# Unary numeral system

Short description: Base-1 numeral system

The unary numeral system is the simplest numeral system to represent natural numbers:[1] to represent a number N, a symbol representing 1 is repeated N times.[2]

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, ...[3]

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.[citation needed]

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.[4] (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.[5][6]

Unary numbers should be distinguished from repunits, which are also written as sequences of ones but have their usual decimal numerical interpretation.

## Operations

Addition and subtraction are particularly simple in the unary system, as they involve little more than string concatenation.[7] 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.Cite error: Closing </ref> missing for <ref> tagCite error: Closing </ref> missing for <ref> tag

## Applications

In addition to the application in tally marks, unary numbering is used as part of some data compression algorithms such as Golomb coding.[8] It also forms the basis for the Peano axioms for formalizing arithmetic within mathematical logic.[9] A form of unary notation called Church encoding is used to represent numbers within lambda calculus.[10]

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. Using a unary representation instead of a decimal number lets the user search for messages with a given rating or higher. For example, searching for **** yield messages with a rating of at least 4.[11]

## References

1. Hodges, Andrew (2009), One to Nine: The Inner Life of Numbers, Anchor Canada, p. 14, ISBN 9780385672665 .
2. Davis, Martin; Sigal, Ron; Weyuker, Elaine J. (1994), Computability, Complexity, and Languages: Fundamentals of Theoretical Computer Science, Computer Science and Scientific Computing (2nd ed.), Academic Press, p. 117, ISBN 9780122063824 .
3. Hext, Jan (1990), Programming Structures: Machines and Programs, 1, Prentice Hall, p. 33, ISBN 9780724809400 .
4. Woodruff, Charles E. (1909), "The Evolution of Modern Numerals from Ancient Tally Marks", American Mathematical Monthly 16 (8–9): 125–33, doi:10.2307/2970818 .
5. Hsieh, Hui-Kuang (1981), "Chinese Tally Mark", The American Statistician 35 (3): 174, doi:10.2307/2683999
6. Lunde, Ken; Miura, Daisuke (January 27, 2016), "Proposal to Encode Five Ideographic Tally Marks", Unicode Consortium, Proposal L2/16-046
7. Sazonov, Vladimir Yu. (1995), "On feasible numbers", Logic and computational complexity (Indianapolis, IN, 1994), Lecture Notes in Comput. Sci., 960, Springer, Berlin, pp. 30–51, doi:10.1007/3-540-60178-3_78, ISBN 978-3-540-60178-4 . See in particular p.  48.
8. "Run-length encodings", IEEE Transactions on Information Theory IT-12 (3): 399–401, 1966, doi:10.1109/TIT.1966.1053907 .
9. Magaud, Nicolas; Bertot, Yves (2002), "Changing data structures in type theory: a study of natural numbers", Types for proofs and programs (Durham, 2000), Lecture Notes in Comput. Sci., 2277, Springer, Berlin, pp. 181–196, doi:10.1007/3-540-45842-5_12, ISBN 978-3-540-43287-6 .
10. Jansen, Jan Martin (2013), "Programming in the λ-calculus: from Church to Scott and back", The Beauty of Functional Code, Lecture Notes in Computer Science, 8106, Springer-Verlag, pp. 168–180, doi:10.1007/978-3-642-40355-2_12, ISBN 978-3-642-40354-5 .
11. http://answers.uillinois.edu/illinois/page.php?id=49002