There are many different numeral systems, that is, writing systems for expressing numbers.
By culture / time period "A base is a natural number B whose powers (B multiplied by itself some number of times) are specially designated within a numerical system." The term is not equivalent to radix, as it applies to all numerical notation systems (not just positional ones with a radix) and most systems of spoken numbers. Some systems have two bases, a smaller (subbase) and a larger (base); an example is Roman numerals, which are organized by fives (V=5, L=50, D=500, the subbase) and tens (X=10, C=100, M=1,000, the base).
By type of notation
Numeral systems are classified here as to whether they use positional notation (also known as place-value notation), and further categorized by radix or base.
Standard positional numeral systems
The common names are derived somewhat arbitrarily from a mix of Latin and Greek, in some cases including roots from both languages within a single name. There have been some proposals for standardisation.
Non-standard positional numeral systems
Bijective numeration Unary, or bijective base‑1, is used in Tally marks, and Counting. Unary numbering is used as part of some data compression algorithms such as Golomb coding. It also forms the basis for the Peano axioms for formalizing arithmetic within mathematical logic. A form of unary notation called Church encoding is used to represent numbers within lambda calculus. Some email spam filters tag messages with a number of asterisks in an e-mail header such as X-Spam-Bar or X-SPAM-LEVEL. The larger the number, the more likely the email is considered spam.
Signed-digit representation
Complex bases
Non-integer bases
Mixed radix Factorial number system {1, 2, 3, 4, 5, 6, ...} Primorial number system {2, 3, 5, 7, 11, 13, ...}
Other Quote notation Redundant binary representation Hereditary base-n notation Asymmetric numeral systems optimized for non-uniform probability distribution of symbols Combinatorial number system
Non-positional notation
All known numeral systems developed before the Babylonian numerals are non-positional, as are many developed later, such as the Roman numerals. The French Cistercian monks created their own numeral system.
See also
References