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History of binary codes

Further information: Binary number § History

Invention

The modern binary number system, the basis for binary code, is an invention by Gottfried Leibniz in 1689 and appears in his article Explication de l'Arithmétique Binaire (English: Explanation of the Binary Arithmetic) which uses only the characters 1 and 0, and some remarks on its usefulness. Leibniz's system uses 0 and 1, like the modern binary numeral system. Binary numerals were central to Leibniz's intellectual and theological ideas. He believed that binary numbers were symbolic of the Christian idea of creatio ex nihilo or creation out of nothing.¹²In Leibniz's view, binary numbers represented a fundamental form of creation, reflecting the simplicity and unity of the divine.³ Leibniz was also attempting to find a way to translate logical reasoning into pure mathematics. He viewed the binary system as a means of simplifying complex logical and mathematical processes, believing that it could be used to express all concepts of arithmetic and logic.⁴

Previous Ideas

Leibniz explained in his work that he encountered the I Ching by Fu Xi⁵ that dates from the 9th century BC in China,⁶ through French Jesuit Joachim Bouvet and noted with fascination how its hexagrams correspond to the binary numbers from 0 to 111111, and concluded that this mapping was evidence of major Chinese accomplishments in the sort of philosophical visual binary mathematics he admired.⁷⁸ Leibniz saw the hexagrams as an affirmation of the universality of his own religious belief.⁹ After Leibniz ideas were ignored, the book had confirmed his theory that life could be simplified or reduced down to a series of straightforward propositions. He created a system consisting of rows of zeros and ones. During this time period, Leibniz had not yet found a use for this system.¹⁰ The binary system of the I Ching is based on the duality of yin and yang.¹¹ Slit drums with binary tones are used to encode messages across Africa and Asia.¹² The Indian scholar Pingala (around 5th–2nd centuries BC) developed a binary system for describing prosody in his Chandashutram.¹³¹⁴

Mangareva people in French Polynesia were using a hybrid binary-decimal system before 1450.¹⁵ In the 11th century, scholar and philosopher Shao Yong developed a method for arranging the hexagrams which corresponds, albeit unintentionally, to the sequence 0 to 63, as represented in binary, with yin as 0, yang as 1 and the least significant bit on top. The ordering is also the lexicographical order on sextuples of elements chosen from a two-element set.¹⁶

In 1605 Francis Bacon discussed a system whereby letters of the alphabet could be reduced to sequences of binary digits, which could then be encoded as scarcely visible variations in the font in any random text.¹⁷ Importantly for the general theory of binary encoding, he added that this method could be used with any objects at all: "provided those objects be capable of a twofold difference only; as by Bells, by Trumpets, by Lights and Torches, by the report of Muskets, and any instruments of like nature".¹⁸

Boolean Logical System

George Boole published a paper in 1847 called 'The Mathematical Analysis of Logic' that describes an algebraic system of logic, now known as Boolean algebra. Boole's system was based on binary, a yes-no, on-off approach that consisted of the three most basic operations: AND, OR, and NOT.¹⁹ This system was not put into use until a graduate student from Massachusetts Institute of Technology, Claude Shannon, noticed that the Boolean algebra he learned was similar to an electric circuit. In 1937, Shannon wrote his master's thesis, A Symbolic Analysis of Relay and Switching Circuits, which implemented his findings. Shannon's thesis became a starting point for the use of the binary code in practical applications such as computers, electric circuits, and more.²⁰

Other forms of binary code

Main article: List of binary codes

The bit string is not the only type of binary code: in fact, a binary system in general, is any system that allows only two choices such as a switch in an electronic system or a simple true or false test.

Braille

Braille is a type of binary code that is widely used by the blind to read and write by touch, named for its creator, Louis Braille. This system consists of grids of six dots each, three per column, in which each dot has two states: raised or not raised. The different combinations of raised and flattened dots are capable of representing all letters, numbers, and punctuation signs.

Bagua

The bagua are diagrams used in feng shui, Taoist cosmology and I Ching studies. The ba gua consists of 8 trigrams; bā meaning 8 and guà meaning divination figure. The same word is used for the 64 guà (hexagrams). Each figure combines three lines (yáo) that are either broken (yin) or unbroken (yang). The relationships between the trigrams are represented in two arrangements, the primordial, "Earlier Heaven" or "Fuxi" bagua, and the manifested, "Later Heaven", or "King Wen" bagua.²¹ (See also, the King Wen sequence of the 64 hexagrams).

Ifá, Ilm Al-Raml and Geomancy

The Ifá/Ifé system of divination in African religions, such as of Yoruba, Igbo, and Ewe, consists of an elaborate traditional ceremony producing 256 oracles made up by 16 symbols with 256 = 16 x 16. An initiated priest, or Babalawo, who had memorized oracles, would request sacrifice from consulting clients and make prayers. Then, divination nuts or a pair of chains are used to produce random binary numbers,²² which are drawn with sandy material on an "Opun" figured wooden tray representing the totality of fate.

Through the spread of Islamic culture, Ifé/Ifá was assimilated as the "Science of Sand" (ilm al-raml), which then spread further and became "Science of Reading the Signs on the Ground" (Geomancy) in Europe.

This was thought to be another possible route from which computer science was inspired,²³ as Geomancy arrived at Europe at an earlier stage (about 12th Century, described by Hugh of Santalla) than I Ching (17th Century, described by Gottfried Wilhelm Leibniz).

Coding systems

ASCII code

The American Standard Code for Information Interchange (ASCII), uses a 7-bit binary code to represent text and other characters within computers, communications equipment, and other devices. Each letter or symbol is assigned a number from 0 to 127. For example, lowercase "a" is represented by 1100001 as a bit string (which is decimal 97).

Binary-coded decimal

Binary-coded decimal (BCD) is a binary encoded representation of integer values that uses a 4-bit nibble to encode decimal digits. Four binary bits can encode up to 16 distinct values; but, in BCD-encoded numbers, only ten values in each nibble are legal, and encode the decimal digits zero, through nine. The remaining six values are illegal and may cause either a machine exception or unspecified behavior, depending on the computer implementation of BCD arithmetic.

BCD arithmetic is sometimes preferred to floating-point numeric formats in commercial and financial applications where the complex rounding behaviors of floating-point numbers is inappropriate.²⁴

Early uses of binary codes

• 1875: Émile Baudot "Addition of binary strings in his ciphering system," which, eventually, led to the ASCII of today.
• 1884: The Linotype machine where the matrices are sorted to their corresponding channels after use by a binary-coded slide rail.
• 1932: C. E. Wynn-Williams "Scale of Two" counter²⁵
• 1937: Alan Turing electro-mechanical binary multiplier
• 1937: George Stibitz "excess three" code in the Complex Computer²⁶
• 1937: Atanasoff–Berry Computer²⁷
• 1938: Konrad Zuse Z1

Current uses of binary

Most modern computers use binary encoding for instructions and data. CDs, DVDs, and Blu-ray Discs represent sound and video digitally in binary form. Telephone calls are carried digitally on long-distance and mobile phone networks using pulse-code modulation, and on voice over IP networks.

Weight of binary codes

The weight of a binary code, as defined in the table of constant-weight codes,²⁸ is the Hamming weight of the binary words coding for the represented words or sequences.

See also

• Binary number
• List of binary codes
• Binary file
• Unicode
• Gray code

External links

• Sir Francis Bacon's BiLiteral Cypher system[usurped], predates binary number system.
• Weisstein, Eric W. "Error-Correcting Code". MathWorld.
• Table of general binary codes. An updated version of the tables of bounds for small general binary codes given in M.R. Best; A.E. Brouwer; F.J. MacWilliams; A.M. Odlyzko; N.J.A. Sloane (1978), "Bounds for Binary Codes of Length Less than 25", IEEE Trans. Inf. Theory, 24: 81–93, CiteSeerX 10.1.1.391.9930, doi:10.1109/tit.1978.1055827.
• Table of Nonlinear Binary Codes. Maintained by Simon Litsyn, E. M. Rains, and N. J. A. Sloane. Updated until 1999.
• Glaser, Anton (1971). "Chapter VII Applications to Computers". History of Binary and other Nondecimal Numeration. Tomash. ISBN 978-0-938228-00-4. cites some pre-ENIAC milestones.
• First book in the world fully written in binary code: (IT) Luigi Usai, 01010011 01100101 01100111 01110010 01100101 01110100 01101001, Independently published, 2023, ISBN 979-8-8604-3980-1. URL consulted September 8, 2023.

References

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3. Leibniz G., Explication de l'Arithmétique Binaire, Die Mathematische Schriften, ed. C. Gerhardt, Berlin 1879, vol.7, p.223; Engl. transl.[1] http://www.leibniz-translations.com/binary.htm ↩

4. Leibniz G., Explication de l'Arithmétique Binaire, Die Mathematische Schriften, ed. C. Gerhardt, Berlin 1879, vol.7, p.223; Engl. transl.[1] http://www.leibniz-translations.com/binary.htm ↩

5. Leibniz G., Explication de l'Arithmétique Binaire, Die Mathematische Schriften, ed. C. Gerhardt, Berlin 1879, vol.7, p.223; Engl. transl.[1] http://www.leibniz-translations.com/binary.htm ↩

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27. Glaser 1971 - Glaser, Anton (1971). "Chapter VII Applications to Computers". History of Binary and other Nondecimal Numeration. Tomash. ISBN 978-0-938228-00-4. ↩

28. Table of Constant Weight Binary Codes http://www.research.att.com/~njas/codes/Andw/ ↩