bitlengthn

Bitlength(n) is the minimal number of binary digits required to represent the integer n in binary form. For a nonzero integer n, bitlength(n) equals floor(log2(|n|)) + 1, i.e., the position of the most significant 1 bit plus one. This definition applies to unsigned integers; when n is signed, bitlength typically refers to the magnitude's bitlength, independent of sign.

For n = 0, conventions vary: many programming languages define bitlength(0) as 0, while some mathematical treatments set it to 1.

Bitlength is different from a fixed word size or bit width used by hardware, which represents the maximum number of bits a value can occupy in a given system. Bitlength is a property of the value itself, whereas bit width is a design parameter. In fixed-width representations such as two's complement, a number may have a smaller bitlength than the system's word size.

Applications include algorithm design and analysis, where running time or space depends on the bitlength of inputs; in cryptography, key sizes are often specified by bitlength (e.g., 2048-bit keys); in data encoding and random number generation, the bitlength determines the amount of information stored.