Babylonian Numbers Calculator — Sexagesimal (Base-60) Converter
Enter any decimal integer up to 215,999 and see its Babylonian base-60 representation step by step — with the cuneiform wedge symbols used in ancient Mesopotamia.
Input (decimal)
Place 60² = 3600
Place 60¹ = 60
Place 60⁰ = 1
Babylonian notation
How does this calculator work?
Convert decimal to Babylonian by repeatedly dividing by powers of 60 (3600, 60, 1). Each quotient digit (0–59) is written using '<' for each ten and 'Y' for each unit; positions are separated by dots. This base-60 system — the ancestor of seconds, minutes, and degree subdivisions — handled place values like modern positional notation.
Formula
How this is calculated
The Babylonian number system, developed in Mesopotamia from around 2000 BCE, is a positional base-60 (sexagesimal) system — the ancestor of today's 60-second minute, 60-minute hour, and 360-degree circle. Like our decimal system, each position represents a power of the base (60⁰ = 1, 60¹ = 60, 60² = 3600, 60³ = 216000), and digits 0–59 are written in each position using just two symbols: a vertical wedge (here shown as 'Y') representing one unit, and a Winkelhaken or corner wedge ('<') representing ten. Larger digits combine both: 32 would be '<<<YY' (three tens + two ones).
To convert from decimal, the algorithm repeatedly divides by the highest power of 60 that fits, records the quotient as the leading digit, and continues with the remainder — the same procedure used to convert decimal to any other base. Missing intermediate places (zero digits) are marked '0' in this representation; historically, the Babylonians lacked a zero symbol for most of their history and had to infer missing places from context.
This calculator uses 'Y' and '<' as the standard ASCII stand-ins for the actual cuneiform characters. The system was positional but only for whole numbers (fractions were also possible but are not covered here). The maximum of 215,999 (= 59 × 60² + 59 × 60 + 59) corresponds to the largest 3-place sexagesimal number.
Frequently asked questions
60 has many factors (1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60), making fractions like one-third and one-quarter exact in the system. The choice may also reflect a combination of finger-joint counting (12 on one hand × 5 fingers on the other = 60). Its legacy survives in time, angles and geography.
A placeholder zero (a special symbol for an empty position) was introduced around 300 BCE in later Babylonian mathematics, but only in the middle of a number — never as a standalone value or at the end. The concept of zero as a number took much longer to develop.
The sexagesimal system directly gives us 60 seconds per minute, 60 minutes per hour, and 360 (= 6 × 60) degrees in a circle. Astronomers and navigators used base-60 fractions (minutes and seconds of arc) for millennia, preserving the Babylonian convention to this day.
TG we-Calculate Editorial Team. (2026). Babylonian Numbers Calculator — Sexagesimal (Base-60) Converter [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/babylonian-numbers-calculator
TG we-Calculate Editorial Team. "Babylonian Numbers Calculator — Sexagesimal (Base-60) Converter." TG we-Calculate. 2026. https://we-calculate.com/calculator/babylonian-numbers-calculator.
TG we-Calculate Editorial Team, "Babylonian Numbers Calculator — Sexagesimal (Base-60) Converter," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/babylonian-numbers-calculator
@misc{wecalculate_babylonian_numbers_calculator, title = {Babylonian Numbers Calculator — Sexagesimal (Base-60) Converter}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/babylonian-numbers-calculator}}, year = {2026}, note = {TG we-Calculate} }
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