The radix or base is the number of unique digits, used to represent numbers.
If the base is larger than $10$ we use caracters 'A', 'B', 'C', ... as the digits 10, 11, 12 ...
To represent a number $n$ in a base $b$ we use the notation :
$$n = \overline{d_{k} d_{k-1} \cdots d_{1}}^{(b)}$$
Examples:
The binary system :
Used internally by nearly all computers, is base 2. The two digits are "0" and "1".
Example : $\overline{31}^{(10)} = \overline{11111}^{(2)}$
The Octal system :
The eight digits are "0"–"7".
Example : $\overline{31}^{(10)} = \overline{37}^{(8)}$
The Decimal system :
The most used system of numbers in the world, is used in arithmetic. Its ten digits are "0"–"9".
The Duodecimal (dozenal) system :
Sometimes advocated due to divisibility by 2, 3, 4, and 6. It was traditionally used as part of quantities expressed in dozens and grosses.
Its 12 digits are "0"–"9" and "A"-"B" or "a"-"b".
Example : $\overline{31}^{(10)} = \overline{27}^{(12)}$
The Hexadecimal system :
Often used in computing as a more compact representation of binary (1 hex digit per 4 bits).
The sixteen digits are "0"–"9" followed by "A"–"F" or "a"–"f".
Example : $\overline{31}^{(10)} = \overline{1F}^{(16)}$
The Vigesimal system (base 20) :
Traditional numeral system in several cultures, still used by some for counting.
Example : $\overline{31}^{(10)} = \overline{1B}^{(20)}$
