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.
^Hodges, Andrew (2009), One to Nine: The Inner Life of Numbers, Anchor Canada, p. 14, ISBN 9780385672665.
^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.
^Hext, Jan (1990), Programming Structures: Machines and Programs, vol. 1, Prentice Hall, p. 33, ISBN 9780724809400.
^Brian Hayes (2001), "Third Base", American Scientist, 89 (6): 490, doi:10.1511/2001.40.3268, archived from the original on 2014-01-11, retrieved 2013-07-28
^Zdanowski, Konrad (2022), "On efficiency of notations for natural numbers", Theoretical Computer Science, 915: 1–10, doi:10.1016/j.tcs.2022.02.015, MR 4410388
^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, JSTOR 2970818.
^Hsieh, Hui-Kuang (1981), "Chinese Tally Mark", The American Statistician, 35 (3): 174, doi:10.2307/2683999, JSTOR 2683999
^Lunde, Ken; Miura, Daisuke (January 27, 2016), "Proposal to Encode Five Ideographic Tally Marks", Unicode Consortium(PDF), Proposal L2/16-046
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