Binomial Coefficients and Nasir Al-Din Al-Tusi

Perspective

Binomial coefficients and Nasir al-Din al-Tusi

Othman Echi

Faculty of Sciences of Tunis, Department of Mathematics, University Tunis-El Manar"Campus Universitaire", 2092,Tunis, TUNISIA. E-mail: [email protected],[email protected].

Accepted 3 November, 2006

A historical note is given about the scientist Nasir al-Din al-Tusi legitimating the introduction of a new concept related to binomial coefficients. Al-Tusi binomial coefficients and binomial formulas are introduced and studied.

Key words: Binomial coefficients, binomial theorem, history of Mathematics.

INTRODUCTION

HISTORICAL NOTE AND NOTATIONS Al-Tusi has wrote important works on Astronomy, logic, Mathematics and Philosophy. The first of these works, Abu Jafar Muhammad Ibn Muhammad Ibn al-Hasan ``Akhlaq-i nasiri", was written in 1232; it was a work on Nasir al-Din al-Tusi was born in Tus, Khurasan (now, ethics which al-Tusi dedicated to the Isma'ili ruler Nasir Iran) in 17 February 1201 and died in Baghdad 25 June ad-Din Abd ar-Rahim. Al-Tusi was kidnaped by the 1274. Al-Tusi was one of the greatest scientists, mathe- Isma'ili Hasan Bin Sabah's agents and sent to Alamut maticians, astronomers, philosophers, theologians and where he remained until its capture by the Mongol physicians of his time. Al-Tusi was an Arabic scholar Halagu Khan. Impressed by Al-Tusi's exceptional abilities whose writings became the standard texts in several and astrological competency, Ilkhanid Halagu Khan disciplines for several centuries. They include editions of appointed him as one of his ministers. Later, has he Euclid's Elements and Ptolemy's Almagest, as well as been designated an administrator of Auqaf. other books on mathematics and astronomy, and books In 1262, al-Tusi built an observatory at Meragha (in the on logic, ethics, and religion. Al-Tusi was known by a Azerbaijan region of north-western Iran) and directed its number of different names during his lifetime such as activity. It was equipped with the best instruments from Muhaqqiq-i Tusi, Khwaja-yi Tusi and Khwaja Nasir. His Baghdad and other Islamic centers of learning. It proper name was Muhammad ibn Muhammad ibn al- contained a twelve-feet wall quadrant made from copper Hasan al-Tusi. and an azimuth quadrant and 'turquet' invented by al- In 1214, when al-Tusi was 13 years old, Genghis Tusi. Other instruments included Astrolabes, represent- Khan, who was the leader of the Mongols, turned away tations of constellation, epicycles, and shapes of sphe- from his conquests in China and began his rapid res. Al-Tusi designed several other instruments for the advance towards the west. Genghis Khan turned his Observatory. After putting his Observatory to good use, attention again towards the east leaving his generals and making very accurate tables of planetary movements, al- sons in the west to continue his conquests. The Mongol Tusi published his work “Zij-i ilkhani (the Ilkhanic Tables)" invasion caused much destruction in both west and east. (which was dedicated to Ilkhanid Halagu Khan), written Fortunately, al-Tusi was able to study more advanced first in Persian and later translated int Arabic. This work topics before seeing the effects of the Mongols con- contains tables for computing the positions of the pla- quests on his own regions. From Tus, al-Tusi went to nets, and it also contains a star catalogue. The tables Nishapur which is 75 km west of Tus; this was a good were developed from observations over a twelve-year choice for al-Tusi to complete his education since it was period and were primarily based on original observations. an important centre of learning. There, al-Tusi studied The major astronomical treatise of al-Tusi was “al- Philosophy, Medicine and Mathematics. Particularly, he Tadhkira fi'ilm al-hay'a". This was written to give serious was taught Mathematics by Kamal al-Din ibn Yunus, who students a detailed acquaintance with astronomical and himself had been a pupil of Sharaf al-Din al-Tusi. In cosmological theory. In this treatise Nasir, gave a new Nishapur, al-Tusi began to acquire a reputation as an model of lunar motion, essentially different from Pto- outstanding scholar and became well known throughout lemy's. In his model Nasir, for the first time in the history the area. of astronomy, employed a theorem invented by himself Echi 029

which, 250 years later, occurred again in Chapter IV of al-Karaji's school. In the manuscript, al-Tusi determined Book III of Copernicus' "De Revolutionibus" (On the the coefficients of the expansion of a binomial to any revolutions of the heavenly spheres). This theorem runs power giving the binomial formula and the Pascal triangle as follows: If a point moves with uniform circular motion relations between binomial coefficients. clockwise around the epicycle while the center of the Edwards (2002) has postulated that the work of al- epicycle moves counterclock-wise with half this speed Karaji in expanding the Binomial Triangle might have along an equal deferent circle, the point will describe a borrowed Brahmegupta's work, given that it was avai- straight-line segment. lable and al-Karaji definitely had read other Hindu texts Many historians claim that the Tusi-couple result was available in Baghdad which was the great cultural and used by Copernicus after he discovered it in al-Tusi's scientific center of Muslims. The binomial coefficients work (see for example, Boyer (1947) and Dreyer (1953). have been studied in cultures around the world, both in However, Veselovsky (1973) shows that it is much more the context of binomial expansions and in the question of plausible to suppose that Copernicus took the argument how many ways to choose k items out of a collection of n he needed from Proclus' "Commentary on the first book things. Note also that, three names from China figure of Euclid", and not from al- Tusi. There are two other prominently in the story of the discovery of the binomial astronomical treatises of al-Tusi. The first treatise, called coefficients. These scientists are Chia Hsien, Yanghui the "Muiniya" and written in 1235, contains a standard (1261) and Chu Shih-chieh (1303). The idea of taking account of Ptolemaic lunar and planetary theory. In the “six tastes one at a time, two at a time, three at a time, second treatise, the "Hall" (written between 1235 and etc." was written down correctly in India 300 years before 1256), al-Tusi uses the plane version of his "Tusi-couple" the birth of Christ in a book called the ``Bhagabati Sutra". to explain the motion in longitude of the epicycle centre Thus the Indian civilization is the earliest one that has an of the moon and the planets. The "Hall" does not yet understanding of the binomial coefficients in their combi- include the spherical version of the "Tusi-couple", which natorial form ``n choose k". is used in the "Tadhkira" to describe the prosneusis of The interest in the binomial coefficients in India dealt the moon and the latitude theory of the planets. It is with choosing and arranging things. However, mathema- worth noting that the term "Tusi couple" is a modern one, ticians in the Middle East were interested almost entirely coined by Edward Kennedy (1966). in expansions of polynomials; the work of the Indian It is also worth noting that al-Tusi has written revised Brahmagupta, which included the expansion of (a + b)3 , Arabic versions of works by Autolycus, Aristarchus, Euc- was available to the scholars of the Middle East. Some lid, Apollonius, Archimedes, Hypsicles, Theodosius, Me- Historians consider that investigations on binomial coef- nelaus and Ptolemy. Ptolemy's Almagest was one of the ficients by mathematicians in the Middle East may well works which Arabic/Muslim scientists studied intently. In have been inspired Brahmagupta's work. There is anot- 1247 al-Tusi wrote ``Tahrir al-Majisti (Commentary on her scientist from the Middle East who worked with the the Almagest)" in which he introduced various trigono- binomial coefficients; namely, al-Samawal (a Jew born in metrical techniques to calculate tables of sines; al-Tusi Baghdad who died in 1180). Omar al-Khayyam is ano- gave tables of sine with entries calculated to three ther famous Persian scientist who makes a claim to sexagesimal places for each half degree of the argu- knowledge; he wrote a letter claiming to have been able ment. Ibadov (1968) has asserted that al-Tusi had found to expand binomials to sixth power and higher, but the the value of the sine of one degree (with the precision up actual work does not survive; in the letter he mentions to the fifth decimal place). Ibadov considers some trigo- that he is aware not only of work done in India, but of nometric propositions used for this purpose by Tusi and Euclid's Elements. Al-Khayyam is best known in the their relation with analogous results obtained by scien- West for his collection of poems “The Rubaiyat", which tists of Central Asia and Western Europe. was translated into English in 1859 by Edward Fitzge- An important mathematical contribution of al-Tusi was rald. the creation of trigonometry as a mathematical discipline In Europe, there are many authors who can fairly lay in its own right rather than as just a tool for astronomical claim to having made a serious study of the binomial applications. In “Treatise on the quadrilateral", al-Tusi coefficients, several of them long before Blaise Pascal gave the first extant exposition of the whole system of was even born. Blaise Pascal was not the first man in plane and spherical trigonometry. This work is really the Europe to study the binomial coefficients, and never clai- first in history on trigonometry as an independent branch med to be such; indeed, both Blaise Pascal and his of pure Mathematics. father Etienne had been in correspondence with Father There is a method due to al-Tusi (dated 1265), from an Marin Mersenne, who published a book with a table of apparently previously unexplored manuscript in Tash- binomial coefficients in 1636. kent, for extracting roots of any order of a number and for It is very reasonable to claim that the manuscripts determining the coefficients of the expansion of a bino- written by Arabic/Muslims scientists have deeply influen- mial to any power (see for example Ahmedov, 1970). ced the philosophical and scientific thoughts during Ren- This work is al-Tusi's version of methods developed by aissance Europe; libraries inherited by Europeans in 030 Sci. Res. Essays

Spain (after the exit of Muslims from Andalusia) is a fied: gigantic scientific treasure that contributed to the deve- Addition: R is an Abelian group relative to addition. lopment of Europe. We think also that, there is no need Multiplication: R is a semigroup relative to multiplication to deny brilliant contributions of medieval Arabic/Muslims (i.e., multiplication satisfies the associativity property). scientists. Undoubtedly Arabic/Muslim civilization has Distribution: For any elements a , b , and c of R , we contributed meaningfully to the human civilization; have the following equalities: contrary to what some Western politicians, who ignore a(b + c) = ab + ac and (b + c)a = ba + ca. the history of civilizations, think. To close this historical note, let us postulate that it is The identity element for addition is denoted by 0 and not easy to give evidence for the claim of precedence for the additive inverse of a is written as ( − a ). If in addi- al-Tusi in finding binomial coefficients and Pascal's tion, multiplication is commutative we say that R is a triangle. Nevertheless, the aim of this paper is the intro- commutative ring. A unitary ring is a ring having a multi- duction of a new type of binomial coefficients that will be plicative identity element (denoted by 1) distinct from 0. called al-Tusi binomial coefficients (in honour to Nasir al- Din al-Tusi). Now, we are in a position to introduce a new concept.

In order to make this paper as self contained as possible (and also relatively accessible to any scientist), Definition 1 here are most of notions and concepts used. Let be a commutative unitary ring and p n! p (R,+,×) Let n, p ∈ N . Set A := , for n ≥ p and An = 0 , n (n − p)! f : IN × IN → R be a map. We say that ( f (n, p)) are al- otherwise; where n!:= n(n −1)...2. Tusi Binomial coefficients (TBCs, for short)}, if the following properties hold: Recall, also, the classical binomial coefficient: other ≈n ’ ∆ ÷ n! for n ≥ p and ≈ n ’ = 0, wise. Then we ∆ ÷ := , ∆ ÷ (i) f (n,0) =1 , for each n ∈ IN ; « p◊ p!(n − p)! « p◊ have the well known recurrence relations: (ii) f (n, p) = 0 , for each p > n ; p p p−1 A = A + pA for , p ≥ 1; and ≈n ’ ≈n −1’ ≈n −1 ’ (iii) There exist α,β ∈ R such that n n−1 n−1 ∆ ÷ = ∆ ÷ + ∆ ÷ « p◊ « p ◊ « p −1◊ f (n, p) = f (n −1, p) + (α + pβ) f (n −1, p −1), foreach n, p ≥1. ``Pascal's formula” for n, p ≥1. These recurrence relations are well known to all school If there is no confusion, f (n, p) will be denoted by T p. children. Looking at the above recurrence relations in two n p variables and in honour to Nasir al-Din al-Tusi, we will It is easily seen that the double sequences (An ) and introduce a new concept. Firstly, as promised, let us ≈n ’ are TBCs. The name ``Binomial Coefficients" allotted ∆ ÷ recall some elementary definitions. « p◊ - By a binary operation on a set S , we mean a to al-Tusi will be justified in Section 2. map*: S × S → S . In mathematics, the binomial theorem is an important - A group is a mathematical system consisting of a formula giving the expansion of powers of sums. The nonempty set G , a binary operation denoted by + (and simplest version of that theorem is the following: let x, y considered as abstract addition) and the axioms: be two elements of a ring (R,+,×) such that xy = yx . (1) (a + b) + c = a + (b + c) , for any elements of G Then we have a well known ``Newton's Binomial (associativity). Formula": n ≈ ’ (2) G contains an element e such that n n i n−i . (x + y) = ƒ∆ ÷x y a + e = e + a = a for each a in G (the element e is i=0«i ◊ called an identity element of the group). The proof is a straightforward induction and can be (3) For each a in G , there is an element −a in G found in all good high school advanced algebra books. In such that a+(−a)=(−a)+a=e . The element −a is called an the following, we generalize the previous binomial formu- (inverse of a ). la in the noncommutative setting.

If the following additional axiom is assumed, the group is said to be a commutative or Abelian group: Definition 2

(4) a + b = b + a, for any elements a and b in G Let R be a ring and (A,+,•,×) be an R -algebra with unit (commutativity). element 1. Let ( F ,n∈ IN ) be a sequence of elements of - A system (R,+,×) consisting of a nonempty set R and n two binary operations, called addition and multiplication, A. We say that ( Fn ,n ∈ IN ) is a Newton's formula if there on R is called a ring if the following conditions are satis- following properties hold: Echi 031

(1) ( Fn ,n∈ IN ) is iterated under * (i.e.; Fn+1 = F1 * Fn , for Thus, clearly, each n ∈ IN ). ≈ n ’ 1 ∆ ÷ -- Suppose that (2) x*1 = x , for each x ∈ A. Tn = n (α + β ) = ∆ ÷(α + β ). «1 ◊ (3) x*(y z) x* y x* z , for each x, y, z A. + = + ∈ ≈ n ’ l l ∆ ÷ for1 ≤ l ≤ k ; and let us (4) x * (λy) = (λx) * y = λ(x * y), for each λ ∈ R . Tn = ∆ ÷(Π (α + iβ )), « l ◊ i =1

k + 1 [Such an operation * will be called a compatible binary compute Tn . operation with the structure of R -algebra on A .] n We will, simply, write Fn = (F1)* . By induction hypothesis, we have In connection with al-Tusi Binomial coefficients and ≈ n −1 ’ k +1 Newton's binomial formula, we introduce the following T k +1 = T k + 1 + ∆ ÷( (α + iβ )), n n −1 Π concept. « k ◊ i =1

Definition 3 which gives immediately the following: n −1 ≈ i ’ k +1 T k +1 ( ∆ ÷( ( i )) Let R be a commutative ring and x, y be two indeter- n = ƒ ∆ ÷ Π α + β i= k « k ◊ i=1 minates over R . We call al-Tusi Binomial formula (TBF, ≈ n ’ k +1 for short) the sequence ( F ( x , y ), n ∈ IN ) in R[x, y] ∆ ÷ n = ∆ ÷(Π (α + iβ )) , i i=1 n « k ◊ defined by i i n−i where (Tn ) are TBCs. Fn (x, y) = ƒTn x y , i=0 The main goal of this paper is to compute TBCs and finishing the induction. show that each TBF is a Newton's formula in the sense of Definition 2. Note that, more details and applications of the conc- THE BINOMIAL FORMULA epts introduced, in this paper, will be considered for publication in a suitable Mathematical Journal. Examples

(1). Let (E,+,•,×) be a unitary algebra over a ring R . BINOMIAL COEFFICIENTS Then × is a compatible binary operation with the The following result computes TBCs. structure of R -algebra on E . (2) Let ( R ,+ ,×) be a commutative ring with unit. Then Theorem 1: Let (R,+,×) be a commutative unitary ring × is a compatible binary operation with the structure of and (T p , n, p ∈ IN ) be a double sequence of elements of n R -algebra on ( R ,+ ,×,×) . If x, y ∈ R are such R such thatT 0 = 1 , for each n∈ IN and T p = 0 , for n n n ≈n’ ∆ ÷ i n−i each p > n . Then the following statements are that xy = xy , we let Bn (x, y) be the sum ƒ∆ ÷x y , «i ◊ equivalent: i=0 p then B (x, y) is iterated by the operation × (according to (i) (Tn ) are TBCs; n the Newton's binomial formula). (ii) There exist α,β in R such that ≈ n ’ p , for each and Naturally, it will be very nice if every TBF is a Newton p ∆ ÷ n ∈ IN Tn = ∆ ÷ (Π (α + iβ )) « i ◊ i =1 formula. This will be carried out in the following result.

p ∈ IN \ { 0}. Theorem 2 Proof: Let R be a commutative ring and x, y be two (ii) (i) . Straightforward. indeterminates over R . Then each TBF is a Newton's (i) (ii) . We use induction on k ∈ IN . Formula in the R -algebra (R[x, y],+,•,×).

1 -- If k = 1, then Tn satisfies the recurrence relation: Proof

1 1 0 Tn = T n + (α + β )T n Let (F (x, y),n ∈ IN) be a TBF in R[x, y]. We are aiming −1 −1 n 1 to construct a binary operation * on R[x, y] such that = T n −1 + (α + β ). 032 Sci. Res. Essays

n REFERENCES Fn = (F1 )* and is compatible with the structure of R - algebra on R[x, y], ,., ) . Ahmedov SA (1970). Extraction of a root of any order and the + × binomial formula in the work of Nasir ad-Din at-Tusi (Russian). Mat. n i V Skole(5): 80-82. Let us write i i n−i where (Tn ) are TBCs. Fn (x, y) = ƒTn x y , di Bono M (1995). Copernicus, Amico, Fracastoro and Tusi's device: i=0 observations on the use and transmission of a model. J. Hist. There existα, β ∈ R such that Astronom. 26 (2): 133-154. k k k−1 Dorofeeva AV (1989). Nasir ad-Din at Tusi (1201-1274) (Russian). Mat. Tn = Tn−1 + (α + kβ )Tn−1 , V Shkole. (3): 145-146. for each n,k 1. Then, for we have Edward AWF (2002). Pascal's Arithmetical Triangle: The story of a ≥ n∈ IN \{ 0}, mathematical idea. Revised reprint of the 1987 original. Johns Hopkins University Press, Baltimore, MD.Hartner W (1969). Nasir al- n Din al-Tusi 's lunar theory, Physis - Riv. Internaz. Storia Sci. 11 (1-4): k k n−k Fn (x, y) = ƒTn x y 287-304. k =0 Ibadov RI (1968). Determination of the sine of one degree by Nasir ad- Din at-Tusi (Russian). Izv. Akad. Nauk Azerbaidzan. SSR Ser. Fiz.- n n Tehn. Mat. Nauk (1): 49-54. n k k n−k k −1 k n−k = y + ƒTn−1x y +ƒ (α + kβ )Tn−1 x y Kennedy ES (1984). Two Persian astronomical treatises by Nasir al-Din k =1 k=1 al-Tusi, Centaurus. 27(2):109-120. Livingston JW (1973). Nasir al-Din al-Tusi's 'al-Tadhkirah’: A category n−1 n of Islamic astronomical literature. Centaurus 17 (4): 260-275. = y n + y( T k x k y (n−1)−k ) + α T k x k y n−k Ragep JF (1987). The two versions of the Tusi couple, in From deferent ƒ n−1 ƒ n−1 to equant (New York, 1987). pp. 329-356. k =1 k=1 Rozenfeld BA (1951). On the mathematical works of Nasir al-Din al- n k −1 k n−k Tusi (Russian). Istor.-Mat. Issled.4: 489-512. + β ƒ kTn−1 x y Saliba G (1951). The role of the 'Almagest' commentaries in medieval k =1 Arabic astronomy : a preliminary survey of Tusi's redaction of Ptolemy's 'Almagest'. Arch. Internat. Hist. Sci. 37 (118): 3-20. n−1 Saliba G (1999) Whose Science is Arabic Science in n n−1 k k +1 (n−1)−k = y + y(Fn−1 (x, y) − y ) + α ƒTn−1 x y Renaissance Europe? k=0 (http://www.columbia.edu/~gas1/project/visions/case1/sci.1.html).Street n−1 T (1995). Tusi on Avicenna's logical connectives. Hist. Philos. Logic k k+1 (n−1)−k 16 (2): 257-268. + β ƒ(k +1)Tn−1x y k =0 Veselovsky IN (1973). Copernicus and Nasir al-Din al-Tusi. J. Hist. Astronom. 4 (2): 128-130.

= yFn−1 (x, y) + αxFn−1 (x, y) + βxFn−1 (x, y)

∂F = (y + (α + β )x)F (x, y) + βx 2 n−1 (x, y) n−1 ∂x ∂F + βx 2 n−1 (x, y) ∂x

F 2 ∂ n−1 = F1(x, y).Fn−1(x, y) + βx (x, y). ∂x

We define *: R[x, y]× R[x, y] → R[x, y], by

∂g f * g = fg + βx2 . Then clearly, * is a binary operation x ∂ which is compatible with the structure of R -algebra on n (R[x, y],+,•,×); and we have F = ( F ) * for n 1 each n ∈ IN \{ 0}. Therefore, (Fn ,n ∈ IN) is a BF.