BİRİNCİ FEN · BİRİNCİ BÂB · ÜÇÜNCÜ FASIL · İkinci Madde
Second Article
The origins and details of the digits of numbers, along with summation (cem‘), subtraction (tefrîk), adding a number to itself (taz‘îf), multiplication (darb), dividing a number into two equal parts (tansîf), division (taksim), multiplying a number by itself, squaring (terbî‘) and square root extraction (teczîr)—these define the results of multiplication and division.
O esteemed one! It is known that arithmetic scholars have stated: Numbers fundamentally possess three places: units, tens, and hundreds. Their details are six, and their total amounts to nine digits. The unit's place is at the rightmost position of the number. These are written with Hindu numerals. The second digit to the left of this first digit represents the tens place (mertebe-i aşerât), and the third digit to the left of it indicates the hundreds place (mertebe-i mi’ât).
The fourth digit to the left of that represents the thousands place (mertebe-i âhâd-ı ulûf), the fifth digit to the left of it, ten thousands place (mertebe-i aşerât-ı ulûf), the sixth digit to the left of it, hundred thousands place (mertebe-i mi’ât-ı ulûf), the seventh digit to the left of that, millions place (mertebe-i âhâd-ı ulûf-ı ulûf), the eighth digit to the left of it, ten millions place (mertebe-i aşerât-ı ulûf-ı ulûf), and the ninth digit to the left of that (mertebe-i mi’ât-ı ulûf-ı ulûf) represents the hundred millions place. We can demonstrate this sequence as follows:
The definitions of the aforementioned operations are briefly as follows: Adding one number on top of another is called summation. For example, the sum of 3 and 5 is 8. Subtracting one number from another is called subtraction (tefrîk). For instance, subtracting 2 from 5 leaves 3. Adding a number to itself is termed “taz‘îf.” The result of adding 1 to itself is 2.
Repeating a number by the value of another number is called multiplication (darb). Multiplying 3 by 5 yields 15, as in 5 x 3 = 15. Dividing a number into two equal parts is called “tansîf.” For example, half of 4 is 2; half of 5 is 2.5. Dividing one number by the value of another is called "kısmet" (division). When 3 is divided by 2, it yields 1.5; when divided by 3, it yields 1; and when divided by 6, it yields a half (0.5).
Multiplying a number by itself is called squaring (terbî‘). The result obtained from this operation is termed the “square” (teczîr). The original number that was multiplied is called the square root (cezr) of that square. For example, multiplying 3 by itself yields the number 9; the square root of this square is 3.
In multiplication, a product is obtained. The ratio of this product to one of its factors is equal to the ratio of the other factor to 1. For instance, when 4 is multiplied by 5 or 5 is multiplied by 4, the number 20 is obtained. The number 4 is one-fifth of 20, just as 1 is one-fifth of 5. Similarly, 5 is one-fourth of 20, and 1 is one-fourth of 4.
Division is the inverse of multiplication. The ratio of the result obtained through division (the quotient) to 1 is equal to the ratio of the dividend to the divisor. For example, when 12 is divided by 4, the result is 3. This quotient is three times 1, just as 12 is three times 4.