Algorism (reckoning with Indian decimal numerals in al-Khwarizmi's book of arithmetic)

Algorism is a way of writing and working numbers with nine numerals and a zero, in which each place takes its value from its position: units, then tens, then hundreds, then thousands. Muhammad ibn Musa al-Khwarizmi wrote a book on it in Baghdad, the Book of Indian Reckoning, through which the Indian numerals and the decimal system entered the Islamic world ¹. Its Arabic original is lost, but a twelfth-century Latin translation survives that opens "Dixit Algorizmi", "al-Khwarizmi said" ¹. From it the West learned arithmetic; al-Khwarizmi's Latin name, "Algorismus", became a name for the numerals themselves, and from it comes the word "algorithm" in the world's languages ².

It is the arithmetic we know today: numbers written with ten symbols, and added, subtracted, multiplied and divided in writing by ordered steps. The Cairo editors of al-Khwarizmi's algebra say that "the Indian method of reckoning, including the use of zero" reached the West through his book of arithmetic ², and the TDV encyclopaedia says that the Indian numerals and the decimal system entered the Islamic world through the same book ¹.

Ibn Khaldun defines the craft of arithmetic as "a practical craft of reckoning numbers by joining and separating". Joining one by one is addition and joining by multiples is multiplication; separating one by one is subtraction and separating into equal parts is division, whether in whole numbers, fractions or roots ³. Indian reckoning is the doing of these four operations with numerals written in their places.

The numerals themselves are Indian, as their name says. Indian reckoning had already come to Baghdad before al-Khwarizmi, in astronomy. TDV records that an Indian delegation came to Baghdad around 154 AH (770–771 CE), under the caliph Abu Ja'far al-Mansur, bringing Brahmagupta's Siddhanta, or a work of the same name derived from it, and that the Arabic word "Sindhind" is a corruption of the Sanskrit "siddhanta" ¹. On that book rest the Sindhind tables that al-Khwarizmi himself later made ¹.

Al-Khwarizmi lived in Baghdad in the time of al-Ma'mun. The Cairo editors quote Ibn al-Nadim's Fihrist: he came from Khwarazm, devoted himself to al-Ma'mun's treasury of wisdom, was one of the astronomers, and people relied on his two tables known as the Sindhind ⁴. From the Fihrist and the preface of the algebra they show that he was al-Ma'mun's contemporary, though the years of his birth and death cannot be fixed ⁵. TDV adds that al-Tabari mentions him among the events of 210 AH, and that he was among the astrologers whom the caliph al-Wathiq called to his sickbed in 232 AH, so he died after 232 AH (847 CE) ¹.

In this setting al-Khwarizmi wrote his book on Indian reckoning. The Cairo editors list a "Book of Arithmetic" among four books of his that Ibn al-Nadim does not mention ⁵. TDV says its Arabic original is lost. A twelfth-century Latin translation was found in the Cambridge library: B. Boncompagni published it as "Algoritmi de numero indorum" (Rome 1857), Kurt Vogel then published a facsimile of the same manuscript (Aalen 1963), and André Allard published a critical edition with a French translation, "Le calcul indien" (Paris 1992) ¹. TDV also names another lost book of his, the "Book of Adding and Subtracting", which Abd al-Qahir ibn Tahir al-Baghdadi quotes in his "al-Takmila fi al-hisab"; the quotation shows that it dealt with finger reckoning ¹.

Al-Khwarizmi did not invent the Indian numerals; he transmitted them and taught reckoning with them. The museum's record therefore names him the "transmitter" of the method: the author of the book through which it entered the Islamic world ¹ and, in its Latin translation, the West ². He opens the preface of his algebra by praising God and asking blessings on the Prophet ﷺ Muhammad and his family, calls al-Ma'mun "the imam al-Ma'mun, commander of the faithful", and closes by asking blessings on all the prophets and messengers ⁶. TDV notes that the historian Toomer took this preface to show that he was a Sunni. It also rejects what Toomer built on a phrase in al-Tabari that calls him "al-Majusi al-Qutrabbulli": that reading confuses two men because the "and" between their names dropped out ¹.

In the preface of his algebra al-Khwarizmi describes the decimal build of numbers himself. He found, he says, that everything people need of reckoning is number, and that all numbers are composed of one. The spoken numbers from one to ten follow the pattern of one; then ten is doubled and tripled as one was, giving twenty and thirty up to a hundred; then a hundred is doubled and tripled up to a thousand; and so the thousand is repeated at every order, to the furthest number that can be grasped ⁷. TDV describes this passage as a short account of how number is expressed in a positional system with base ten ¹.

The Indian numerals put this build on paper: nine signs for the units, a zero to keep an empty place, and each numeral taking its value from its position, so that the same numeral means one, ten or a hundred according to its place. The four operations Ibn Khaldun lists, addition, multiplication, subtraction and division ³, then become steps repeated place by place. TDV says al-Khwarizmi applied to algebra a method like the one he applied to Indian arithmetic, setting the operations in an ordered sequence, and that this method was later called "algoritma" after him: an ordered technique of calculation ¹.

Without the Arabic original we cannot trace its copies in the Islamic world, but TDV makes this book the road by which the Indian numerals and the decimal system entered the Islamic world ¹. Ibn Khaldun says the craft of arithmetic arose from the need of reckoning in dealings, that people wrote much on it, and that it was passed on in the cities by teaching it to children ⁸. In the twelfth century al-Khwarizmi's book was translated into Latin ¹, and Westerners learned arithmetic from it and from books built on it ².

• The Indian numerals and the decimal place-value system with zero: they entered the Islamic world through al-Khwarizmi's book ¹, and through it the Indian method, zero included, reached the West ².
• Brahmagupta's Siddhanta: brought to Baghdad by an Indian delegation around 154 AH; al-Khwarizmi's Sindhind tables rest on it ¹.

• The Latin translation, "Algoritmi de numero Indorum", of the twelfth century, and its editions by Boncompagni, Vogel and Allard ¹.
• The "Carmen de Algorismo" of Alexander de Villa Dei, about 1220, and the "Algorismus vulgaris" of John of Halifax, about 1250: both rest on Muhammad ibn Musa's book of arithmetic, and both remained references for teaching arithmetic for centuries ⁹.
• "Algorismus" as the Latin name of the numerals until the early eighteenth century, and the Spanish word "guarismo" for a digit ².
• The word "algorithm" for an ordered way of solving problems ².

Arithmetic is an everyday need. Ibn Khaldun says the craft of arithmetic arose because reckoning was needed in dealings, that it was taught to children in the cities, and that starting with it is among the best ways to teach, because its knowledge is clear and its proofs are orderly ⁸. Among the books on it in the Maghrib of his time he names al-Hassar's small book and the summary by Ibn al-Banna al-Marrakushi with his commentary "Raf' al-hijab" ⁸. It is the same need al-Khwarizmi names when he says he wrote his algebra for the reckoning people require in their inheritances, bequests, trade and the measuring of land ⁷.

The Cairo editors say Westerners learned arithmetic from the Latin translation of al-Khwarizmi's book and from books built on it, and that the books of Alexander de Villa Dei and John of Halifax remained references for teaching it for centuries ⁹. They conclude that al-Khwarizmi "founded algebra and taught it, and taught arithmetic, to all people" ¹⁰. The numerals the world writes today passed from India to Baghdad, and from Baghdad to Europe, through a book that carries his name.

The museum records al-Khwarizmi's work here as transmission, not invention. The numerals and the decimal system are Indian, and Indian reckoning was in Baghdad before him with Brahmagupta's Siddhanta ¹. What is al-Khwarizmi's is a book teaching reckoning with these numerals, through which they entered the Islamic world ¹ and reached the West ², until his name became theirs.

Most of the world today writes numbers with the ten decimal numerals and calculates with them by the same ordered steps. The word "algorithm", a corruption of al-Khwarizmi's name ², has become the name of every ordered set of steps a computer follows. The Latin translation of his book is in print in the editions of Boncompagni (1857), Vogel (1963) and Allard (1992) ¹.

The Arabic original is lost. The oldest surviving witness is the twelfth-century Latin translation in the Cambridge library ¹. The image on this page is the opening page of that manuscript (Cambridge University Library, Ii.6.5), scanned from Vogel's facsimile, beginning "DIXIT algorizmi" ¹¹.

Al-Khwarizmi's name became a word in the dictionaries of most of the world's languages. English has "algorithm" for a set way of solving problems, and the English poet Chaucer used "augrim" for zero; the numerals were called "Algorismus" in Latin until the early eighteenth century, and Spanish calls a digit "guarismo" ². TDV notes that Latin sources write his name as Alkarismi, Algoritmi, Algorismi and Algorism ¹.

About twelve hundred years ago a scholar in Baghdad named al-Khwarizmi wrote a book teaching people to calculate with nine numerals that came from India, together with zero. With these numerals you only need to move a digit to change its value: a one can mean one, ten or a hundred! His book was translated into Latin, people in Europe learned to calculate from it, and his name became a word we use today: "algorithm", the ordered steps a computer follows ^{1 2}.

• İhsan Fazlıoğlu, "HÂRİZMÎ, Muhammed b. Mûsâ", TDV Encyclopedia of Islam, vol. 16 (Istanbul 1997), pp. 224–227, online edition. ¹
• Muhammad ibn Musa al-Khwarizmi, Kitab al-Jabr wal-Muqabala, ed. Ali Mustafa Mosharafa and Muhammad Mursi Ahmad, Cairo 1939: editors' introduction, scan pp. 22–26; author's preface, scan pp. 29–30. ¹²
• Abd al-Rahman ibn Khaldun, al-Muqaddima, pp. 494–495, section "The science of arithmetic". ¹³
• Wikimedia Commons, "Dixit algorizmi.png", public domain, Cambridge University Library Ii.6.5. ¹¹