Numerals and Arithmetic

The Claim in One Sentence

The decimal positional number system with zero was created in India; scholars of the Islamic world adopted it, completed it with decimal fractions, perfected its written algorithms, and delivered the finished instrument to Europe, which is why the world calls its digits "Arabic numerals" and its procedures "algorithms."

The Chain of Development

Arrival (770s). Indian astronomical texts reach Baghdad; al-Fazari and colleagues render them into Arabic. The system's advantages, nine digits plus zero, value by position, uniform algorithms, are immediately clear to a bureaucratic empire that computes for a living.

Codification (c. 820s). al-Khwarizmi writes the treatise on Hindu reckoning that, in its Latin afterlife (Dixit Algorizmi), names the entire subject. The algorithms it teaches, digit-by-digit addition, subtraction with borrowing, multiplication, division, root extraction, are recognizably the ones taught in elementary schools today.

Decimal fractions (952/953). al-Uqlidisi in Damascus extends place value below the units place, marking the decimal point's position with a stroke, and adapts all algorithms to pen and paper. The number line's other half opens for business.

Textbook maturity (10th–11th c.). Kushyar ibn Labban and al-Nasawi write the clean classroom expositions; Ibn Tahir al-Baghdadi surveys and compares the competing systems (finger reckoning, sexagesimal, decimal).

The Maghrebi refinements (12th–15th c.). al-Hassar introduces the horizontal fraction bar; Ibn al-Banna writes the region's standard textbook; checking rules (casting out nines, sevens, elevens) become systematic; approximation rules for roots are refined.

Completion (1427). al-Kashi's Key of Arithmetic treats decimal fractions as a complete general-purpose system, mirroring every sexagesimal technique in base ten, and demonstrates the system's power with computations (16 decimal places of \(2\pi\)) that would remain unmatched in Europe for 170 years.

Sexagesimal: The Professional's Second Base

Alongside decimal arithmetic, astronomers computed in base 60 inherited from Babylon via Greek astronomy, degrees, minutes, seconds, exactly as angles and time are still divided today. Treatises taught both bases and conversion between them; al-Kashi's decimal fractions were consciously modeled on sexagesimal fractional practice. The modern clock face and the 360° circle are living fossils of this dual-base world.

Mental and Finger Arithmetic

A third stream, hisab al-yad (finger reckoning) and mental-arithmetic methods used by merchants and secretaries, was also codified in treatises (Abu al-Wafa wrote one for scribes, including rules for negative intermediate results treated as debts). The tradition's arithmetic culture was thus three-layered: mental/finger methods for the market, written Indian reckoning for general calculation, sexagesimal for the observatory.

Transmission to Europe

The Latin algorismus corpus of the 12th–13th centuries (from al-Khwarizmi's name) taught decimal computation in the universities; Fibonacci's Liber Abaci (1202) taught it to the commercial world, fraction bar and all; the Italian abacus schools mass-produced numerate merchants; and Simon Stevin's De Thiende (1585) finally gave Europe the decimal fractions the Islamic world had used for six centuries. The path from a Damascus copyist's stroke over a digit in 952 to the decimal point on every calculator on Earth runs through this literature.

Connection to Modern Mathematics

Positional notation is the substrate of modern computation in a strict sense: the digit-by-digit algorithms codified in this tradition are, complexity-theoretically, the "schoolbook" algorithms against which fast multiplication (Karatsuba, FFT-based methods) is measured, and base-positional representation generalizes directly to base 2, the hardware representation of all modern computing. "Algorithm" as the name for a finite effective procedure, the central object of computer science, descends by direct etymological and conceptual lineage from these arithmetic texts.

Sources: Saidan, The Arithmetic of al-Uqlidisi; Berggren, Episodes, ch. 2; Katz; Levey & Petruck; Djebbar on the Maghrebi school.