Moses ibn Tibbon’s Hebrew Translation of al-Hassar's Kitab al Bayan - Division

Al-Hassar's chapter, "Denomination," considered on the preceding page, is subtitled “Division of a Small Number by a Larger One.” The reverse operation, “Division of a Large Number by a Smaller One,” is taken up in the chapter headed “Division” (the seventh of the ten chapters in Part One of the Kitāb al-Bayān) and it is here that al-Hassar first demonstrates an application of his new notation. The context is how to deal systematically with the remainder when the dividend is not an integer multiple of the divisor. The example he gives (at fol. 5v in the Christ Church manuscript and fol. 11r in the Vatican manuscript) is the division of ninety-eight thousand seven hundred and forty-six (98746) by thirty-six (36).

The compound number thirty-six has the factors four and nine, so first divide the ninety-eight thousand seven hundred and forty-six by four. This gives twenty-four thousand six hundred and eighty-six with two remaining over the four. Now divide the twenty-four thousand six hundred and eighty-six by nine. This gives two thousand seven hundred and forty-two with eight remaining over the nine \([98746\div 4 = 24686\,{\rm{R}}\,2\) and \(24686\div 9 = 2742\,{\rm{R}}\,8].\)

Al-Hassar now shows how the two remainders, the two over four from the first division and the eight over nine from the second, can be combined using his new notation. Mixed numbers comprising an integer and a simple fraction are written with the integer to the right of the fraction.

Draw a horizontal line and write the nine and the four beneath it with the nine on the right and the four on the left. Now place the eight over the nine and the two over the four. Placing the integer [part of the quotient] to its right gives the final answer, two thousand seven hundred and forty-two and eight ninths and two fourths of a ninth \[\frac{2\quad 8}{4\quad 9}\,2742\implies 2742+\frac{8}{9}+\frac{2}{4\times 9}=2742\,\frac{17}{18}.\]

(See Note.) In this neat way, the remainders from any sequence of divisions by the factors of a compound divisor can be combined using al-Hassar’s new notation. He now goes on to show how, having obtained this expression, the answer can be checked using the technique of “casting out sevens.”

The answer will be correct if dividing both the dividend and the quotient by seven leaves the same remainder. Dividing the dividend by seven gives a remainder of four \([98746\div 7 = 14106\,{\rm{R}}\,4].\)

Dividing the integer part of the quotient by seven leaves a remainder of five \([2742\div 7 = 391\,{\rm{R}}\,5].\)

Dividing the nine below the line in the fractional part of the quotient by seven leaves a remainder of two and multiplying this by the remainder five from the integer part gives ten \([9\div 7 = 1\,{\rm{R}}\,2\) and \(2\times 5=10].\)

Dividing this ten by seven leaves a remainder of three. To this add the remainder of one that is left from dividing the eight that is above the nine by seven; this gives four \([10\div 7 = 1\,{\rm{R}}\,3,\,\,8\div 7 = 1\,{\rm{R}}\,1,\) and \(1 + 3=4].\)

Multiply this four by the other number below the line, namely, the four; this gives sixteen. Dividing the sixteen by seven leaves a remainder of two which when added to the two above the four in the expression gives four \([4\times 4=16,\,\,16\div 7 = 2\,{\rm{R}}\,2,\) and \(2+2=4].\)

Thus, when they are divided by seven, the dividend and quotient leave the same remainder, i.e., four, which bears out the correctness of the result.

Note. Again, the symbol \(\implies\) indicates replacement of al-Hassar's representation of arithmetic operations by modern notation.