Numeral base

In mathematical numeral systems, the radix or base is the number of unique digits, including zero, used to represent numbers in a positional numeral system. For example, for the decimal system (the most common system in use today) the radix is ten, because it uses the ten digits from 0 through 9.

In any standard positional numeral system, a number is conventionally written as (x)_y with x as the string of digits and y as its base, although for base ten the subscript is usually assumed (and omitted, together with the pair of parentheses), as it is the most common way to express value. For example, (100)_dec = 100 (in the decimal system) represents the number one hundred, while (100)₂ (in the binary system with base 2) represents the number four.

is a Latin word for "root". Root can be considered a synonym for base in the arithmetical sense.

In the system with radix 13, for example, a string of digits such as 398 denotes the number 3 × 13² + 9 × 13¹ + 8 × 13⁰ = 625.

More generally, in a system with radix b (b > 1), a string of digits d₁ … d_n denotes the number d₁bⁿ⁻¹ + d₂bⁿ⁻² + … + d_nb⁰, where 0 ≤ d_i < b.

Commonly used numeral systems include:

For a larger list, see List of numeral systems.

The octal and hexadecimal systems are often used in computing because of their ease as shorthand for binary. Every hexadecimal digit corresponds to a sequence of four binary digits, since sixteen is the fourth power of two; for example, hexadecimal 78₁₆ is binary 1111000₂. A similar relationship holds between every octal digit and every possible sequence of three binary digits, since eight is the cube of two.

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