Numeração Babilônica

How the sexagesimal system actually works

The Babylonian number system is a place-value system based on sixty. It was developed by the Sumerians and later adopted by the Babylonians around 3000 BCE, which is why the numeração babilônica is also called the sexagesimal system. Two basic wedge-shaped marks represent the digits: a single vertical wedge ( ) stands for 1, and a corner-shaped chevron ( ) stands for 10. Values from 1 to 59 are built by combining these symbols horizontally, grouped by tens and ones. What makes it a true place-value system is that the same symbol can represent different magnitudes depending on its position from right to left. The rightmost position counts units (1s), the next position to the left counts sixties (60s), then the third position counts 3,600 (60 × 60), and so on. There was no consistent symbol for zero in the earliest versions. Scribes sometimes left a gap to indicate an empty place, but this caused ambiguity that nobody solved properly until much later, when a closing wedge pair was occasionally used in later periods.

Converting numeração babilônica to decimal: the method

Let me walk through a concrete example because that is where most people get tripped up. Suppose you see three groups of symbols arranged left to right and you need to read them as a single number. Write each group's value, multiply the leftmost by 60, and add the rightmost. Consider this sequence: two chevrons in the left position and four vertical wedges in the right position. The left group equals 20 (two tens). The right group equals 4. Applying place value: 20 × 60 + 4 = 1,204 in decimal. That is the conversion in one clean step.

A slightly harder example. Left position shows one chevron plus three wedges (13). Right position shows three chevrons plus five wedges (35). Calculation: 13 × 60 + 35 = 815. If there were a third position, say a single chevron (10), the full value would be 10 × 3,600 + 13 × 60 + 35 = 36,815. The pattern is always the same: each position is worth sixty times the one to its right.

Practical problems I ran into

I spent time reading reproduced cuneiform tablets a few years ago, and one thing hit me immediately: the spacing between groups is not reliable. In some copies, a missing space between the sixties place and the units place makes it impossible to tell whether you are reading 1,4 or 61. I encountered this on a tablet that listed what looked like the number 70 at first glance. Once I checked the context and realized the scribe had left no gap, the correct reading was 1 in the sixties column and 10 in the units column, which gives 70 in decimal. But without context, you cannot know for certain which interpretation is intended. This is a well-documented ambiguity in Old Babylonian mathematical texts. My workaround was simple: cross-reference the number with surrounding text. If the passage discussed a field area or a grain ration, I could check whether 70 or 1,10 made sense as a plausible quantity. Context resolved the ambiguity 90 percent of the time. The rest required noting the uncertainty and moving on.

Why this system still matters today

The reason the numeração babilônica is not just a historical footnote is that its base-60 structure survived into modern measuring systems. We divide an hour into 60 minutes and a minute into 60 seconds because the Babylonians did. We divide a circle into 360 degrees, another multiple of 60. When you look at a clock or a map, you are reading the direct descendant of this system. There is a practical advantage to base 60 that base 10 lacks: 60 has twelve divisors (1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60), while 10 has only four. This makes fractions much easier to work with. A half, a third, a quarter, and a fifth all resolve to whole numbers in base 60. That is likely why Babylonian astronomers and administrators preferred it over other candidate bases.

Common misconceptions

People often assume the Babylonians had a fully developed zero. They did not. The empty-place notation appeared only late in the tradition, and even then it was optional and inconsistent. This means any number you read from an early tablet could theoretically be off by a factor of 60, 3,600, or higher, depending on where the scribe intended the place values to start. Scholars resolve this by using contextual clues and known astronomical or economic values. If you are doing casual conversion practice, you can ignore this issue, but it is worth knowing when you encounter original sources. Another frequent mistake is treating the wedges as a simple tally system. They are not. The combination of horizontal grouping within each position and vertical place-value positioning is what makes this a positional system. A single symbol like (four chevrons and three wedges) does not mean 43 by additive counting alone. It means 43 within one position, and its actual magnitude depends entirely on where it sits relative to other positions.

Quick reference table

Babylonian group	Value
	1
	2
	3
	4
	10
	20
	30
	40
	50
	59

For multi-position numbers, multiply each group by the appropriate power of 60 and sum the results. There is no special rule beyond that, but the ambiguity problem means you should always verify your reading against the surrounding context whenever possible. If the text gives no context and the gaps are unclear, note both interpretations rather than committing to one without evidence.