Radix

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In mathematical numeral systems, the base or radix for the simplest case is the number of unique digits, including zero, that a positional numeral system uses to represent numbers. 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 numeral system, the base is written as "10". In a base ten numeral system, "10" represents the number ten; in a base two system, "10" represents the number two.

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Etymology

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

In numeral systems

In the system with radix 13, for example, a string of digits such as 398 denotes the decimal number 3 \times 13^2 %2B 9 \times 13^1 %2B 8 \times 13^0. More generally, in a system with radix b (b > 1), a string of digits d_1 \ldots d_n denotes the decimal number d_1 b^{n-1} %2B d_2 b^{n-2} %2B \cdots %2B  d_n b^0.

Commonly used numeral systems include:

The octal, hexadecimal and base-64 systems are often used in computing because of their ease as shorthand for binary. For example, every hexadecimal digit has an equivalent 4 digit binary number.

Radices are usually natural numbers. However, more sophisticated positional systems are possible, e.g. golden ratio base (whose radix is a non-integer algebraic number), and negative base (whose radix is negative).

In exponentiation

In exponentiation, the base refers to the number b in an expression of the form bn. The number n is called the exponent and the expression is known formally as exponentiation of b by n or the exponential of n with base b. It is more commonly expressed as "the nth power of b", "b to the nth power" or "b to the power n".

When the nth power of b equals a number a, or  a = bn , then b is called an "nth root" of a. The term power strictly refers to the entire expression, but is sometimes used to refer to the exponent.

The inverse function to exponentiation with base b (when it is well-defined) is called the logarithm to base b, denoted logb. Thus:

\log_b(b^n) = b^{\log_b(n)} = n. \,

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