To understand how medieval Arab mathematicians wrote numbers and operations, let us begin with a practical example. In his Book on the Arithmetic Necessary for Scribes and Merchants (Kitāb fi ma yaḥtāju ilayhi al-kuttāb wa l-‘ummāl min ‘ilm al-ḥisāb), the mathematician Abū l-Waf ā al-Buzjanī (died 998) describes how to multiply two integers, illustrating the method with 7,857 multiplied by 3,465. He begins: "To multiply seven and fifty and eight hundred and seven thousand by five and sixty and four hundred and three thousand, one must begin by multiplying seven thousand by five [units]." He does not represent any numbers with numerals; instead, he writes out the name of each digit in full, along with the corresponding place values (units, tens, hundreds or thousands).
For 7,857 × 3,465, he needs sixteen operations, whose results are laboriously arranged over several lines while respecting the hierarchy of the place values. The entire operation fills two pages.
Yet Abū l-Waf ā knew the Indian numerals and could have used them. He chose not to because, as he explains: "Anyone who masters this type of multiplication can dispense with the system used by the Indians in their cosmographical calculations, since this type of calculation can be checked neither by finger reckoning nor by mental arithmetic. This method is easier to use than one requiring sand and a board (takht) or ink, since sand and a board are not available everywhere, and not everyone practises this [Indian] arithmetic. What we propose [here] allows them to dispense with all that."
In his book, Abū l-Waf ā addresses both experienced and apprentice scribes, as well as property managers and merchants; he does not seek to offer them new techniques for conducting their transactions.
In the passage quoted above, Abū l-Waf ā mentions finger reckoning and a form of arithmetic used by specialists in astronomy. What exactly were they?
