baserepresentation

Base representation, or base representation of numbers, is the expression of numbers in terms of powers of a fixed base b. In a base-b system, any nonnegative integer n can be written uniquely as n = d_k b^k + d_{k-1} b^{k-1} + ... + d_0, where 0 ≤ d_i < b and d_k ≠ 0. The sequence of digits (d_k ... d_0) is the base-b representation of n. For example, decimal (base 10) representation of 255 is 255, binary (base 2) representation is 11111111, and hexadecimal (base 16) representation is FF.

To obtain the base-b representation, one performs repeated division of n by b, recording the remainders. The remainders give the digits from least significant to most significant. The representation of 0 is 0. If the number is negative, a minus sign is placed in front.

Base representations extend to fractions, where the base-b expansion of a rational number is finite if and only if the denominator divides a power of b; otherwise it is infinite and eventually periodic. For example, in base 2, 1/2 = 0.1, while in base 10, 1/3 = 0.333... with a repeating block.

Common bases include binary (base 2), octal (base 8), decimal (base 10), and hexadecimal (base 16). In higher bases, digits beyond 9 are represented with letters (for instance, A–F for base 16). Base representation is fundamental in computing, digital encoding, and mathematical numeration systems, providing efficient methods for arithmetic, data storage, and number interpretation.