Although al-Umawi lived in Damascus in Syria, he came from Andalusia in the south of Spain. The name Andalusia comes from the Arabic "Al-Andalus" given to this district by the Muslims who conquered it in the 8th century. The unified Spanish Muslim state broke up in the early 11th century but Muslims from Africa kept Spanish Islam strong into the 14th century. Indeed al-Umawi was a Muslim but the mathematical scholarship of the Muslim world at this time was certainly not uniform. There were differences in the numerals used in western areas (which al-Umawi came from) and those used in the east. Indeed some scholars find it surprising that al-Umawi as a westerner wrote an arithmetic text for those in the east. The usual perception is that, at this time. the arithmetical skills of the east exceeded those of the west.
Two texts by al-Umawi which have survived are Marasim al-intisab fi'ilm al-hisab (On arithmetical rules and procedures), and Raf'al-ishkal fi ma'rifat al-ashkal which is a work on mensuration. It is the first of these two works which contains the 1373 date referred to in the first paragraph and it is the most interesting of the two texts.
Before describing the Marasim we should make some brief comments about al-Umawi's work calculating lengths and areas. In it al-Umawi gives rules for calculating: lengths of chords and lengths of arcs of circles (using Pythagoras's theorem); areas of circles, areas of segments of circles, areas of triangles and quadrilaterals; volumes of spheres, volumes of cones and volumes of prisms. It is not a work of any great importance and Saidan, writes in [1] that:-
... it is a small treatise of seventeen folios in which we find nothing on mensuration that the arithmeticians of the East did not know.Let us now return to the more important treatise on arithmetical rules and procedures. This is the earliest surviving arithmetical treatise written by an Arab from Spain, so it is interesting to see the content of the work. After describing the very briefly the basic arithmetical operations of addition and multiplication, al-Umawi moves on to discuss the summation of series.
Among the series al-Umawi considers are arithmetic and geometric series. He considers the sum of the first n polygonal numbers, that is 1 + (r - 1)d summed from r = 1 to r = n. These sums of polygonal numbers are called pyramidal numbers and al-Umawi then considers the sums of the first n pyramidal numbers. In discussing ∑ r3, ∑ (2r+1)3, and ∑ (2r)3 al-Umawi was giving results which al-Karaji had proved geometrically 400 years earlier.
Al-Umawi then describes casting out sevens, eights, nines, and elevens. Although he only gives these special cases, the general rule which they all obey is the following: take a number n written in decimal notation as
n = aq +10a1 + 102a2 + 103a3 + ...Let rj = 10j(mod t) where, as far as al-Umawi is concerned, t = 7, 8, 9, or 11. Then if Sajrj is divisible by t so is n. This theorem is attributed to Pascal three hundred years after al-Umawi, and indeed al-Umawi only gives the special cases mention here. However, he does note that the sequence r1, r2, r3, r4, r5, ... recurs after finitely many steps in each of the cases he considers.
Some results appearing in this work by al-Umawi are not found in any other Arabic arithemetics. He gives some interesting conditions for the decimal representation of a number n to be a square:
n must either end in 00, 1, 4, 5, 6, or 9;None of these results are hard to prove today (try them!) with our understanding of the decimal representation of numbers. One has to remember that these results are about decimal representations rather than about numbers themselves and show how an understanding of the decimal system was progressing at a time when Christian Europe (if I may call it that) had little interest in anything beyond the mathematics of the ancient Greeks.if n ends in 6, the 10's place is odd, otherwise the 10's place is even;
if n ends in 5 then the 10's place must be 2;
n must leave a remainder of 0, 1, 2, or 4 on division by 7;
n must leave a remainder of 0, 1, or 4 on division by 8;
n must leave a remainder of 0, 1, 4, or 7 on division by 9.
Al-Umawi gives similar results for n to be a cube including:
n must leave a remainder of 0, 1, or 6 on division by 7;
n must leave a remainder of 0, 1, 3, 5, or 7 on division by 8;
n must leave a remainder of 0, 1, or 8 on division by 9.
If you have enjoyed proving these results due to al-Umawi then here is one more he gives in the Marasim. If the integer n is a square and its final digit is 1, then either the 100's place and 1/2 the 10's place are both even or they are both odd.
Article by: J J O'Connor and E F Robertson
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