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The ancient Babylonians, writing in cuneiform on clay tablets nearly 4,000 years ago, used a base-60 (sexagesimal) positional number system — one of the earliest known positional systems in history, where a digit's value depends on its place, much like our own base-10 system. Remarkably, base 60 never fully disappeared: it survives today in how the modern world tells time (60 seconds per minute, 60 minutes per hour) and measures angles (360 degrees, with each degree split into 60 minutes and each minute into 60 seconds), a direct legacy of Babylonian mathematics and astronomy passed down through Greek and later scholarship. This converter shows the same positional logic the Babylonians used, but with today's Hindu-Arabic decimal digits standing in for cuneiform wedge-and-hook symbols, making the base-60 place-value structure easy to read and verify. It's a useful tool for history-of-mathematics study, for understanding why 60 was chosen as a base in the first place (it divides evenly by 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, and 30 — far more divisors than 10 has), and for anyone curious how a number like a clock time or an angle measurement would have been written thousands of years ago.
How it works
Choose a conversion direction. For decimal → Babylonian, enter a whole number; the calculator repeatedly divides it by 60, keeping track of each remainder from smallest (rightmost) to largest (leftmost) place value — exactly the same repeated-division process used to convert decimal to any other base, just with 60 as the base instead of the usual 10 or 2. For Babylonian → decimal, enter the base-60 "digits" separated by commas (each between 0 and 59); the calculator multiplies each digit by the appropriate power of 60 for its position (60⁰ for the rightmost, 60¹ for the next, 60² after that, and so on) and sums the results.
- Enter conversion direction.
- Enter decimal number (decimal → Babylonian).
- Enter babylonian digits, comma-separated (Babylonian → decimal).
- Click Calculate to see your results.
Examples
Decimal 5091 → Babylonian
5091 = 1×3600 + 24×60 + 51 (since 3600 = 60²), so the Babylonian notation is "1,24,51".
Decimal 90 → Babylonian
90 = 1×60 + 30, so the Babylonian notation is simply "1,30" — a two-digit sexagesimal number, since 90 exceeds a single base-60 place (0-59).
Babylonian "2,15,0" → decimal
2×3600 + 15×60 + 0×1 = 7200 + 900 + 0 = 8100.
Common mistakes to avoid
- Entering a base-60 digit of 60 or more, which isn't a valid single digit in base 60 — it should instead carry into the next position, the same way "10" isn't a valid single digit in base 10.
- Assuming the comma-separated digits are meant to be read as separate decimal numbers rather than as place values in a single base-60 number.
- Forgetting that the leftmost digit represents the highest power of 60, not the lowest — reading the digits in the wrong order flips the entire value.
- Confusing this positional sexagesimal system with the unrelated concept of "Babylonian" cuneiform tally marks, which used a completely different, non-positional method for smaller everyday counts.