Convert decimal numbers to Babylonian base-60 (sexagesimal) notation with cuneiform symbols, positional breakdown, and conversion tables showing the legacy of base-60 in modern time and angles.
The Babylonian number system, developed in ancient Mesopotamia around 1800 BCE, was the world's first positional number system — and it used base 60 (sexagesimal) instead of our familiar base 10. This seemingly unusual choice was actually brilliant: 60 has twelve factors (1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60), far more than 10's four factors (1, 2, 5, 10), making fractions and division much cleaner.
Babylonian numbers were written using just two cuneiform symbols pressed into clay tablets: a vertical wedge (𒐕) for 1 and a corner wedge (𒌋) for 10. To write any digit from 1 to 59, you combined these marks — for example, 23 was written as two corner wedges and three vertical wedges (𒌋𒌋𒐕𒐕𒐕). For numbers 60 or above, the same digit groups were separated by spaces in columns representing powers of 60, just as our digits represent powers of 10.
The legacy of base-60 pervades modern life: 60 seconds in a minute, 60 minutes in an hour, 360 (6 × 60) degrees in a circle, and the division of degrees into 60 arcminutes and 60 arcseconds all trace directly to Babylonian mathematics. This calculator converts any decimal number to Babylonian sexagesimal notation with authentic cuneiform symbols, step-by-step breakdowns, and cultural context connecting 4,000-year-old mathematics to the clocks and compasses we use today.
Converting between decimal and base-60 requires repeated division by 60 and mapping remainders to cuneiform symbols — a laborious process that is easy to botch for large numbers. This calculator shows every positional digit, its power-of-60 contribution, and the corresponding cuneiform wedge marks in one glance. History of mathematics students use it to visualise how the world's first positional system worked, teachers project the cuneiform symbols in class, and anyone curious about why clocks and compasses use 60 can see the direct connection.
Base-60 positional: value = d₀×60ⁿ + d₁×60ⁿ⁻¹ + … + dₙ×60⁰, where each digit dᵢ is 0–59. Cuneiform: each digit uses 𒌋 (×10) and 𒐕 (×1) symbols.
Result: 3, 25, 45 → 𒐕𒐕𒐕 · 𒌋𒌋𒐕𒐕𒐕𒐕𒐕 · 𒌋𒌋𒌋𒌋𒐕𒐕𒐕𒐕𒐕
12345 = 3×3600 + 25×60 + 45. In base-60: [3, 25, 45]. The cuneiform uses wedge symbols for each digit within its position.
In a positional system the value of a digit depends on *where* it sits, not just what it is. In base-10, the number 325 means 3×100 + 2×10 + 5×1. In base-60 the columns represent powers of 60: the rightmost column is 60⁰ = 1, the next is 60¹ = 60, then 60² = 3,600, then 60³ = 216,000, and so on. Each column holds a "digit" from 0 to 59. The Babylonians wrote each digit with only two mark types — vertical wedges for units and corner wedges for tens — then separated columns by a small space. This made large numbers surprisingly compact: the number 86,400 (seconds in a day) is just four base-60 digits: 24, 0, 0, 0.
Virtually every system that divides a unit into 60 parts traces back to Mesopotamia. Hours divide into 60 minutes, minutes into 60 seconds. Degrees divide into 60 arcminutes, arcminutes into 60 arcseconds. Navigators still express latitude and longitude in degrees-minutes-seconds. Ptolemy's *Almagest* (c. 150 CE) inherited sexagesimal fractions from Babylonian astronomy and passed them to Islamic and then European scholars. The notation survived because 60 is evenly divisible by 2, 3, 4, 5, 6, 10, 12, 15, 20, and 30, so thirds, quarters, and fifths of an hour or degree are all whole numbers of the sub-unit.
A fun exercise is to have students convert their birthday year or the current date into sexagesimal. For example, 2025 = 33×60 + 45, so it is [33, 45] in base-60 — just two digits. Then ask: how many base-10 digits does it take to represent the same range? (4.) This highlights why base-60 is compact for large values. Another activity is reading cuneiform: display a sequence of wedge marks and have students decode it by counting corners (×10) and verticals (×1) in each group, then multiply each group by its place value. These exercises build intuition for positional systems *in general*, reinforcing how base-10 and base-2 work by comparison.
Base 60 has more factors than base 10 (twelve vs. four), making fractions cleaner. Some historians also suggest it arose from counting on finger joints (12 per hand) combined with 5 fingers on the other hand (12 × 5 = 60).
They used two cuneiform symbols on clay tablets: a vertical wedge (𒐕) for 1 and a corner wedge (𒌋) for 10. Numbers 1–59 are written by combining these marks. Larger numbers use positional notation with columns for 60⁰, 60¹, 60², etc.
Not initially. For centuries, an empty position was simply left blank, which caused ambiguity. Around 300 BCE, they introduced a placeholder symbol (𒑊), but it was not used at the end of numbers and was not a true zero concept.
Directly from the Babylonian base-60 system. Greek astronomers (particularly Ptolemy) adopted Babylonian sexagesimal fractions for calculations, and this convention passed through medieval astronomy into modern timekeeping.
The Babylonians approximated the year as 360 days and divided circles accordingly. Since 360 = 6 × 60, it fits neatly into their base-60 system and is divisible by 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, and 180.
In base-10, each position represents a power of 10 (1, 10, 100, 1000…). In base-60, each position represents a power of 60 (1, 60, 3600, 216000…). Both are positional systems — the Babylonian system was the first in history.