Babylonian cuneiform numerals, also used in

Assyria Assyria (Neo-Assyrian cuneiform: , ''māt Aššur'') was a major ancient Mesopotamian civilization that existed as a city-state from the 21st century BC to the 14th century BC and eventually expanded into an empire from the 14th century BC t ...

and

Chaldea Chaldea () refers to a region probably located in the marshy land of southern Mesopotamia. It is mentioned, with varying meaning, in Neo-Assyrian cuneiform, the Hebrew Bible, and in classical Greek texts. The Hebrew Bible uses the term (''Ka� ...

, were written in

cuneiform Cuneiform is a Logogram, logo-Syllabary, syllabic writing system that was used to write several languages of the Ancient Near East. The script was in active use from the early Bronze Age until the beginning of the Common Era. Cuneiform script ...

, using a wedge-tipped reed stylus to print a mark on a soft clay tablet which would be exposed in the sun to harden to create a permanent record. The Babylonians, who were famous for their astronomical observations, as well as their calculations (aided by their invention of the

abacus An abacus ( abaci or abacuses), also called a counting frame, is a hand-operated calculating tool which was used from ancient times in the ancient Near East, Europe, China, and Russia, until the adoption of the Hindu–Arabic numeral system. A ...

), used a

sexagesimal Sexagesimal, also known as base 60, is a numeral system with 60 (number), sixty as its radix, base. It originated with the ancient Sumerians in the 3rd millennium BC, was passed down to the ancient Babylonians, and is still used—in a modified fo ...

(base-60)

positional numeral system Positional notation, also known as place-value notation, positional numeral system, or simply place value, usually denotes the extension to any base of the Hindu–Arabic numeral system (or decimal system). More generally, a positional system ...

inherited from either the

Sumer Sumer () is the earliest known civilization, located in the historical region of southern Mesopotamia (now south-central Iraq), emerging during the Chalcolithic and Early Bronze Age, early Bronze Ages between the sixth and fifth millennium BC. ...

ian or the Akkadian civilizations. Neither of the predecessors was a positional system (having a convention for which 'end' of the numeral represented the units).

Origin

This system first appeared around 2000 BC; its structure reflects the decimal lexical numerals of

Semitic languages The Semitic languages are a branch of the Afroasiatic languages, Afroasiatic language family. They include Arabic, Amharic, Tigrinya language, Tigrinya, Aramaic, Hebrew language, Hebrew, Maltese language, Maltese, Modern South Arabian language ...

rather than Sumerian lexical numbers. However, the use of a special Sumerian sign for 60 (beside two Semitic signs for the same number) attests to a relation with the Sumerian system.

Symbols

The Babylonian system is credited as being the first known

, in which the value of a particular digit depends both on the digit itself and its position within the number. This was an extremely important development because non-place-value systems require unique symbols to represent each power of a base (ten, one hundred, one thousand, and so forth), which can make calculations more difficult. Only two symbols (𒁹 to count units and 𒌋 to count tens) were used to notate the 59 non-zero digits. These symbols and their values were combined to form a digit in a sign-value notation quite similar to that of

Roman numerals Roman numerals are a numeral system that originated in ancient Rome and remained the usual way of writing numbers throughout Europe well into the Late Middle Ages. Numbers are written with combinations of letters from the Latin alphabet, eac ...

; for example, the combination 𒌋𒌋𒁹𒁹𒁹 represented the digit for 23 (see table of digits above). These digits were used to represent larger numbers in the base 60 (sexagesimal) positional system. For example, 𒁹𒁹 𒌋𒌋𒁹𒁹𒁹 𒁹𒁹𒁹 would represent 2×60²+23×60+3 = 8583. A space was left to indicate a place without value, similar to the modern-day

zero 0 (zero) is a number representing an empty quantity. Adding (or subtracting) 0 to any number leaves that number unchanged; in mathematical terminology, 0 is the additive identity of the integers, rational numbers, real numbers, and compl ...

. Babylonians later devised a sign to represent this empty place. They lacked a symbol to serve the function of

radix point alt=Four types of separating decimals: a) 1,234.56. b) 1.234,56. c) 1'234,56. d) ١٬٢٣٤٫٥٦., Both a full_stop.html" ;"title="comma and a full stop">comma and a full stop (or period) are generally accepted decimal separators for interna ...

, so the place of the units had to be inferred from context: 𒌋𒌋𒁹𒁹𒁹 could have represented 23, 23×60 (𒌋𒌋𒁹𒁹𒁹␣), 23×60×60 (𒌋𒌋𒁹𒁹𒁹␣␣), or 23/60, etc. Their system clearly used internal

decimal The decimal numeral system (also called the base-ten positional numeral system and denary or decanary) is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers (''decimal fractions'') of th ...

to represent digits, but it was not really a mixed-radix system of bases 10 and 6, since the ten sub-base was used merely to facilitate the representation of the large set of digits needed, while the place-values in a digit string were consistently 60-based and the

arithmetic Arithmetic is an elementary branch of mathematics that deals with numerical operations like addition, subtraction, multiplication, and division. In a wider sense, it also includes exponentiation, extraction of roots, and taking logarithms. ...

needed to work with these digit strings was correspondingly sexagesimal. The legacy of sexagesimal still survives to this day, in the form of degrees (360° in a

circle A circle is a shape consisting of all point (geometry), points in a plane (mathematics), plane that are at a given distance from a given point, the Centre (geometry), centre. The distance between any point of the circle and the centre is cal ...

or 60° in an

angle In Euclidean geometry, an angle can refer to a number of concepts relating to the intersection of two straight Line (geometry), lines at a Point (geometry), point. Formally, an angle is a figure lying in a Euclidean plane, plane formed by two R ...

of an

equilateral triangle An equilateral triangle is a triangle in which all three sides have the same length, and all three angles are equal. Because of these properties, the equilateral triangle is a regular polygon, occasionally known as the regular triangle. It is the ...

arcminute A minute of arc, arcminute (abbreviated as arcmin), arc minute, or minute arc, denoted by the symbol , is a unit of angular measurement equal to of a degree. Since one degree is of a turn, or complete rotation, one arcminute is of a tu ...

s, and

arcsecond A minute of arc, arcminute (abbreviated as arcmin), arc minute, or minute arc, denoted by the symbol , is a unit of angular measurement equal to of a degree. Since one degree is of a turn, or complete rotation, one arcminute is of a tu ...

s in

trigonometry Trigonometry () is a branch of mathematics concerned with relationships between angles and side lengths of triangles. In particular, the trigonometric functions relate the angles of a right triangle with ratios of its side lengths. The fiel ...

and the measurement of

time Time is the continuous progression of existence that occurs in an apparently irreversible process, irreversible succession from the past, through the present, and into the future. It is a component quantity of various measurements used to sequ ...

, although both of these systems are actually mixed radix. A common theory is that 60, a superior highly composite number (the previous and next in the series being 12 and 120), was chosen due to its

prime factorization In mathematics, integer factorization is the decomposition of a positive integer into a product of integers. Every positive integer greater than 1 is either the product of two or more integer factors greater than 1, in which case it is a comp ...

: 2×2×3×5, which makes it divisible by 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, and 60.

Integer An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...

s and

fraction A fraction (from , "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight-fifths, thre ...

s were represented identically—a radix point was not written but rather made clear by context.

Zero

The Babylonians did not technically have a digit for, nor a concept of, the number

. Although they understood the idea of nothingness, it was not seen as a number—merely the lack of a number. Later Babylonian texts used a placeholder () to represent zero, but only in the medial positions, and not on the right-hand side of the number, as is done in numbers like .

References

Bibliography

* *

External links

Babylonian numerals

* ttp://it.stlawu.edu/%7Edmelvill/mesomath/tablets/YBC7289.html Photograph, illustration, and description of the ''root(2)'' tablet from the Yale Babylonian Collection
Babylonian Numerals
by Michael Schreiber,

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