# Unary numeral system

__: Base-1 numeral system__

**Short description**Numeral systems |
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Hindu–Arabic numeral system |

East Asian |

Alphabetic |

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Positional systems by base |

Non-standard positional numeral systems |

List of numeral systems |

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>`

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## 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]}

## See also

- Unary coding
- One-hot encoding

## References

- ↑ 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. - ↑ 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. - ↑ Hext, Jan (1990),
*Programming Structures: Machines and Programs*,**1**, Prentice Hall, p. 33, ISBN 9780724809400. - ↑ 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. - ↑ Hsieh, Hui-Kuang (1981), "Chinese Tally Mark",
*The American Statistician***35**(3): 174, doi:10.2307/2683999 - ↑ 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 - ↑ 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. - ↑ "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/. - ↑ 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. - ↑ 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. - ↑ http://answers.uillinois.edu/illinois/page.php?id=49002

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