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Tally marks

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Title: Tally marks  
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Subject: Prehistoric numerals, Numeral system, List of numeral systems, Hanakapiai Beach, Hangman (game)
Collection: Elementary Mathematics, Korean Language, Mathematical Notation, Numeral Systems, Numerals
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Tally marks

Counting using tally marks at Hanakapiai Beach. The number shown is 82.

Tally marks, also called hash marks, are a unary numeral system. They are a form of numeral used for counting. They are most useful in counting or tallying ongoing results, such as the score in a game or sport, as no intermediate results need to be erased or discarded. However, because of the length of large numbers, tallies are not commonly used for static text. Notched sticks, known as tally sticks, were also historically used for this purpose.


  • Early history 1
  • Clustering 2
  • Writing systems 3
  • See also 4
  • References 5

Early history

Counting aids other than body parts appear in the Upper Paleolithic. The oldest tally sticks date to between 35,000 and 25,000 years ago, in the form of notched bones found in the context of the European Aurignacian to Gravettian and in Africa's Late Stone Age.

The so-called Wolf bone is a prehistoric artifact discovered in 1937 in Czechoslovakia during excavations at Vestonice, Moravia, led by Karl Absolon. Dated to the Aurignacian, approximately 30,000 years ago, the bone is marked with 55 marks which may be tally marks. The head of an ivory Venus figurine was excavated close to the bone.[1]

It has been claimed that the Ishango Bone, found in the Ishango region of what is today the Democratic Republic of Congo and dated to over 20,000 years old, portrays a series of prime numbers. In the book How Mathematics Happened: The First 50,000 Years, Peter Rudman argues that the development of the concept of prime numbers could only have come about after the concept of division, which he dates to after 10,000 BC, with prime numbers probably not being understood until about 500 BC. He also writes that "no attempt has been made to explain why a tally of something should exhibit multiples of two, prime numbers between 10 and 20, and some numbers that are almost multiples of 10."[2]


Various ways to cluster the number 8. The first or fifth mark in each group may be written at an angle to the others for easier distinction. In the fourth example, the fifth stroke "closes out" a group of five, forming a "herringbone". In the fifth row (used in Brazil, France, and the United States) the fifth mark crosses diagonally, forming a "five-bar gate".

Tally marks are typically clustered in groups of five for legibility. The cluster size 5 has the advantages of (a) easy conversion into decimal for higher arithmetic operations and (b) avoiding error, as humans can far more easily correctly identify a cluster of 5 than one of 10.

Writing systems

Brahmi numerals (lower row) in India in the 1st century CE

Roman numerals, the Chinese numerals 一 二 三, and rod numerals were derived from tally marks, as possibly was the ogham script.

See also


  1. ^ *Graham Flegg, Numbers: their history and meaning, Courier Dover Publications, 2002 ISBN 978-0-486-42165-0, pp. 41-42.
  2. ^ Rudman, Peter Strom (2007). How Mathematics Happened: The First 50,000 Years. Prometheus Books. p. 64.  
  3. ^ Hsieh, Hui-Kuang (1981) "Chinese tally mark", The American Statistician, 35 (3), p. 174, doi:10.2307/2683999
  4. ^ Schenck, Carl A. (1898) Forest mensuration. The University Press. (Note: The linked reference appears to actually be "Bulletin of the Ohio Agricultural Experiment Station", Number 302, August 1916)
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