Babylonian Numeral Converter
Converts automatically as you type or tap.
About Babylonian Numerals
The Babylonian numeral system was sexagesimal (base-60), written as cuneiform wedge marks on clay tablets. This tool uses ▼ for the unit wedge (1) and ◄ for the corner wedge (10); groups of them form digits from 1 to 59.
Place value: groups are separated by spaces. The rightmost group counts units (60⁰), the next counts sixties (60¹), then 3,600s (60²), and so on.
The empty place: use ○ when a place holds nothing. Example: 3,601 is ▼ ○ ▼ — one 3,600, zero 60s, and one unit. Without the placeholder, ▼ ▼ would read as 61, which is exactly the ambiguity that pushed later Babylonian scribes to invent a placeholder sign.
Example: ▼ ◄▼▼ = (1 × 60) + (12 × 1) = 72. This converter covers 1 to 12,959,999 (four sexagesimal places).
Babylonian Numeral Calculator
How It Works
The calculator converts both Babylonian numerals to decimal, performs the operation, then converts the result back to Babylonian. The decimal equivalent of each input appears live underneath it as you type.
Input format: ▼ = 1, ◄ = 10, ○ = an empty place, spaces between place-value groups. Each group must be ○ alone, or a combination worth 1–59 (at most five ◄ and nine ▼).
Results must be whole numbers from 1 to 12,959,999 — the Babylonian system as used here has no sign for zero on its own, fractions, or negative numbers. Division shows the whole-number quotient plus any remainder.
Babylonian Numeral Reference
Basic Symbols
| Symbol | Value |
|---|---|
| ▼ | 1 |
| ◄ | 10 |
| ○ | 0 (empty-place marker) |
▼ and ◄ combine additively inside a place-value group, up to 59. ○ stands alone to mark a place that holds nothing.
Place Value (Base-60)
Groups of symbols are the digits (1–59) of a base-60 number. Spaces separate the places.
Rightmost group: × 60⁰ (× 1)
Next group left: × 60¹ (× 60)
Next group left: × 60² (× 3,600)
Next group left: × 60³ (× 216,000)
Example: ▼ ◄▼▼ = (1 × 60) + (12 × 1) = 72
Examples (1–59)
| Decimal | Babylonian |
|---|---|
| 1 | ▼ |
| 9 | ▼▼▼▼▼▼▼▼▼ |
| 10 | ◄ |
| 11 | ◄▼ |
| 23 | ◄◄▼▼▼ |
| 59 | ◄◄◄◄◄▼▼▼▼▼▼▼▼▼ |
Examples (> 59)
| Decimal | Babylonian | Calculation |
|---|---|---|
| 60 | ▼ ○ | (1 × 60) + 0 |
| 61 | ▼ ▼ | (1 × 60) + (1 × 1) |
| 70 | ▼ ◄ | (1 × 60) + (10 × 1) |
| 135 | ▼▼ ◄▼▼▼▼▼ | (2 × 60) + (15 × 1) |
| 3,600 | ▼ ○ ○ | (1 × 3,600) + 0 + 0 |
| 3,601 | ▼ ○ ▼ | (1 × 3,600) + 0 + (1 × 1) |
| 3,671 | ▼ ▼ ◄▼ | (1 × 3,600) + (1 × 60) + (11 × 1) |
Symbol Economy: wedges needed for each digit 1–59
Every base-60 digit is built from ◄ tens and ▼ ones, so the wedge count climbs in a sawtooth: it rises toward each multiple of ten, then drops when a ◄ replaces ten ▼. Teal bars mark the multiples of ten. Hover or tap a bar to inspect it.
Learn: Why Base 60?
A 4,000-year-old positional system
Babylonian mathematics, written in cuneiform from around 2000 BCE, was the world's first true positional number system — the same big idea behind the decimal numbers you use today, just with a base of 60 instead of 10. Scribes pressed wedge-shaped marks into wet clay: a narrow vertical wedge for 1 and a wide corner wedge for 10.
Because the system was positional, the same group of wedges could mean 1, 60, or 3,600 depending on where it sat. Early tablets relied on context (and careful spacing) to tell those apart; by the Seleucid era, scribes added a dedicated placeholder sign for an empty place — one of history's first steps toward the concept of zero. This tool's ○ plays that role.
Base 60 never left
- 60 seconds in a minute — straight from Babylonian astronomy.
- 60 minutes in an hour and in a degree of arc.
- 360 degrees in a circle — 6 × 60, the Babylonian division of the sky.
- 12 zodiac signs of 30° each, mapped by Babylonian astronomers.
The case for 60: it divides cleanly
60 is a "highly composite" number: more whole-number divisors than any smaller number. Halves, thirds, quarters, fifths, sixths, tenths — all exact, no fractions. That made trade, surveying, and astronomy far easier on a clay tablet. Compare how many divisors each candidate base has:
Reading a numeral, step by step
1. Split the numeral at the spaces — each chunk is one base-60 digit.
2. Read each chunk additively: every ◄ adds 10, every ▼ adds 1. A lone ○ means that place is empty.
3. Multiply the rightmost digit by 1, the next by 60, the next by 3,600, the next by 216,000.
4. Add the products. Try it in the Converter tab — the place-value breakdown chart animates each step for you.