Babylonian Numerals Converter
Convert numbers to Babylonian base-60 numerals and back, fractions included, and see them drawn as cuneiform wedges. Uses the modern notation in which √2 from tablet YBC 7289 is 1;24,51,10.
How to Use
- Choose decimal to Babylonian or Babylonian to decimal.
- Type a number such as 2026 or 1/7, or sexagesimal digits such as 33,46 or 1;24,51,10.
- Read the sexagesimal digits, the decimal value and whether the fraction ends or repeats.
- The drawing shows each digit in cuneiform: a corner wedge for each ten, an upright wedge for each unit.
- Show Work divides by 60 for the whole part and multiplies by 60 for the fraction.
Worked Example
2026 in base 60. 2026 ÷ 60 = 33 remainder 46, so the digits are 33 and 46: 33,46. In cuneiform, 33 is three corner wedges (tens) and three upright wedges (units); 46 is four corner wedges and six upright wedges. Check: 33 × 60 + 46 = 2,026.
Reading √2 from YBC 7289. The tablet shows 1;24,51,10. That is 1 + 24/60 + 51/3,600 + 10/216,000 = 1 + 0.4 + 0.0141667 + 0.0000463 = 1.4142130, against the true √2 = 1.4142136, about 3,800 years ago.
The common mistake: reading the digits as decimal. 1,24,51,10 is not one million and something: each comma-separated group is one base-60 digit worth 60 times the next. And without the semicolon the scribes’ own writing was ambiguous: the same wedges could mean 1;24 or 1,24 (84).
Show Work
Formulas
Wedges in Clay
The Sumerians and then the Babylonians wrote by pressing a reed stylus into wet clay, and by about 2000 BC their scribes used a true place-value system in base 60, the oldest known. Thousands of school and scholarly tablets survive: multiplication tables, tables of reciprocals for division, and problem texts. Plimpton 322 lists sides of right triangles, and YBC 7289, a small round school tablet at Yale, gives √2 to the equivalent of six decimal places.
Division was done by multiplying by a reciprocal from a table, which is why the scribes favoured “regular” numbers whose reciprocals end in base 60. Greek astronomers took over sexagesimal fractions for their calculations, and through them came our 60 minutes and 60 seconds.
About This Calculator
This converter writes whole numbers, decimals and fractions in base 60 using the modern scholarly notation, exactly, marking repeating sexagesimal fractions, and reads such notation back. It draws each digit as the Babylonians did, with corner wedges for tens and upright wedges for units, and shows the divisions and multiplications.
Everything runs in your browser; nothing is sent anywhere. Empty places are drawn as ∅ for clarity, though Babylonian scribes simply left a gap.
Related calculators: Maya Numerals, Roman Numeral Converter, and Mixed Radix Converter.
Frequently Asked Questions
How did Babylonian numbers work?
They were written in base 60 with just two signs pressed into clay: an upright wedge for 1 and a corner wedge for 10. A digit from 1 to 59 is a cluster of up to five tens and nine units, and the clusters are placed side by side like our digits, each worth 60 times the next. 2026 is 33 sixties and 46 units: 33,46.
What does the notation 1;24,51,10 mean?
It is how historians write base-60 numbers today: commas separate the digits and a semicolon marks where the whole part ends. 1;24,51,10 is 1 + 24/60 + 51/3,600 + 10/216,000 = 1.41421296…, the value of √2 on the tablet YBC 7289, correct to about six decimal places.
Did the Babylonians have a zero?
Not at first. An empty place was just a gap, and the scribes did not mark where the whole part ended, so the same wedges could mean 1, 60 or 1/60 and context decided. By the Seleucid period (after about 300 BC) a separator sign was used for an empty place inside a number, one of the earliest zeros.
Why base 60?
Probably because 60 divides by 1, 2, 3, 4, 5, 6, 10, 12, 15, 20 and 30, so many fractions are exact: 1/3 = 0;20, 1/4 = 0;15 and 1/5 = 0;12. Base 60 makes division by most small numbers easy, which mattered for trade, astronomy and land measurement.
Do we still use base 60?
Yes, every day: 60 seconds in a minute, 60 minutes in an hour, and 60 minutes of arc in a degree all come from Babylonian astronomy by way of Greek astronomers such as Ptolemy. Writing 2:26:40 for a time is sexagesimal notation.
How do I use the Babylonian Numerals Converter?
Just type or paste your value. The answer shows up right away — there is no button to press. Change anything and it updates by itself.
Do I need to install or sign up for anything?
Not at all — it runs in the browser with nothing to install and no account. After it loads once, it even works without an internet connection.
Is my information private?
Yes. Everything happens in your browser. Nothing you type is sent to a server or saved anywhere.
Common Use Cases
Years
2026 = 33,46: 33 sixties and 46.
Ancient mathematics
YBC 7289’s √2, 1;24,51,10, is 1.414212963.
Fractions
1/3 = 0;20 and 1/8 = 0;7,30 end; 1/7 = 0;8,34,17 repeats.
Time and angles
3,600 = 1,0,0: one hour of 60 minutes of 60 seconds.
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