FIRST SCIENCE · FIRST CHAPTER · THIRD SECTION · First Article
First Article
It briefly presents the definition of number, whole numbers (ṣaḥīḥ), fractions (küsûr-ı tis‘a) with numerator 1 (one) and denominators from 2 to 10, rational numbers (muntak), irrational numbers (asam), perfect numbers (tām), abundant numbers (ziyāde), and deficient numbers (nākıs).
O dear one! It should be known that mathematicians have said as follows:
In arithmetic, unknown numbers are found with the help of known, definite numbers.
Number is a quantity given as a name to 1 (one) and those derived from it. And if the number is absolute, that is, not ascribed to another number, it is called a whole number (ṣaḥīḥ). And if 1 (one) is ascribed to another assumed number, it is called a fraction. For example, a half, that is, ½. The numerator of a half is 1 (one), and its denominator is 2 (two). Fractions (küsûr-ı tis‘a) with numerator 1 and denominators from 2 to 10 are: a half, a third, a quarter, a fifth, a sixth, a seventh, an eighth, a ninth, and a tenth. The denominator of a half is 2, the denominator of a third is 3, the denominator of a quarter is 4, the denominator of a fifth is 5, the denominator of a sixth is 6, the denominator of a seventh is 7, the denominator of an eighth is 8, the denominator of a ninth is 9, and the denominator of a tenth is 10.
If an absolute and whole number has a fraction or a root, it is called “muntak” (rational number). If not, this number is called “asam” (irrational number). For example, the square root of four is two, the square root of nine is three. Otherwise, as in the case of the number 11 among asam numbers, eleven has neither a fraction nor a square root.
And if an absolute number is equal to the sum of its own parts (cüz), this number is called a perfect number; like the number 6. Because its half is 3, its third is 2, and its sixth is 1. The sum of these is 6.
And if a muntak number is less than the sum of its own parts, it is called a zāid number (abundant number); like the number 12. Because half of 12 is 6, its third is 4, its quarter is 3, and its sixth is 2.
The sum of these [21/a] is 15. Since this 15 is greater than 12, it is called a zāid number. And if the muntak number is greater than the sum of its own parts, it is called a nākıs number (deficient number); like the number 8. Because half of 8 is 4, its quarter is 2, and its eighth is 1. The sum of these is 7.
Therefore, the number 8, which is being divided, is called a nākıs number.