BİRİNCİ FEN · BİRİNCİ BÂB · ÜÇÜNCÜ FASIL · Sekizinci Madde
The Eighth Article
Concerning the Easiest Way of Division
O esteemed one! Know that mathematicians have stated: The easiest way to perform division is to seek a number such that when multiplied by the divisor, it equals the dividend or leaves a remainder less than the divisor.
If this number is equal to it [the dividend], then the assumed number is the quotient (of the division). If the result obtained is less than the dividend [23/b] and the remainder is also less than the divisor, you divide the remainder by the divisor. The result of this ratio, together with the aforementioned number, is the quotient of the division.
For example: When you wish to divide 13 by 4, the resulting quotient is 3. When you multiply this quotient by the divisor, which is 4, the result is less than the dividend. That is, the remainder is deficient from the dividend. Therefore, if you proportion the remaining number to the divisor in a one-quarter manner, the quotient will be three whole and one quarter (3 ¼). And if the dividend were 14, the quotient would be three and a half (3 ½ ).
An example of an operation where the product of the quotient and the divisor equals the dividend is the division of 12 by 4. For in this operation, the quotient is 3. Mathematicians have stated that the most known and preferred method for dividing multi-digit numbers is the four-place method (buyût-i erba’a).
This process is performed as follows: First, you write the dividend and draw a horizontal line beneath it, beginning with the units digit, as previously described. When you reach the end of the digits, you turn slightly upwards and extend it again to the left. Then, you write the divisor below this oblique line on the left side of the dividend. Next, you take twice the divisor and write it underneath. Then, you take twice this new number and write it again underneath. Subsequently, you take twice that number as well and write it underneath. This is how division is performed using the four-place method (buyût-i erba’a).
The first place in this process is the divisor. The second place is twice its value. The third place is twice the value of that, and the fourth place is twice the number taken twice. Then, starting from the left, you look at the digit on the left side of the dividend, beginning with the last digit. You subtract the largest possible number from this digit among the four places. If a carry-over remains, you write it above the next digit and erase that digit. You write the same number of the place being subtracted beneath the line. And if it is not possible to subtract from the last digit, you add the digit to its right and continue the process as described above.
If it is not possible to subtract by adding a digit, then you add another digit and proceed in this manner until it is possible to remove one of the four places. You write the number in the place being subtracted beneath the rightmost digit of the digits added. You continue this process until you reach the last digit of the dividend sequence.
[At the end of these operations] if any remainder remains from the dividend, it is no longer possible to subtract the divisor from it. In this case, the remaining number is a fraction, and its denominator is the divisor.
If none of the numbers in the four places mentioned above are found at the alignment of a digit of the dividend beneath the division line, you write zero there. Then, you sum up the numbers obtained below the division line to find the quotient.
For example: When you divide 9789 by 14, the quotient is 699 and 3 remains. [24/a] This remaining remainder is a fraction with a denominator of 14.
[The method of performing division using the four-place method, whose details we have explained above, is as follows.]
The verification of division is as follows: Multiply the divisor’s verification by the quotient and add the remaining number to the result of this multiplication. If the verification of the sum you made equals the verification of the division, then the division process is correct. Otherwise, the operation is deficient and incorrect, and the process must be repeated.