Unary numeral system

Short description: Base-1 numeral system

It has been suggested that Unary coding be merged into this article. (Discuss) Proposed since July 2026.

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Numeral systems

Hindu–Arabic numeral system

* Western Arabic
* Eastern Arabic

* Bengali
* Devanagari
* Gurmukhi
* Sinhala
* Tamil

* Balinese
* Burmese
* Dzongkha
* Gujarati
* Javanese
* Khmer
* Lao
* Mongolian
* Thai

East Asian

* Chinese
* Suzhou
* Hokkien
* Japanese
* Korean
* Vietnamese
* Tangut

* Counting rods

Alphabetic

* Abjad
* Armenian
* Āryabhaṭa
* Cyrillic
* Ge'ez
* Georgian
* Greek
* Hebrew
* Roman

Former

* Aegean
* Attic
* Babylonian
* Brahmi
* Chuvash
* Egyptian
* Etruscan
* Glagolitic
* Inuit
* Kharosthi
* Mayan
* Muisca
* Quipu

* Prehistoric

Positional systems by base

* 2
* 3
* 4
* 5
* 6
* 8
* 10
* 12
* 16
* 20
* 60

Non-standard positional numeral systems

* Bijective numeration (1)
* Signed-digit representation (Balanced ternary)
* factorial
* negative
* Complex-base system (2i)
* Non-integer representation (φ)
* mixed
* Asymmetric numeral systems

List of numeral systems

* v
* t
* e

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, although it has sometimes been described as "base 1",[4] it differs in some important ways from positional notations, in which the value of a digit depends on its position within a number. For instance, the unary form of a number can be exponentially longer than its representation in other bases.[5]

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.[6] (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.[7][8]

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

## Contents

* 1 Operations
* 2 Applications
* 3 See also
* 4 References
* 5 External links

## Operations

Addition and subtraction are particularly simple in the unary system, as they involve little more than string concatenation.[9] 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>` tag However, this is potentially misleading. Using a unary input is slower for any given number, not faster; the distinction is that a binary (or larger base) input is proportional to the base 2 (or larger base) logarithm of the number while unary input is proportional to the number itself. Therefore, while the run-time and space requirement in unary looks better as function of the input size, it does not represent a more efficient solution.[10]

In computational complexity theory, unary numbering is used to distinguish strongly NP-complete problems from problems that are NP-complete but not strongly NP-complete. A problem in which the input includes some numerical parameters is strongly NP-complete if it remains NP-complete even when the size of the input is made artificially larger by representing the parameters in unary. For such a problem, there exist hard instances for which all parameter values are at most polynomially large.[11]

## Applications

In addition to the application in tally marks, unary numbering is used as part of some data compression algorithms such as Golomb coding.[12] It also forms the basis for the Peano axioms for formalizing arithmetic within mathematical logic.[13] A form of unary notation called Church encoding is used to represent numbers within lambda calculus.[14]

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.[15]

## See also

* Unary coding
* One-hot encoding

## References

1. ↑ Hodges, Andrew (2009), One to Nine: The Inner Life of Numbers, Anchor Canada, p. 14, ISBN 9780385672665, https://books.google.com/books?id=UCuwrtBax7AC&pg=PA14 .
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, https://books.google.com/books?id=GRWOqKwZGRAC&pg=PA117 .
3. ↑ Hext, Jan (1990), Programming Structures: Machines and Programs, 1, Prentice Hall, p. 33, ISBN 9780724809400 .
4. ↑ Brian Hayes (2001), "Third Base", American Scientist 89 (6): 490, doi:10.1511/2001.40.3268, http://www.americanscientist.org/issues/pub/2001/11/third-base, retrieved 2013-07-28
5. ↑ Zdanowski, Konrad (2022), "On efficiency of notations for natural numbers", Theoretical Computer Science 915: 1–10, doi:10.1016/j.tcs.2022.02.015
6. ↑ 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, https://books.google.com/books?id=JggPAAAAIAAJ&pg=PA125 .
7. ↑ Hsieh, Hui-Kuang (1981), "Chinese Tally Mark", The American Statistician 35 (3): 174, doi:10.2307/2683999
8. ↑ Lunde, Ken; Miura, Daisuke (January 27, 2016), "Proposal to Encode Five Ideographic Tally Marks", Unicode Consortium, Proposal L2/16-046, https://www.unicode.org/L2/L2016/16046-ideo-tally-marks.pdf
9. ↑ 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, https://archive.org/details/logiccomputation0000unse/page/30 . See in particular p. 48.
10. ↑ Moore, Cristopher; Mertens, Stephan (2011), The Nature of Computation, Oxford University Press, p. 29, ISBN 9780199233212, https://books.google.com/books?id=z4zMiZyAE1kC&pg=PA29 .
11. ↑ "'Strong' NP-completeness results: Motivation, examples, and implications", Journal of the ACM 25 (3): 499–508, 1978, doi:10.1145/322077.322090 .
12. ↑ "Run-length encodings", IEEE Transactions on Information Theory IT-12 (3): 399–401, 1966, doi:10.1109/TIT.1966.1053907, http://urchin.earth.li/~twic/Golombs_Original_Paper/ .
13. ↑ 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 .
14. ↑ 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 .
15. ↑ Email, Spam Control, How to get service for departmental email servers, http://answers.uillinois.edu/illinois/page.php?id=49002

## External links

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* OEIS sequence A000042 (Unary representation of natural numbers)

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