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Tally marks, also called hash marks, are a form of numeral used for counting. They can be thought of as a unary numeral system.
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.
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 Dolní Věstonice, 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]
The Ishango bone, found in the Ishango region of the present-day Democratic Republic of Congo, is dated to over 20,000 years old. Upon discovery, it was thought to portray 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] Alexander Marshack examined the Ishango bone microscopically, and concluded that it may represent a six-month lunar calendar. [3]
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.[ citation needed ]
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Roman numerals, the Brahmi and Chinese numerals for one through three (一 二 三), and rod numerals were derived from tally marks, as possibly was the ogham script. [7]
Base 1 arithmetic notation system is a unary positional system similar to tally marks. It is rarely used as a practical base for counting due to its difficult readability.
The numbers 1, 2, 3, 4, 5, 6 ... would be represented in this system as [8]
Base 1 notation is widely used in type numbers of flour; the higher number represents a higher grind.
In 2015, Ken Lunde and Daisuke Miura submitted a proposal to encode various systems of tally marks in the Unicode Standard. [9] However, the box tally and dot-and-dash tally characters were not accepted for encoding, and only the five ideographic tally marks (正 scheme) and two Western tally digits were added to the Unicode Standard in the Counting Rod Numerals block in Unicode version 11.0 (June 2018). Only the tally marks for the numbers 1 and 5 are encoded, and tally marks for the numbers 2, 3 and 4 are intended to be composed from sequences of tally mark 1 at the font level.
Counting Rod Numerals [1] [2] Official Unicode Consortium code chart (PDF) | ||||||||||||||||
0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | A | B | C | D | E | F | |
U+1D36x | 𝍠 | 𝍡 | 𝍢 | 𝍣 | 𝍤 | 𝍥 | 𝍦 | 𝍧 | 𝍨 | 𝍩 | 𝍪 | 𝍫 | 𝍬 | 𝍭 | 𝍮 | 𝍯 |
U+1D37x | 𝍰 | 𝍱 | 𝍲 | 𝍳 | 𝍴 | 𝍵 | 𝍶 | 𝍷 | 𝍸 | |||||||
Notes |
A numeral system is a writing system for expressing numbers; that is, a mathematical notation for representing numbers of a given set, using digits or other symbols in a consistent manner.
The unary numeral system is the simplest numeral system to represent natural numbers: to represent a number N, a symbol representing 1 is repeated N times.
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The Chinese Character Code for Information Interchange or CCCII is a character set developed by the Chinese Character Analysis Group in Taiwan. It was first published in 1980, and significantly expanded in 1982 and 1987.
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The Ishango bone, discovered at the "Fisherman Settlement" of Ishango in the Democratic Republic of Congo, is a bone tool and possible mathematical device that dates to the Upper Paleolithic era. The curved bone is dark brown in color, about 10 centimeters in length, and features a sharp piece of quartz affixed to one end, perhaps for engraving. Because the bone has been narrowed, scraped, polished, and engraved to a certain extent, it is no longer possible to determine what animal the bone belonged to, although it is assumed to belong to a mammal.
In computing, a Unicode symbol is a Unicode character which is not part of a script used to write a natural language, but is nonetheless available for use as part of a text.
The Chinese, Japanese and Korean (CJK) scripts share a common background, collectively known as CJK characters. During the process called Han unification, the common (shared) characters were identified and named CJK Unified Ideographs. As of Unicode 15.1, Unicode defines a total of 97,680 characters.
Number systems have progressed from the use of fingers and tally marks, perhaps more than 40,000 years ago, to the use of sets of glyphs able to represent any conceivable number efficiently. The earliest known unambiguous notations for numbers emerged in Mesopotamia about 5000 or 6000 years ago.
A numeral is a character that denotes a number. The decimal number digits 0–9 are used widely in various writing systems throughout the world, however the graphemes representing the decimal digits differ widely. Therefore Unicode includes 22 different sets of graphemes for the decimal digits, and also various decimal points, thousands separators, negative signs, etc. Unicode also includes several non-decimal numerals such as Aegean numerals, Roman numerals, counting rod numerals, Mayan numerals, Cuneiform numerals and ancient Greek numerals. There is also a large number of typographical variations of the Western Arabic numerals provided for specialized mathematical use and for compatibility with earlier character sets, such as ² or ②, and composite characters such as ½.
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In the Unicode standard, a plane is a continuous group of 65,536 (216) code points. There are 17 planes, identified by the numbers 0 to 16, which corresponds with the possible values 00–1016 of the first two positions in six position hexadecimal format (U+hhhhhh). Plane 0 is the Basic Multilingual Plane (BMP), which contains most commonly used characters. The higher planes 1 through 16 are called "supplementary planes". The last code point in Unicode is the last code point in plane 16, U+10FFFF. As of Unicode version 15.1, five of the planes have assigned code points (characters), and seven are named.
Counting rods are small bars, typically 3–14 cm long, that were used by mathematicians for calculation in ancient East Asia. They are placed either horizontally or vertically to represent any integer or rational number.
A tally stick was an ancient memory aid device used to record and document numbers, quantities and messages. Tally sticks first appear as animal bones carved with notches during the Upper Palaeolithic; a notable example is the Ishango Bone. Historical reference is made by Pliny the Elder about the best wood to use for tallies, and by Marco Polo (1254–1324) who mentions the use of the tally in China. Tallies have been used for numerous purposes such as messaging and scheduling, and especially in financial and legal transactions, to the point of being currency.
Counting Rod Numerals is a Unicode block containing traditional Chinese counting rod symbols, which mathematicians used for calculation in ancient China, Japan, Korea, and Vietnam. The orientation of the Unicode characters follows Song dynasty convention, with digits represented as horizontal lines, and tens represented as vertical lines, which differs from Han dynasty practice which represented digits as vertical lines, and tens as horizontal lines.