Babylonian Numbers Converter (base-60 cuneiform numerals ↔ decimal)

Understanding the Babylonian numbers converter — concept and principle

This converter turns a whole number into Babylonian base-60 (sexagesimal) digits and wedge signs, or turns base-60 digits back into an ordinary decimal number. It shows the place values and how each digit is built from wedges. Use it for history of mathematics lessons, puzzles or to understand where 60 minutes and 360 degrees come from.

How it is calculated

Decimal → base 60: divide by 60 repeatedly; the remainders, read from last to first, are the digits.

Base 60 → decimal: value = Σ digit × 60^place, with places 1, 60, 3,600, 216,000, …

• Each digit from 1 to 59 is written with two signs: ▾ for 1 (up to 9 times) and ◂ for 10 (up to 5 times).
• | separates places; an empty place is shown as ·.
• Modern transliteration writes the digits separated by commas, such as 1, 6, 40.

The principle

Our decimal system is positional with base 10; the Babylonian system was positional with base 60. Each place is worth 60 times the one to its right. Within a place, though, the Babylonians did not have 59 separate symbols: they grouped tens and ones, so each digit is itself a small base-10 tally. For most of their history they had no zero symbol, so an empty place was shown by a gap or not at all, which made 1, 60 and 3,600 look the same; context told readers which was meant. Sixty divides evenly by 2, 3, 4, 5, 6, 10, 12, 15, 20 and 30, which made fractions convenient.

Worked example

Default inputs: decimal → base 60, number 4,000.

1. 4,000 ÷ 60 = 66 remainder 40.
2. 66 ÷ 60 = 1 remainder 6; 1 ÷ 60 = 0 remainder 1.
3. Digits, highest place first: 1, 6, 40, using 3 places.
4. Check: 1 × 3,600 + 6 × 60 + 40 × 1 = 3,600 + 360 + 40 = 4,000.
5. Wedges: 1 = ▾; 6 = ▾▾▾▾▾▾; 40 = ◂◂◂◂, written ▾ | ▾▾▾▾▾▾ | ◂◂◂◂.

How to read the result

Read the comma-separated digits the way you read hours, minutes and seconds: 1, 6, 40 is like 1:06:40, which is 4,000 seconds. That is no coincidence: our time and angle units are a direct legacy of base 60.

Common mistakes

• Writing a digit 60 or more in one place; each place holds 0–59.
• Forgetting empty places, such as 1, 0, 5 for 3,605.
• Reading the wedge groups as separate places.
• Assuming the Babylonians had a zero like ours.

Frequently asked questions

Why did the Babylonians use base 60?

Historians do not know for certain; a practical advantage is that 60 has many divisors, which makes fractions easy.

How do you write 60 in Babylonian numerals?

As 1, 0: one ▾ in the sixties place and an empty ones place.

Did the Babylonians have zero?

Not for most of their history; a placeholder sign appeared only late, and an empty place was usually a gap.

Sources: OpenStax, Contemporary Mathematics (2023), §4.2 Early Numeration Systems