| Invention Name | Babylonian Base-60 Number System (Sexagesimal) |
|---|---|
| Short Definition | A positional numeral system that groups values by 60 instead of 10. |
| Approximate Date / Period | Old Babylonian period, c. 2000–1600 BCE Approximate |
| Geography | Mesopotamia (southern Iraq region) |
| Inventor / Source Culture | Anonymous (Babylonian scribal culture) |
| Category | Mathematics, accounting, astronomy, measurement |
| Importance |
|
| Need / Reason It Emerged | Reliable accounting; practical division; standard measures |
| How It Works | Place-value base-60; digits 1–59; meaning set by position/context |
| Material / Tech Basis | Clay tablets; cuneiform wedges; reed stylus |
| First Use Context | Administrative records; schooling; mathematical problem texts |
| Spread Path | Mesopotamia → wider Near East scholarly traditions |
| Derived Developments | Reciprocal tables; advanced problem solving; refined measures |
| Areas Influenced | Science; commerce; education; surveying; calendars |
| Debates / Different Views | Origins and early notation conventions Discussed |
| Precursors + Successors | Earlier Mesopotamian counting → later scholarly sexagesimal notation |
| Key Cultures | Sumerian and Akkadian roots; Babylonian schools |
| Notable Variants | Administrative metrology systems; later placeholder for empty places |
A base-60 number system can sound exotic, yet its fingerprints are familiar. Every hour and degree quietly carries the same idea: numbers arranged by sixties, not tens. In ancient Mesopotamia, this approach grew into a flexible tool for records, teaching, and calculation. The result was a compact way to write large values and fine fractions with the same set of signs.
Table Of Contents
What It Is
The Babylonian system is a positional numeral system built around 60. Each “digit” can represent values from 0 to 59, and the meaning shifts with position, much like decimal place value. The writing uses a small set of wedge-shaped signs pressed into clay, giving it a clean visual rhythm and fast repeatability.
- One place can represent units (1–59).
- The next place can represent sixties (60–3540).
- Places continue as powers of 60 for large values, and as fractions for small ones.
Origins and Evidence
Sexagesimal thinking grew from earlier Mesopotamian counting and measurement habits, then matured into a school-based tradition. Clay tablets show that scribes treated numbers as working tools, not decoration. They recorded counts, shares, lengths, and computed results in neat columns, making the tablet itself a durable “work surface” for math.
A Strong Artifact Signal
One well-known example from the Yale Babylonian Collection is a mathematical school tablet labeled YBC 7289. It includes a highly accurate sexagesimal value linked to the square root of 2, showing how confidently scribes handled place value and fractions. Details
The Medium That Made It Practical
Numbers were pressed into moist clay with a cut reed, creating the wedge strokes known as cuneiform. This writing tradition is documented as beginning before 3200 BCE, and the same tablet format supported both text and calculation. Details
How It Works
In a sexagesimal place-value system, each position can stand for a power of 60. A sequence of signs can represent a large whole number, a fraction, or a mix of both. The key is that Babylonian notation often relied on context to infer scale, since early writing did not consistently mark an absolute “point” or a full zero symbol.
Digits, Not Dozens Of New Symbols
Instead of needing sixty separate characters, scribes used two basic signs (for 1 and 10) and combined them like tidy tallies to build 1–59. Larger values came from position. The same idea extended naturally to fractions, because places to the right could represent 1/60, 1/3600, and so on. Details
A Small Example Of Place Value
Modern writers sometimes show sexagesimal with separators to make structure obvious. In that style, 1,6,40 means 1×60² + 6×60 + 40. The same pattern works for tiny fractions when places represent negative powers, so one compact string can carry both magnitude and precision.
| Idea | Decimal Habit | Sexagesimal Habit |
|---|---|---|
| Base | 10 per place | 60 per place |
| Fraction Steps | 1/10, 1/100, 1/1000 | 1/60, 1/3600, 1/216000 |
| Clean Shares | Best for 2 and 5 | Best for many divisors (2, 3, 4, 5, 6, 10, 12, 15…) |
Why Sixty
The appeal of 60 is plain in daily math. Sixty has many divisors, so common shares land on tidy endings. Halves, thirds, quarters, fifths, and sixths can be expressed with short sexagesimal fractions, which keeps records and computations compact and easy to check.
Fractions That Stay Neat
- 1/2 becomes 0;30
- 1/3 becomes 0;20
- 1/4 becomes 0;15
- 1/5 becomes 0;12
- 1/6 becomes 0;10
That regularity supported a strong culture of checking and recomputing, since many results stayed short enough to rewrite cleanly.
Tables That Made Division Practical
Many tablets rely on reciprocal tables, turning division into multiplication. In one scholarly framing, numbers with prime factors only 2, 3, and 5 have finite sexagesimal reciprocals and were treated as especially friendly for exact work. When a number was not “regular,” results could be recorded as approximations to a few sexagesimal places. Details
Variants and Legacy
“Babylonian numbers” can refer to more than one practice. Alongside the well-known sexagesimal place-value used in mathematical texts, Mesopotamia also used metrology-based counting systems in administration, where measures and goods shaped notation choices. Over time, scribes refined conventions, including ways to show an empty place more clearly in some contexts.
Main Forms Seen In Texts
- Mathematical sexagesimal: a compact place-value notation for computation and teaching.
- Administrative metrology: numbers tied closely to units of capacity, weight, area, and labor.
- Later clarifications: increased care in separating groups and indicating missing positions in some writing habits.
Where The Legacy Still Lives
Modern life still leans on sexagesimal structure in familiar places. Time uses 60 seconds per minute and 60 minutes per hour. Angle measure uses 360 degrees, with minutes and seconds as subdivisions. This is not a museum curiosity; it is a living format that remains natural for division and fine measurement.
Why It Stayed Useful
- Many clean splits make everyday fractions manageable.
- Place value scales smoothly from small to huge numbers.
- Table culture supports repeatable computation and verification.
FAQ
Why is the Babylonian system called “base-60”?
Its place values scale by 60. Each position represents a power of 60, and each digit can represent values from 0 to 59.
Did Babylonians have a zero?
Early sexagesimal notation did not consistently use a dedicated zero symbol for empty places. Meaning often depended on spacing and context, and some later traditions introduced clearer placeholders for missing positions.
How did base-60 help with fractions?
Sixty has many divisors, so common shares produce short sexagesimal fractions. This made trade, allocation, and table-based computation more compact than a base with fewer divisors.
What is the strongest single piece of evidence for advanced use?
Several tablets illustrate sophisticated practice. A well-known example is YBC 7289, which shows a highly accurate sexagesimal value connected to the diagonal of a square.