertingDecimal โ†” Sexagesimal
๐Ÿ“œ
ANCIENT MATHEMATICSMesopotamian Cuneiform

Babylonian Numbers โ€” The Sexagesimal Legacy

Convert between decimal and base-60 notation. Discover why we have 60 seconds in a minute and 360 degrees in a circle โ€” gifts from ancient Mesopotamia.

Mayan Numerals โ†’Roman Numerals โ†’

Conversion Direction

Input Value

For educational and informational purposes only. Verify with a qualified professional.

๐Ÿงฎ Fascinating Math Facts

โฑ๏ธ

60 seconds in a minute and 60 minutes in an hour come from the Babylonian sexagesimal system

โ€” Timekeeping

๐Ÿ“

The 360-degree circle equals 6 ร— 60 โ€” a Babylonian invention still used today

โ€” Geometry

๐Ÿ“‹ Key Takeaways

  • โ€ข Babylonian numbers use base-60 (sexagesimal), unlike our base-10 decimal system.
  • โ€ข Only two symbols: vertical wedge (๐’น) for 1 and chevron (๐’Œ‹) for 10.
  • โ€ข Each position represents a power of 60; place value increases left to right.
  • โ€ข Our 60 seconds/minute and 60 minutes/hour come from the Babylonian system.
  • โ€ข The 360-degree circle (6 ร— 60) is another Babylonian legacy.

๐Ÿ’ก Did You Know?

โฑ๏ธ60 seconds in a minute and 60 minutes in an hour come directly from the Babylonian sexagesimal systemSource: Timekeeping
๐Ÿ“The 360-degree circle equals 6 ร— 60 โ€” a Babylonian invention still used todaySource: Geometry
๐Ÿ“œPlimpton 322 tablet (~1800 BCE) contains Pythagorean triples in sexagesimal notationSource: Math History
๐Ÿ”ข60 has 12 divisors (1,2,3,4,5,6,10,12,15,20,30,60), making fractions easierSource: Number Theory
๐’นBabylonians used only two cuneiform symbols: ๐’น for 1 and ๐’Œ‹ for 10Source: Cuneiform
๐Ÿ“ŠBabylonians calculated โˆš2 as 1;24,51,10 โ€” accurate to 6 decimal placesSource: Precision
๐ŸŒSexagesimal survives in geographic coordinates (lat/long) and nautical navigationSource: Navigation

๐Ÿ“– How It Works

To convert decimal to Babylonian: repeatedly divide by 60 and collect remainders in reverse order. To convert Babylonian to decimal: multiply each digit by its place value (60^position) and sum.

Example: 150 in decimal = 2ร—60 + 30 = 2,30 in Babylonian. Use commas to separate sexagesimal digits.

๐ŸŽฏ Expert Tips

๐Ÿ’ก Use Commas or Semicolons

Babylonian notation uses 1,30 or 1;30 โ€” both are valid separators.

๐Ÿ’ก Digits 0โ€“59 Only

Each sexagesimal digit must be between 0 and 59.

๐Ÿ’ก No Negative Numbers

Ancient Babylonians did not use negative numbers.

๐Ÿ’ก Fractions

1,30 can mean 1 + 30/60 = 1.5 in decimal.

โš–๏ธ Comparison Table

DecimalBabylonian
601,0
1001,40
36001,0,0
1502,30

โ“ Frequently Asked Questions

Why did Babylonians use base-60?

60 has many divisors (1,2,3,4,5,6,10,12,15,20,30,60), making fraction calculations easier. It may have merged earlier base-10 and base-6 systems.

How did Babylonians write fractions?

Using the same sexagesimal system: 1,30 could mean 1 + 30/60 = 1.5 in decimal.

Did Babylonians have zero?

Early Babylonian math lacked zero. By ~700 BCE they used a spacer for empty positions, but not at the end of numbers.

How accurate were Babylonian calculations?

Very accurate. They calculated โˆš2 as 1;24,51,10 (โ‰ˆ1.41421296), correct to 6 decimal places.

What is cuneiform?

Wedge-shaped writing made by pressing a reed stylus into clay. The name comes from Latin cuneus meaning "wedge".

Where is sexagesimal used today?

Time (60 sec/min, 60 min/hr), angles (360ยฐ = 6ร—60), and geographic coordinates.

๐Ÿ“Š Key Stats

60
Base
2
Basic Symbols
~2000 BCE
Origin
360ยฐ
Circle

โš ๏ธ Disclaimer: This calculator provides educational conversions. For historical research, consult academic sources. Babylonian notation can have multiple interpretations for fractional values.

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