Zero and nothing in medieval Arabic arithmetic

Abstract

Whether explaining calculations with decimal or sexagesimal notation, arithmetic books composed in Arabic beginning in the ninth century CE consistently describe the zero (ṣifr) as a sign indicating an empty place where there is no number. And yet we find that some arithmeticians explicitly performed operations on this zero. To understand how the zero was conceived and manipulated in medieval Arabic texts we first address the way that numbers themselves were conceived and how ‘nothing’ entered into arithmetical problem-solving. From there we examine arithmetic books for their treatment of zero. We find that there is no inconsistency in operating on what is literally nothing, and thus there was no motive for arithmeticians to regard zero as a number.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 In this article both Muslim and Christian dates are given in most instances.

2 Talkhīṣ aʿmāl al-ḥisāb (#696 [M1]), published with a French translation in (Ibn al-Bannāʾ 1969). The entire text is included in the commentary by al-Hawārī, which is published with an English translation in (Abdeljaouad and Oaks 2021). ‘#696 [M1]’ means book [M1] by author #696 in (Rosenfeld and İhsanoğlu 2003). Similarly for subsequent references of this form.

3 Kitāb fī uṣūl ḥisāb al-hindī (#308 [M1]), published with an English translation in (Kūshyār ibn Labbān 1965).

4 The introductions to (Abdeljaouad and Oaks 2021) and (al-Uqlīdisī 1978) give overviews of Arabic arithmetic.

5 Kitāb fīmā yaḥtāju ilayhi al-kuttāb wa l-ʿummāl wa ghayruhum min ʿilm al-ḥisāb (#256 [M2]), published in (Saidan 1971).

6 al-Kitāb fī l-ḥisāb al-Hindī (#41 [M1]). The redaction of the Latin translation with a German translation is published in (al-Khwārazmī 1997). An English translation of roughly the first half is published in (Crossley and Henry 1990).

7 al-Fuṣūl fī l-ḥisāb al-Hindī (#232 [M1]), published in (al-Uqlīdisī 1984). An English translation is published in (al-Uqlīdisī 1978).

8 (Berkey 1992; Chamberlain 1994).

9 (Ibn Sīnā 2005, 18), Marmura’s translation.

10 For the first three authors, these are the books already mentioned above. The other two are (al-Samawʾal MS A) and (al-Kāshī 1969; al-Kāshī 2019).

11 (al-Nīsābūrī MS, 7.1; al-Ḥaṣṣār MS, fol. 3v.4). It is only after describing the nine numerals that the Latin text of al-Khwārazmī gives what amounts to a paragraph to explaining that one is the origin of number and that a number is ‘a collection of ones’. The fact that this paragraph ends with ‘But now let us return to the book’ indicates that it was added later by an editor (Crossley and Henry 1990, 110–111).

12 Asās al-qawāʿid fī uṣūl al-fawāʾid (#674 [M2]), published in (al-Fārisī 1994).

13 Maqālāt fī l-ḥisāb (#696 [M2]), published in (Ibn al-Bannāʾ 1984).

14 Rafʿ al-ḥijāb ʿan wujūh aʿmāl al-ḥisāb (#696 [M8]), published in (Ibn al-Bannāʾ 1994).

15 (Ibn al-Bannāʾ 1994, 207; Aballagh 1988, 142). Compare with (Ibn Sīnā 2005, 91).

16 Kitāb al-tafhīm li-awāʾil ṣināʿat al-tanjīm (#348 [A2]), published with an English translation in (al-Bīrūnī 1934).

17 (al-Bīrūnī 1934, 24), modified from Wright’s translation.

18 Iḥṣāʾ al-ʿulūm, published with a French translation in (al-Fārābī 2015). In this section he considers practical numbers that come from counting. In the next section he distinguishes between practical and theoretical geometry, and there he addresses non-integral measure.

19 (al-Fārābī 2015, 127.5). A dinar was a denomination of gold coin. He then goes on to discuss theoretical numbers, which coincide with the intelligible numbers of the Greek texts.

20 This use of the word ‘species’ is not the same as the species (kinds) of number in Greek number theory, which begin with the even and the odd. Three horses and five horses are the same species by our meaning because they are both horses, while they are of the same species according to Nicomachus because they are both odd.

21 al-Maʿunah fī ʿilm al-ḥisāb al-hawāʿī (#783 [M1]), published in (Ibn al-Hāʾim 1988).

22 (Ibn al-Hāʾim 1988, 91.1). He gives essentially the same explanation in (Ibn al-Hāʾim 1999, 107).

23 The phrase can also be translated ‘divide ten dirhams [equally] among five men’.

24 An irdabb (pl. arādib) is a dry measure equivalent to a little over 5 bushels.

25 (Ibn Ghāzī 1983, 102; Sibṭ al-Māridīnī 2004, 109). Ibn Ghāzī’s book was completed in 890/1483. He quotes Ibn al-Bannāʾ in his definitions.

26 For a more complete discussion of plus/minus vs. and/less, see (Oaks 2009; Abdeljaouad and Oaks 2021, 214ff).

27 See (Oaks and Alkhateeb 2007; Oaks 2009, 2010, 2017) for studies of the nature of Arabic algebraic expressions.

28 al-Fakhrī fī ṣināʿat al-jabr wa l-muqābala (#309 [M2]), in manuscript (al-Karajī MS) and published in (Saidan 1986).

29 While it is true that we can also view our $12x$ as being twelve copies of the $x$, this premodern way does not make sense for terms that are just a little more intricate, like $\sqrt{5}x$, $\pi r^2$, or 3xy. The aggregations interpretation of polynomials is presented in (Oaks 2009, 2010, 2017; Abdeljaouad and Oaks 2021, 214–219).

30 Kitāb fī l-jabr wa l-muqābala (#124 [M1]), published with a French translation in (Abū Kāmil 2012).

31 Here the word māl takes its usual meaning as ‘money’, and is not the name of the second power in algebra.

32 (Abū Kāmil 2012, 427.12). If we convert this to a modern algebra problem, it asks for $a$ and $b$ such that $a+b=10$ and ${\displaystyle \frac{a}{b}+{\displaystyle \frac{b}{a}=\sqrt{5}}}$. The enunciation and first solution are translated in (Oaks 2017, 145–146).

33 Talqīḥ al-afkār fī l-ʿilm bi-rushūm al-ghubār (#521 [M3]), published in (Zemouli 1993).

34 (Zemouli 1993, 209.18). Converting this arithmetic problem into an algebra problem, it asks for x such that ${\displaystyle \frac{1}{3}x+{\displaystyle \frac{1}{4}x+2=2x}}$. The problem is worked out twice, the first time by single false position and the second time by algebra.

35 Murshidat al-ṭālib ilā asnā al-maṭālib fī ʿilm al-ḥisāb (#783 [M5]), published in (Ibn al-Hāʾim 1999).

36 That is, a third and a fourth of the sum, not of the original quantity.

37 The common denominator of ‘a third and a fourth’ is 12. This is repeated, so presuming a value of 144 will result in a whole number after passing it through the operations in the enunciation, disregarding the operations on the added dirham. The terms ‘numerator’ and ‘denominator’ are used because these operations result in ${\displaystyle \frac{95}{144}}$ of the unknown.

38 The third and a fourth are removed from the first dirham, then the remainder is subtracted from ‘the subtracted dirham’, making the new remainder a third and a fourth of a dirham.

39 To ‘denominate’ (from sammā) 84 ‘from’ (min) 95 means to make the name, or species, of the 84 ‘95ths’. This forms the fraction 84/95. In Arabic arithmetic one ‘divides’ (from qasama) a greater number by a smaller number, but one ‘denominates’, or ‘relates’ (from nasaba), a smaller number from a greater number.

40 These books are: #124 Abū Kāmil [M1], #179 Liber augmenti et dimintionis (a Latin translation of an Arabic book) [M1], #256 Abū l-Wafāʾ [M2], #267 ʿAlī al-Sulamī [M1], #309 al-Karajī [M2], [M3], #310 Ibn al-Samḥ [M1], #521 Ibn al-Yāsamīn [M3], #587 Ibn Badr [M1], #657 Ibn al-Khawwām [M2] (for his [M1], see al-Fārisī), #674 al-Fārisī [M2], #696 Ibn al-Bannāʾ [M6], #780 Ibn al-Qunfūdh [M1], #783 Ibn al-Hāʾim [M1], [M5], #1045 al-Qabāqibī [M1], #873 Sibṭ al-Māridīnī [M7], #924 Zakarīyā al-Anṣārī [M2], #997 al-Ghazzī [M1], #1058 al-ʿĀmilī [M1], #1066 Nūr al-Dīn al-Anṣārī [M2]. If an author repeats a problem in two different books, I only count it once.

41 Kitāb al-uṣūl wa l-muqaddimāt fī l-jabr wa l-muqābala (#696 [M6]), published in (Saidan 1986). The other three are in the books of Ibn Badr, al-Fārisī, and Ibn al-Qunfūdh.

42 This ‘restore and confront’, conjugated from al-jabr wa l-muqābala, means to simplify the equation by restoration and/or confrontation (Oaks and Alkhateeb 2007, §4.1).

43 Ḥaṭṭ al-niqāb ʿan wujūh aʿmāl al-ḥisāb (#780 [M1]), in manuscript (Ibn al-Qunfūdh, MS).

44 For an overview of this algebraic notation, see (Oaks 2012).

45 The 35 books are: #41 al-Khwārazmī [M1], #218 al-Nīsābūrī [M1], #232 al-Uqlīdisī [M1], #308 Kūshyār ibn Labbān [M1], #320 al-Baghdādī [M1], #411 al-Ṣardafī [M1], #487 al-Samawʾal [M4], #521 Ibn al-Yāsamīn [M3], #532 al-Ḥaṣṣār [M1], #566 Ibn Munʿim [M1], #606 Naṣīr al-Dīn al-Ṭūsī [M17], #682 al-Abharī [M1], #696 Ibn al-Bannāʾ [M1], [M2], #747 al-Hawārī [M1], al-Mawāḥidī (#M176 in (Lamrabet 2014)), #780 Ibn al-Qunfūdh [M1], #783 Ibn al-Hāʾim [M5], [M7], #802 al-Kāshī [M1], #815 Ibn al-Majdī [M3], #865 al-Qalaṣādī [M3], [M7], #980 Raḍī al-Dīn ibn al-Ḥanbalī [M3], #997 al-Ghazzī [M1], #1004 Taqī al-Dīn [M2], #1006 Yaḥyā al-Ruʿaynī [M1], #1026 al-Sakhāwī [M1], #1045 al-Qabāqibī [M1], #1058 al-ʿĀmilī [M1], #1066 Nūr al-Dīn al-Anṣārī [M2], #1186 Jawād ibn Saʿd ibn Jawād al-Kāẓimī [M1], #1355 Ḥusayn al-Maḥallī [M2], the guide by the lexicographer al-Khwārazmī (1895), and the anonymous Paris, BnF arabe 4441, copied in 1572.

46 Bulūgh al-ṭullāb fī ḥaqāʾiq ʿilm al-ḥisāb (#218 [M1]), in manuscript (al-Nīsābūrī MS).

47 (al-Nīsābūrī MS, 20; Ibn al-Qunfūdh MS, 12).

48 (Crossley and Henry 1990, 111), adjusted slightly from their translation.

49 (al-Uqlīdisī 1978, 42), adjusted slightly from Saidan’s translation.

50 Kitāb al-bayān wa l-tadhkār fī ṣanʿat ʿamal al-ghubār (#532 [M1]), in manuscript (al-Ḥaṣṣār MS).

51 Jāmiʿ al-ḥisāb bi l-takht wa l-turāb (#606 [M17]), published in (Saidan 1967).

52 Miftāḥ al-ḥisāb (#802 [M1]), published in al-Kāshī 1969; (al-Kāshī 2019), the latter covering the first, arithmetical part together with an English translation.

53 (al-Kāshī 1969, 46.16; 2019, 34.15). Our references to (al-Kāshī 2019) are for the page and line number of the Arabic.

54 Fatḥ al-wahhāb ʿalā nuzhat al-ḥussāb (#1066 [M2]), in manuscript (al-Anṣārī MS).

55 This is an oral instruction not to misspell ṣifr (‘zero’) as sifr (‘book of scripture’).

56 Ibn al-Bannāʾ’s [M1], Ibn al-Hāʾim’s [M5], and al-Qalaṣādī’s [M7] do not explain the meaning of zero, but their books [M2], [M7], and [M3] respectively do. Ḥusayn al-Maḥallī’s [M2] is a commentary on al-Sakhāwī’s [M1], and Jawād ibn Saʿd ibn Jawād al-Kāẓimī’s [M1] is a commentary on al-ʿĀmilī’s [M1]. Al-Kāẓimī was a student of al-ʿĀmilī.

57 al-Lubāb fī sharḥ Talkhīṣ aʿmāl al-ḥisāb (#747 [M1]), published with an English translation in (Abdeljaouad and Oaks 2021).

58 al-Takmila fī l-ḥisāb (#320 [M1]), published in (al-Baghdādī 1985).

59 (al-Uqlīdisī 1978, 46), Saidan’s translation.

60 Kitāb al-tabṣira fī ʿilm al-ḥisāb (#487 [M4]), in manuscript (al-Samawʾal MS A).

61 (Zemouli n.d., 11.5). Note that it is only the quantity, and not the quantity-species, of the multiplier that is applied. As some authors note, this definition only makes sense for whole numbers. Finger-reckoners had their own definition of multiplication based in proportion that is valid for fractional numbers. The Euclidean definition suffices for operating on Indian numerals because the nine possible entries are all whole numbers.

62 Mukhtaṣar al-hindī fī ʿilm al-ḥisāb (#411 [M1]), in manuscript (al-Ṣardafī MS).

63 (al-Uqlīdisī 1978, 190), Saidan’s translation.

64 (Crossley and Henry 1990, 116), their translation.

65 Fiqh al-ḥisāb (#556 [M1]), published in (Ibn Munʿim 2005).

66 (al-Kāshī 1969, 48.9; 2019, 38.15), Aydin’s and Hammoudi’s translation.

67 (Saidan 1967, 254). The ‘02’ is mistakenly shown as ‘03’ in the text. The word burj designates 1/12 of the circle, or 30 degrees, and takes the meaning of ‘sign’ of the zodiac. It is not a problem that the minuend is smaller than the subtrahend in this example. He adds 12 signs to the minuend and then performs the subtraction.

68 (al-Bīrūnī 1934, 42), Wright’s translation.

69 (al-Uqlīdisī 1978, 87, 88), adjusted from Saidan’s translations. For the second quotation, the figure again shows a pair of zeros.

70 (al-Kāshī 1969, 104.6; 2019, 178.18), Aydin’s and Hammoudi’s translation.

71 Translation by Toomer, adjusted slightly (Ptolemy 1898, 500.20; 1998, 295). One 0 is to be placed in the first line of the table and the other in the last line.

72 I thank Jan Hogendijk for pointing out this passage to me, and the translation is his.

73 al-Badīʿ fī l-ḥisāb (#309 [M3]), published with a French summary in (al-Karajī 1964).

74 al-Bāhir fī ʿilm al-ḥisāb (#487 [M1]), in manuscript (al-Samawʾal MS B) and published with a French translation in (Rashed 2021).

75 (al-Samawʾal MS B, fol. 28r; Rashed 2021, 59). We reproduce this table and the next based on MS Aya Sofia 2718, the only manuscript which shows the tables. We do not know why zeros are shown for only some empty places in the manuscript.

76 (al-Samawʾal MS B, fol. 29r; Rashed 2021, 61).

77 Sufficient introduction on calculation by algebra and what one can learn from its examples (al-Muqaddima al-kāfiyya fī ḥisāb al-jabr wa l-muqābala wa mā yuʿrafu bihi qiyāsuhū min al-amthila) (#267 [M1]), in manuscript (ʿAlī al-Sulamī).

78 (Ibn al-Bannāʾ 1994, 246.15; Abdeljaouad and Oaks 2021, 52). In our translation in the latter reference we wrote ‘deleted’ for ‘lacking’. Either translation fits the meaning.

79 Al-Hawārī writes of ‘twenty-four starlings’ (Abdeljaouad and Oaks 2021, 110), al-Karajī of ‘two hours and ten parts of nineteen parts of an hour’ (Saidan 1986, 185.16), and Abū l-Wafāʾ of ‘twenty-nine inches and two thirds of an inch’ (Saidan 1971, 205.11). ‘Ten dirhams’ and ‘three fourths of a unit’ are common.