BİRİNCİ FEN · BİRİNCİ BÂB · ÜÇÜNCÜ FASIL · Dokuzuncu Madde
**Article Nine: Concerning the Calculation of Square Roots and Fractions, and the Easiest Method for Obtaining Results**
It is known that the masters of arithmetic have said: If the desired number is small and regular (whole), then its square root is easily taken. For example, the square root of 4 is 2, the square root of 9 is 3, the square root of 16 is 4, the square root of 25 is 5, the square root of 36 is 6, the square root of 49 is 7, the square root of 64 is 8, the square root of 81 is 9, and the square root of 100 is 10. All these are regular numbers; therefore, their square roots are whole numbers. If the given number is an irrational number (whose square root is not a whole number), then the easiest way to find its square root is as follows:
Take from the said irrational number the whole number whose square root can be taken from it, and subtract that from the number whose square root you wish to find. Then divide the remaining number by the square root of the number whose square root has been taken. This division will yield an approximate value for the square root of that irrational number.
For example, if one wishes to take the square root of 5, then subtract from it the whole number whose square root can be taken immediately before it, which is 4; this leaves a remainder of 1, which is the numerator. Add 1 to twice the square root of the subtracted number (4), yielding 5. This 5 is the denominator. Thus, the square root of 5 becomes 2 and one-fifth (two whole numbers and one-fifth).
For the number 6, the nearest number whose square root can be taken is 4. Subtracting it from 6 leaves a remainder of 2, which is the numerator. Add 1 to twice the square root of the subtracted number (4), which is 2, yielding 5. This 5 is the denominator. Thus, the square root of 6 becomes 2 and two-fifths. By this analogy, the square root of 7 is 2 and three-fifths, and the square root of 8 is 2 and four-fifths.
Because these irrational numbers have 4 as the nearest number from which a square root can be taken. However, if one wishes to take the square root of 10, then 9 is the closest number whose square root can be taken. Subtracting 9 from 10 leaves a remainder of 1, which is the numerator. Add this number (1) to twice the square root of 9, yielding 7. This 7 is the denominator. Thus, the square root of 10 becomes 3 and one-seventh.
Continue adding 1/7 for each value greater than 10 to find its square root. The square root of 15 is found to be 3 and six-sevenths [there are five numbers between 10 and 15; add five more 1/7s, resulting in 6/7]. If you want to take the square root of 16, [add one more 1/7], it becomes 7/7, which is a whole number; therefore, the square root of 16 is the whole number 4. We can find other numbers and their square roots by comparing them with these using the same method.
**Concerning the Calculation of Fractional Numbers:** Two numbers are said to be commensurable if they are not equal but are not different from one another (i.e., have a common factor). If the smaller of two numbers divides the larger number exactly, then that number is divisible (a whole multiple of the other).
If two numbers can be simplified by a third number, these two numbers are said to be compatible. The result obtained when dividing them by this third number is the simplest form of the fraction.
If two numbers have no common divisor other than 1—that is, they are relatively prime—then these numbers are dissimilar (mutbayin). However, in others, divide the larger number by the smaller number. If there is no remainder, then these two numbers are divisible (whole multiples of each other). If there is a remainder after the operation, continue the process until the remainder is eliminated by dividing the divisor into the remainder. These two numbers are compatible. If the result of the operation is 1, then these two numbers are dissimilar.
**Fractional Numbers:** Fractions can be either regular (whole), which are the previously mentioned “ninths,” or irrational (called parts). Each of these is either singular (like 1/3 and 1/11), repeated (like 2/3 and 2/11), compound (like one-half of 1/6 and one-eleventh of 1/13), or attached (like one-half and 1/3 with 1/11 and 1/13).
When writing mixed numbers, follow this method: If there is a whole number, write it above the fraction; place the numerator below it, and write the denominator above. If the number is not a whole number, substitute zero in its place. For attached fractions, insert “v (+)” between them; for compound irrational numbers, insert "min (=)".
Write 1 and 2/3 as follows:
1
2 (1 2/3)
3
Write 1 and 1/3 as follows:
1
1 (1 1/3)
3
Write one-third as follows:
0
1 (1/3)
3
Write one-half of five-sixths as follows:
0
1
2 (1/2 x 5/6)
5
6
Write two-fifths and three-fourths as follows:
0 0
2
3 (2/5 + 3/4)
5 4
Write one-eleventh of one-thirteenth as follows:
1 1
(1/13 x 1/11)
11 13
The denominator of the fraction is called a perfection or a prime. This small number forms a fraction. The denominator of a singular fraction is obvious. The denominator of a repeated fraction is also the same. For example, the denominator of one-fourth is 4. The denominators of twice one-fourth and three times one-fourth are also 4.
The denominator of compound fractions is equal to the product of their denominators. Whether the denominators are dissimilar, compatible, or divisible, they are multiplied by each other. For example, the common denominator of one-fifth and one-sixth is 30. The common denominator of one-sixth and one-eighth is 48. The common denominator of one-fourth and one-eighth is 32.
For attached fractions, compare two denominators. If they are dissimilar, multiply them by each other; or if they are compatible, multiply one by the simplified form of the other. If they are divisible, take the larger denominator. Then accept the result of the multiplication as the denominator of a third fraction and continue the process until the attached fractions are resolved. The resulting value is the denominator of that fraction.
The rule for finding the denominator of "ninths" is as follows: Multiply two dissimilar numbers, 2 and 3, by each other; the resulting number (6) is multiplied by one-half of 4, which is compatible with it; the result (12) is then multiplied by 5, which is dissimilar, yielding 60. Sixty is a perfect divisor of the final result.
Here follows the translation of the provided text:
When a result is obtained, and it is found to be divisible by 7, multiply it by 7; when the resulting number (420) is compatible with 8, multiply it by one-fourth of 8. Again, when the resulting number (840) is compatible with 9, multiply it by one-third of 9. If 10 is a perfect divisor of the result, perform no operation; the product of 3 and 840, which is 2520, becomes the denominator of this "ninth fraction."
**Conversion (Tecnis)**
Conversion refers to transforming a mixed number into a complex fraction. The method is as follows: Multiply the “whole” part of the mixed number by the denominator of the fraction; add the result to the numerator. The resulting sum becomes the denominator of the complex fraction. For example, the numerator of 2 and one-fourth is 9. The numerator of 6 and three-fifths is 33. The numerator of 4 and one-seventh of one-third is 85.
**Reduction (Ref‘)**
Reduction refers to converting fractions into whole numbers. The method is as follows: Divide the number whose numerator is greater than its denominator by the latter’s denominator. The quotient is the whole number part of the mixed number. If there is a remainder, that remainder is the numerator. For example, the reduction of 15/4 is 3 and three-fourths.
**Addition and Doubling in Fractions**
The method for addition and doubling is as follows: Find the common denominator of the fractions whose sum and double are desired; divide the numerators by the common denominator. If the numerators are greater than the common denominator, the quotient becomes the whole number part of the result. The quotient of the remainder divided by the common denominator becomes the fractional part.
If the numerators are less than the common denominator, express them as ratios to the common denominator. If the numerators are equal to the common denominator, the result is 1.
Example:
– Adding one-half, one-third, and one-fourth results in half of one and one-sixth (1 1/12).
– The sum of one-sixth and one-third is one-half (1/2).
– The sum of one-half, one-third, and one-sixth is 1.
– Double three-fifths is one and one-fifth (1 1/5).
**Halving Fractional Numbers (Tansîf-i Küsûr)**
If the numerators of fractional numbers are even, divide by two. If they are odd, multiply the denominator by two and express the numerator as a ratio to it; this is well known.
**Subtraction in Fractions (Tefrîk-i Küsûr)**
Find the common denominator of the two fractions and subtract one from the other. Express the remainder as a ratio to the common denominator.
For example, if you subtract one-fourth from one-third, you get half of one-sixth (1/12). Because their common denominator is 12. One-fourth of this number 12 is 3, and one-third of it is 4. Subtracting 3 from 4 leaves 1. This 1 is half of one-sixth (1/12).
**Multiplication in Fractions (Darb-i Küsûr)**
The multiplication of fractional numbers is performed as follows: If a mixed number or a non-whole fraction is multiplied by a whole number, the mixed number is converted to a complex fraction; the whole number and the numerator of this complex fraction are multiplied. If the product is greater than or equal to the denominator of the fraction, the result is divided by the denominator of the fraction; otherwise, express the result as a ratio to the denominator.
For example, if 2 and three-fifths are multiplied by 4, you get 52; this number is divided by the denominator of the fraction, which is 5. The result of the multiplication is 10 and two-fifths. Another example: If three-fourths is multiplied by 7, the result is 21. This number is divided by the denominator of the fraction, which is 4, resulting in 5 and one-fourth.
If both numbers being multiplied are fractions, or if neither is a mixed number (complex fraction), multiply their numerators together; or multiply the numerator of the complex fraction by the numerator of the non-whole fraction; or multiply the numerators of the non-whole fractions with each other. This is the first result. Then multiply the denominators together; this is the second result. Then divide the first result (the result obtained from multiplying the numerators) by the second (the result obtained from multiplying the denominators). If the first is greater than the second, divide the numerator by the denominator; if the second is less than the first, the resulting number is the desired result.
Example: Multiply 2 and one-half by 3 and one-third to get 8 and one-third.
Multiply 2 and one-fourth by five-sixths to get 1 and seven-eighths.
Multiply three-fourths by five-sevens to get one-half and one-seventh of one-fourth.
**Division in Fractions (Kısmet-i Küsûr)**
Division has eight categories. The dividend is either a whole number, a fraction, or a mixed number. The divisor is either a whole number, a fraction, or a mixed number. Eight classes of division can be performed with these. Excluding the division of a whole number by a whole number, there remain eight classes of division.
The method of division is as follows: If both the dividend and the divisor are fractions, multiply each by the common denominator. If one is a fraction and the other is a whole number, multiply the dividend and the divisor by the denominator of the existing fractional number; then, if the product is greater, divide the products by each other; if it is less, express the result as a ratio to the divisor.
If you divide 5 and one-fourth by 3, you get 1 and three-fourths. If you were to reverse this operation and divide 3 by 5 and one-fourth, the result would be four-sevens. If you divide two-sixths by one-sixth, the result is 2. Having exhausted these examples, we shall stop here.