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Concept
Unary numeral system
Summary
The unary numeral system is the simplest numeral system to represent natural numbers: to represent a number N, a symbol representing 1 is repeated N times.
In the unary system, the number 0 (zero) is represented by the empty string, that is, the absence of a symbol. Numbers 1, 2, 3, 4, 5, 6, ... are represented in unary as 1, 11, 111, 1111, 11111, 111111, ...
Unary is a bijective numeral system. However, because the value of a digit does not depend on its position, it is not a form of positional notation, and it is unclear whether it would be appropriate to say that it has a base (or "radix") of 1, as it behaves differently from all other bases.
The use of tally marks in counting is an application of the unary numeral system. For example, using the tally mark | (𝍷), the number 3 is represented as |||. In East Asian cultures, the number 3 is represented as 三, a character drawn with three strokes. (One and two are represented similarly.) In China and Japan, the character 正, drawn with 5 strokes, is sometimes used to represent 5 as a tally.
Unary numbers should be distinguished from repunits, which are also written as sequences of ones but have their usual decimal numerical interpretation.
Addition and subtraction are particularly simple in the unary system, as they involve little more than string concatenation. The Hamming weight or population count operation that counts the number of nonzero bits in a sequence of binary values may also be interpreted as a conversion from unary to binary numbers. However, multiplication is more cumbersome and has often been used as a test case for the design of Turing machines.
Compared to standard positional numeral systems, the unary system is inconvenient and hence is not used in practice for large calculations. It occurs in some decision problem descriptions in theoretical computer science (e.g. some P-complete problems), where it is used to "artificially" decrease the run-time or space requirements of a problem.
Official source
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