radixes

Radix, in numeral notation, is the base of a numeral system—the number of digits available in each position. In a positional system with radix r, the value of a numeral equals the sum of its digits multiplied by powers of r, i.e., digits d_k with 0 ≤ d_k < r in positions k carry weights r^k. The radix thus determines both the digit alphabet and the place-value structure of the numeral.

Common radices include base-2 (binary), base-8 (octal), base-10 (decimal), and base-16 (hexadecimal). Binary uses digits 0 and 1 and powers of 2; decimal uses 0–9 and powers of 10; hexadecimal uses 0–9 and A–F and powers of 16. For example, the decimal number 255 equals 11111111 in binary and FF in hexadecimal.

Historically and technically, other radices have been used. Sexagesimal, with base 60, appeared in ancient Mesopotamia and structured time and angles; unary is sometimes described as a degenerate, non-positional base-1 system. Negative-radix systems, such as base −2, provide a positional representation of integers without a separate sign.

Radices are fundamental in computing, where hardware and software perform base conversions, encoding, and arithmetic across bases. The term radix also appears in logarithms, where it denotes the base of the logarithm.