II.

Place Values

The position of a symbol denotes the value of that symbol in terms of exponential values of the base. That is, in the decimal system, the quantity represented by any of the ten symbols used—0, 1, 2, 3, 4, 5, 6, 7, 8, and 9—depends on its position in the number. Thus, the number 3,098,323 is an abbreviation for (3 × 10⁶) + (0 × 10⁵) + (9 × 10⁴) + (8 × 10³) + (3 × 10²) + (2 × 10¹) + (3 × 10⁰, or 3 × 1). The first 3 (reading from right to left) represents 3 units; the second 3, 300 units; and the third 3, 3 million units.

Two digits—0, 1—suffice to represent a number in the binary system; 6 digits—0, 1, 2, 3, 4, 5—are needed to represent a number in the senary (base-6) system; and 16 digits—0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A (ten), B (eleven), C (twelve), ... , F (fifteen)—are needed to represent a number in the hexadecimal system. The number 30155 in the senary system is the number (3 × 6⁴) + (0 × 6³) + (1 × 6²) + (5 × 6¹) + (5 × 6⁰) = 3959 in the decimal system; the number 2EF in the hexadecimal system is the number (2 × 16²) + (14 × 16¹) + (15 × 16⁰) = 751 in the decimal system.

To write a given base-10 number n as a base-b number, divide (in the decimal system) n by b, divide the quotient by b, the new quotient by b, and so on until the quotient 0 is obtained. The successive remainders are the digits in the base-b expression for n. For example, to express 3959 (base 10) in the base 6, one writes