As easy as 1,2,3…

In every day life we all do our calculations, whether for the taxman, our purchases, paying the household bills or in some academic discipline, using the place value decimal number system. It consists of just ten symbols (numerals) – 1,2,3,4,5,6,7,8,9,0 – with which we can express any number, of any size that we may require. The value of the symbol changes according to its position – place – within the number that we write. This is an incredibly powerful and efficient method of writing numbers and the algorithms that it uses also make it a very efficient system for conducting calculations. The numerals are usually referred to as Arabic numerals or more correctly as Hindu-Arabic numerals because we Europeans inherited them and the entire system of how to use them from the Islamic Empire in the High Middle Ages, which in turn had inherited them from India in the Early Middle Ages, where they originated. In what follows I shall sketch the path that this number system took from India to medieval Europe, a path that has several twist and turns.

The history of the early development of the place value decimal number system is long, complicated and full of holes and I shan’t be dealing with it here. It also throws up some important and unanswered questions. The Babylonians developed a place value number system as early as the beginning of the second millennium BCE but it was a sexagesimal or base sixty number system rather than a decimal base ten one. The Babylonian system even had a placeholder zero in its later versions. This poses the question whether the Indians got the idea of a place value system from the Babylonians but it is simply not known. The Chinese also had a place value decimal number system but whether the Chinese influenced the Indians, the Indians the Chinese or both developed their systems independently is also not known.

There are three principle figures, who played a central role in the transmission of the place value decimal number system and the first of these is the Indian astronomer Brahmagupta (c.598–c.668 CE), who lived most of his life in Bhillamala (modern Bhinmal) in North-western India. He wrote his Brāhma-sphuṭa-siddhānta a treatise on astronomy written in verse, with 24 chapters and 1008 verses, in 628 CE. Writing scientific works in verse in ancient cultures was probably in order to make them easier to memorise in predominantly oral societies. Although an astronomical work Brahmagupta devotes several chapters to mathematics. Chapter 12 is devoted to arithmetic and introduces the basic arithmetical operations. In chapter eighteen he deals with negative numbers and with zero, not as a placeholder but as a number. He defines zero as that which results from subtracting a number from itself and gives the correct rules for addition, subtraction and multiplication with zero. Unfortunately he defines zero divided by zero as zero and gives a term for a number divided by zero without saying what the result would be. We, of course, now say division by zero is not defined. Brahmagupta’s use of zero as a number is the earliest known such use but this doesn’t mean that he invented zero as a number. His description suggests that this is already common usage. We know that zero as a number doesn’t appear in the astronomical text Aryabhatiya of Aryabhata (476–550 CE), which Brahmagupta criticises, so we can assume that zero as a number was developed in the period between the two works. The Brāhma-sphuṭa-siddhānta also contains description of what we would call algebra the details of which needn’t interest us here although we will meet them again. The Brāhma-sphuṭa-siddhānta was translated into Arabic in the eighth century CE and became one of the principle sources in the Islamic Empire for the Indian number system.

Our second principle figure is the eighth-century Persian mathematician Muḥammad ibn Mūsā al-Khwārizmī (c.780–c.850 CE), who produced two works influenced by Brahmagupta’s Brāhma-sphuṭa-siddhānta, one on algebra and one on arithmetic. The more famous is his Al-kitāb al-mukhtaṣar fī ḥisāb al-ğabr wa’l-muqābala (The Compendious Book on Calculation by Completion and Balancing) from which we get the word algebra (al-ğabr) and from his name we also get the term algorithm (a corruption of al-Khwārizmī). However it is his second work on arithmetic that interests us here. There is no known extant Arabic original of this work but it was translated into Latin in the twelfth century, possibly by Adelard of Bath,[1] under the title Algorithmo de Numero Indorum. This was the first introduction of the Hindu-Arabic numerals and the place value decimal number system into Europe. This introduction was realised at the early medieval universities, where the place value decimal number system was taught under the name algorism, as part of the discipline of computos, the calculation of the date of Easter, an important branch of mathematics at the Catholic universities. John of Sacrobosco wrote a widely read text book Algorismus aka De Arte Numerandi aka De Arithmetica in the early thirteenth century. However the use of the Hindu-Arabic numerals did not spread outside of the university.

In the Arabic world the books on algebra and arithmetic, and al-Khwārizmī’s were by no means the only ones, were largely aimed at merchants and traders. They were what we would term books on commercial arithmetic teaching bookkeeping, calculation of interest, calculation of profit shares in joint business ventures, division of property in testaments etc. and it is from this area that the Hindu-Arabic numbers and the place value decimal number system was finally introduced into Europe by the third of our principle figures Leonardo Pisano or Leonardo of Pisa (c.1175–c.1250).

Leonardo is more generally incorrectly known today by the name Fibonacci. This name, which translates as the son of Bonacci, was, however the creation of the French historian, Guilluame Libri in in 1838. Leonardo’s father Guilichmus or Guilielmo was a merchant who became a customs official. Bonacci was a general family name and not the name of his father his book the Liber Abbaci, to which we will turn shortly, starts:

Here begins the Book of Calculations

Composed by Leonardo Pisano, Family Bonacci

In the Year 1202

As with both Brahmagupta and al-Khwārizmī we know next to nothing about Leonardo personally, the only information that we have is in the introduction to the Liber Abbaci:

As my father was a public official away from our homeland in the Bugia [Now Béjaïa in Algeria] customshouse established for the Pisa merchants who frequently gathered there, he had me in my youth brought to him, looking to find for me a useful and comfortable future; there he wanted me in the study of mathematics and to be taught for some days; there from a marvellous instruction in the art of the nine Indian figures, the introduction and knowledge of the art pleased me so much above all else, and I learned from them, whoever was learned in it, from nearby Egypt, Syria, Greece, Sicily, and Provence, and their various methods, to which locations of business I travelled considerably afterwards for much study, and I learned from the assembled disputations. But this, on the whole, the algorithm and even the Pythagorean arcs, I still reckoned almost an error compared to the Indian method. Therefore strictly embracing the Indian method, and attentive to the study of it, from mine own sense adding some, and some more still from the subtle geometrical art, applying the sum that I was able to perceive to this book, I worked to put it together in xv distinct chapters, showing certain proof for almost everything that I put in, so that further, this method perfected above the rest, this science is instructed to the eager, and to the Italian people above all others, who up to now are found without a minimum. [i.e. with no knowledge of this method] If by chance, something less or more proper or necessary I omitted, your indulgence for me in entreated, as there is no one who is without fault, and in all things is altogether circumspect.

Leonardo obviously used numerous sources for his extensive book but, which sources he used is not known for certain; it is not even known if he could read Arabic and used original Arabic sources or whether he relied on the Latin translations that already existed. We do however know from textual analysis that he did use al-Khwārizmī’s book on algebra as one of his sources.

The Liber Abbaci is a book written by a merchant for merchants and it is as commercial arithmetic that the Hindu-Arabic numerals finally made it onto the big stage in medieval Europe. Abbaci, with two ‘bs’, and not one as it is often falsely written, comes from abbaco meaning to reckon or calculate in Italian and has nothing to do with abacus. Leonardo’s book might not have had the impact that it did if it had not appeared at roughly the same time as another innovation, double entry bookkeeping. The combination of the Hindu-Arabic numerals and double entry bookkeeping become the engine room to the so-called medieval economic revolution that saw the invention of banking and the rise of large scale international trading centred round the economic power house of Northern Italy. Leonardo’s book triggered a whole abbaco industry in Northern Italy.

To teach the new Indian arithmetic small abbaco schools (scuole d’abbaco or botteghe d’abbco) were established in the towns, where teenagers, who were apprentice traders or merchants, were taught commercial arithmetic and double entry bookkeeping. The teachers, who ran these establishments, maestri d’abbaco, wrote their own textbooks, a genre known as Libri d’abbaco, (abbacus books). The first ever printed mathematics book was an abbacus book, the so-called Treviso Arithmetic or Arte dell’Abbaco written in vernacular Venetian and published in Treviso in 1478. These schools and their textbooks spread to the trading cities of Southern Germany, such as Augsburg, Regensburg and Nürnberg, and from there throughout Europe. In German we have Rechenmeister, Rechenschule and Rechenbucher, in English reckoning masters, reckoning schools and reckoning books. Arithmetic and algebra remained in the province of the traders and merchants as commercial arithmetic until the middle of the fifteenth century. Gerolemo Cardano is credited with bringing algebra into the realm of mathematics with his Artis magnae, sive de regulis algebraicis liber unus published by Johannes Petreius in Nürnberg in 1545 but he also started his career as a mathematical author with an abbacus book, his Practica arithmetice et mensurandi singularis published in Milano in 1538.

The introduction of the Indian numerals into Northern Italy didn’t go entirely unopposed. In 1299 a local law was passed in Florence banning the use of them in bookkeeping, Statuto dell’Arte del Cambio, with the argument that they were easier to change, thus falsifying the accounts, than Roman numerals or written number words. Many modern authors claim that reckoning with the Hindu-Arabic numerals was faster and simpler than using the abacus or reckoning board but I don’t think this is true and I strongly suspect that most merchants continued to do their reckoning on a counting board reserving the new arithmetic for their written bookkeeping.

Leonardo was not just the man, who introduced the place value decimal number system into Europe with his Liber Abbaci, but was also the author of several other important mathematical works establishing him as an important mathematician in thirteenth-century Italy. In 1240 he was even invited to an audience with the Holy Roman German Emperor Frederick II, who was an avid patron of the sciences. The most famous judgement on the introduction of the place value decimal number system is that of the eighteenth-century French polymath Simon Laplace:

“It is India that gave us the ingenious method of expressing all numbers by means of ten symbols, each symbol receiving a value of position as well as an absolute value; a profound and important idea which appears so simple to us now that we ignore its true merit. But its very simplicity and the great ease which it has lent to computations put our arithmetic in the first rank of useful inventions; and we shall appreciate the grandeur of the achievement the more when we remember that it escaped the genius of Archimedes and Apollonius, two of the greatest men produced by antiquity.”

It is Leonardo Pisano to whom we own our thanks for having introduced this invention into Europe. If you want to know more about the man and his book then I recommend Keith Devlin, The Man of Numbers: Fibonacci’s Arithmetic Revolution, Bloomsbury, London, 2011 from which the long quote from the Liber Abbaci is taken.

The theme of this post was requested by one of my anonymous €30 plus GoFundMe donors. It’s slightly different to what he suggested but I hope he’s satisfied with the end result. I wait for other donors to claim their right to negotiate a post theme.

[1] The secondary sources I have consulted say, unknown translator, probably Adelard of Bath, Robert of Chester (who definitely did translate the algebra) and John of Seville, so take your pick. Interestingly several of them name Adelard of Bath but my biography of Adelard says that the attribution is probably false.



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20 responses to “As easy as 1,2,3…

  1. vinjk

    Fascinating history of numbers!

    I’m a first time visitor. I think I will be a regular here from now on.

  2. Laurence Cox

    William H. Goetzmann in “Money changes everything”, a history of the influence of finance on civilisation from Babylonian times onwards, also credits Leonardo of Pisa with introducing the concept of what we now call Net Present Value in one of his Liber Abbaci problems.

  3. the eighth-century Persian mathematician Muḥammad ibn Mūsā al-Khwārizmī (c.780–c.850 CE)

    A question about this: is writing a book in Arab and using an Arab name by a Persian here the equivalent of the European use of Latin as the language for science?

    • Yes, we talk about the Islamic Empire of which the academic language was Arabic but the population of that Empire consisted of many different nationalities.

      • Jeb

        A trick I just learned from David Bloch (translator of Aristotle) is to use the naming convention to deal with historical complexity. He distinguishes Avicenna from Ibn Sīnā noting that the Latin author is more extensively researched than the Arabic author.

        He makes a distinction with naming depending on wither the context is Latin or Islamic.

        Basically it allowed deal with complexity and issues of translation/ current research in a couple of sentences. Terse and to the point.

    • Jacob Rus

      From what I understand al-Khwārizmī was at the time living/working in Baghdad at the House of Wisdom, an institution of the Abbasid Caliphate.

      • The House of Wisdom is a complete myth; it never existed. Read Dimitri Gutas, “Greek Thought, Arabic Culture: The Graeco-Arabic Translation Movement In Baghdad and Early ‘Abbāsid Society (2nd-4th/8th-10th centuries)”. The Wikipedia article is a mess and is largely based on the books of Jim Al-Khalili and Jonathan Lyons both of which are historical crap.

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  11. Jacob Rus

    > abbaco meaning to reckon or calculate in Italian and has nothing to do with abacus

    What is the etymology if this word didn’t come from “abacus”?

  12. It was a nice article and I enjoyed reading it. I may mention that Aryabhatta, a great Indian scholar wrote a treatise “Aryabhatiya” around 500 CE and zero was not mentioned there. Brahmagupta (598–c.670 CE) at the age of 30, wrote “Brahmasphuṭasiddhanta”, the first book that mentions zero as a number. Since we don’t find mention of any notable Indian mathematian in between, it is reasonable to assume that Zero was invented by Brahmagupta somewhere around 628 CE. I hate to mention but there is an article by me where an elaborate history of zero is given.

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