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# Al-Marrakushi ibn Al-Banna

## (29 Dec 1256 in Marrakesh, Morocco- 1321 in Marrakesh)

Ibn al-Banna is also known as Abu'l-Abbas Ahmad ibn Muhammad ibn Uthman al-Azdi. It is a little unclear whether al-Banna was born in the city of Marrakesh or whether it was the region of Marrakesh which was named Morocco by Europeans. There is a claim that al-Banna was born in Granada in Spain and moved to North Africa for his education. What is certain is that he spent most of his life in Morocco.

The Marinids tribe were allies of the Umayyad caliphs of Córdoba. The tribe lived in eastern Morocco then, under their ruler Abu Yahya, they began to conquer the region. The Marinids captured Fez in 1248 and made it their capital. They captured Marrakesh from the ruling Almohads tribe in 1269, thus taking control of the whole of Morocco. Having conquered Morocco, the Marinids tried to help Granada to prevent the Christian advance through their country. The strong link between Granada and Morocco may account for the confusion as to which country al-Banna was a native.

Morocco was certainly the country that al-Banna was educated in, learning the leading mathematical skills of the period. He studied geometry in general andEuclid's Elements in particular. He also studied fractional numbers and learnt much of the impressive contributions that the Arabs had made to mathematics over the preceding 400 years. The Marinids had a strong culture for learning and Fez became their centre of learning. At the university in Fez Al-Banna taught all branches of mathematics, which at this time included arithmetic, algebra, geometry and astronomy. Fez was a thriving city with a new quarter being built housing the Royal Palace and the adjoining Great Mosque. Many students studied under al-Banna in this thriving academic community.

It is clear that al-Banna wrote a large number of works, in fact 82 are listed by Renaud (see for example [1]). Not all are on mathematics, but the mathematical texts included an introduction toEuclid's Elements, an algebra text and various works on astronomy. One difficulty with the works on mathematics is knowing how much of the material which al-Banna presents is original and how much is simply his version of work by earlier Arab mathematicians which has been lost. We should certainly say that al-Banna does not claim any originality and, indeed, the style of his writing would suggest that he is collecting together ideas that he has learnt from other mathematicians.

Two "firsts" for al-Banna are that he seems to have been the first to consider a fraction as a ratio between two numbers (see [2] for more details) and he is the first to use the expression almanakc (in Arabic al-manakh meaning weather) in a work containing astronomical and meteorological data.

Perhaps al-Banna's most famous work is Talkhis amal al-hisab (Summary of arithmetical operations) and the Raf al-Hijab which is al-Banna's own commentary on the Talkhis amal al-hisab. It is in this work that al-Banna introduces some mathematical notation which has led certain authors to believe that algebraic symbolism was first developed in Islam by ibn al-Banna andal-Qalasadi (see for example [3]). We refer the reader to the biography ofal-Qalasadi where we present arguments to show that neither al-Banna noral-Qalasadi were the inventors of mathematical notation.

There are, however, many interesting mathematical ideas and results which appear in the Raf al-Hijab. For example it containscontinued fractions[4] and they are used to compute approximate square roots. Other interesting results on summing series are the results

13+ 33+ 53+ ... + (2n-1)3= n2(2n2- 1) and

12+ 32+ 52+ ... + (2n-1)2= (2n + 1)2n(2n - 1)/6.

Perhaps the most interesting of all is the work onbinomial coefficients[5] which is described in detail in [6] and [7]. If we denote the binomial coefficient p choose k bypCk then al-Banna shows that

pC2= p(p-1)/2

and then that

pC3=pC2(p-2)/3.

He writes (see for example [6] or [7]):-

... the ternary combination is thus obtained by multiplying the third of the third term preceding the given number; and so we always multiply the combination that precedes the combination sought by the number that precedes the given number, and whose distance to it is equal to the number of combinations sought. From the product, we take the part that names the number of combinations.

Although this is a little difficult to interpret, what al-Banna is stating here is that

pCk=pCk-1(p - (k - 1) )/k.

He then goes on to give the familiar (to us) result

pCk= p(p - 1)(p - 2)...(p - k + 1)/(k !)

As Rashed points out in [6], this is only a small step from thePascal triangle[8] results given three hundred years earlier byal-Karaji, then still one hundred years before al-Banna byal-Samawal. However Rashed writes:-

... in our opinion, there is something more fundamental than [the Pascal triangle] results; it is precisely the combinatorial appearance of ibn al-Banna's exposition, together with the relation he partially establishes betweenpolygonal numbers[9] andcombinations[10]. It concern, in the first place,triangular numbers[11] and combinations of p objects in twos, and then polygonal numbers of order 4 and combinations of p objects in threes.

## Article by: J J O'Connor and E F Robertson

## Notes:

1-H P J Renaud, Ibn al-Banna de Marrakech, sufi et mathématicien, Hesperis

## 25

(1938), 13-42.2-M Zarruqi, Fractions in the Morroccan mathematical tradition between the 12th and 15th centuries A.D. as found in anonymous manuscripts (Arabic), in Deuxième Colloque Maghrebin sur l'Histoire des Mathématiques Arabes (Tunis, 1990), A97-A109.

3-G Arrighi, Review of some mathematical symbols (Italian), Physis - Riv. Internaz. Storia Sci. 27 (1-2) (1985), 163-179.

4

## -

The continued fraction expansion of a number r is an expression of the form:If r is arational number this expansion terminates.

5

## -

The binomial coefficients are the coefficients of powers of x in the expansion of (1 + x)n.

We have

where the binomial coefficient is the number of ways of choosing an (unordered) subset of size k from a set of size n.

The binomial coefficients are the entries in thePascal triangle.

6-R Rashed, The development of Arabic mathematics : between arithmetic and algebra (London, 1994).

7-R Rashed, Entre arithmétique et algèbre: Recherches sur l'histoire des mathématiques arabes (Paris, 1984).

8-The Pascal triangle (actually known long before Pascal) is a table of thebinomial coefficients where the (n, k)th entry is

Each entry is the sum of the pair immediately above it.

9-A polygonal number is the number of dots that may be arranged in a regular polygon.

As, for exampletriangular numbers,square numbers, ...

10-A combination is a subset of a given set where the order of elements is ignored (as distinct from apermutation).

For example there are 6 combinations of size 2 from the set {1, 2, 3, 4}:

{1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 4}, {3, 4}.

The number of combinations of size k from a set of size n is thebinomial coefficient also writtennCk.

11-A triangular number is the number of dots that may be arranged in an equilateral triangle: 1, 3, 6, 10, ...

In general n(n + 1)/2.

Taken from:

http://www-history.mcs.st-andrews.ac.uk/Mathematicians/Al-Banna.html