BİRİNCİ FEN · BİRİNCİ BÂB · ÜÇÜNCÜ FASIL · Yedinci Madde
**Seventh Section**
This describes the easiest way to perform multiplication. O esteemed one! It is known that
arithmeticians have said: Multiplication occurs in three ways: Firstly,
multiplying a single number by another single number. Secondly,
multiplying a single number by a composite number.
Thirdly, multiplying composite numbers by other composite numbers.
The method for the first of these is threefold: First, multiplying the units digit by the units digit, which is seen in the multiplication table. [434]
The second method: Multiplying by tens, hundreds, and thousands. The third method:
multiplying outside the units digit, that is, multiplying a single number by another single number. The easiest way to perform this multiplication with these last two methods is as follows: you send the numbers other than the units digit in these two parts to the units digit. Then you multiply what is found in the units digit by ones and record the result.
Thereafter, you add the digits of both multiplicands, accepting any "carry-over" as belonging to the number at the beginning of the other place value, and add it. In the second part, you follow this method. For example: if you want to multiply the number 4 by 50 or 3 by 400, first you consider the 20 from the tens place. Because the total number of digits is three, and the second one is the tens place. In the second way (3 x 400), you accept 12 from the hundreds place. Because the total number of digits is four, and the third one is the hundreds place. Regarding the third part: if you need to multiply 30 by 40 or 40 by 500, for the first operation you consider 12 as the hundreds place. Because there are four digits, the third of which is the hundreds place. For the second operation, you count the 20 from the thousands place. Because there are five digits and the fourth one is the thousands place.
Multiplication using the second and third methods: The composite number is written in terms of single numbers, and the first method is used to perform the operation. You multiply the single numbers with each other and add the results. In the second method, for example, if you need to multiply 6 by 54 or 20 by 64, for the first operation you multiply 6 by each of them separately and add the two products obtained. This becomes 324. For the second operation: you multiply 20 by each of these separately and add the two products. Thus, 1280 is obtained. According to the third method, if you want to multiply 14 by 25, first multiply 4 by 5, then 20 by it, then 10 by 5, and then 20 by it. You add the result; it becomes 350. [435] [23/a]
Rule: If you take one or more multiples of one factor and half of the other factor (by dividing by two), you multiply the results obtained from these operations with each other. The result of that multiplication is the solution to the operation.
For example, if you need to multiply 25 by 16, taking twice the multiple of the first, which is 25, and taking half of the second number, which is 16, twice, then multiplying the resulting 4 with the previous result of 100 gives 400. This rule is a clear guide, so that those who know this operation can quickly do their calculations. However, if the digits of the numbers being multiplied are numerous and the operation is difficult, it is easier to perform the operation with the help of a pen.
If you want to multiply a single number by a composite number, write both of them using Hindi numerals. Then multiply the single number by its corresponding digit and write the result below the units digit. For each remaining ten, hold 1 in mind. If there is a subsequent digit, add that held number to the product of that digit.
If there is a 0 (zero) afterwards, write that ten below the zero. If the product is a multiple of 10 and higher, write 0 below the units digit because the units digit is zero. Again, hold 1 in mind for each ten and complete the operation in this way.
If you multiply a single number by 0 (zero), do not place the multiplicand below the 0 (zero) but write 0 (zero). If there are zeros with a single number, write these zeros to the rightmost digits. For example: if you multiply the single number 5 by the composite number 63043, the operation is done as follows:
63043 x 5 = 315215
Verification: 8/8
If the factor is 50, write a zero first below the line of operations; if the factor is 500, add two zeros. The operation is as follows:
63043 x 500 = 31521500
Verification: 8/8
The verification of multiplication is obtained by multiplying the verification of the factors. If this multiplication is equal to its verification, the operation is correct; otherwise it is incorrect.
A Remark
Often when you multiply the number of days in months that are mostly thirty with the number of months in a year, you obtain 360, which you then multiply by the number of days in a week, which is 7. Thus, you reach the common denominator of "küsûr-ı tis‘a," which is 2520. Indeed, when Hazret-i Ali (r. a.) was asked about the result of “küsûr-ı tis‘a,” he said, “Multiply the days of the week by the days of the year.”
And if you multiply the denominators of fractions containing the letter of the month with each other, you perform the operation in the same way. Because [according to their Arabic written form] the numbers containing the letter of the month are erba‘a (4), seb‘a (7), tis‘a (9) and aşere (10). You multiply 4 by 7, then multiply that product by 9, then multiply the second result by 10 to obtain 2520. This is the common denominator of the aforementioned “küsûr-ı tis‘a.”