A binary number is a number expressed in the base-2 numeral system or binary numeral system, a method for representing numbers that uses only two symbols for the natural numbers: typically (zero) and (one). A binary number may also refer to a rational number that has a finite representation in the binary numeral system, that is, the quotient of an integer by a power of two.

The base-2 numeral system is a positional notation with a radix of 2. Each digit is referred to as a bit, or binary digit. Because of its straightforward implementation in digital electronic circuitry using logic gates, the binary system is used by almost all modern computers and computer-based devices, as a preferred system of use, over various other human techniques of communication, because of the simplicity of the language and the noise immunity in physical implementation.

History

The modern binary number system was first studied in Europe in the 16th and 17th centuries by Thomas Harriot, and decades later by Gottfried Leibniz, who is credited for the invention. However, systems related to binary numbers have appeared earlier in multiple cultures including ancient Egypt, China, Europe and India, e.g. in relation to divination using binary lots.

Egypt

thumb|left|Arithmetic values thought to have been represented by parts of the [[Eye of Horus]]

The scribes of ancient Egypt used two different systems for their fractions, Egyptian fractions (not related to the binary number system) and Horus-Eye fractions (so called because some historians of mathematics believed that the symbols used for this system could be arranged to form the eye of Horus, although this has been disputed). Horus-Eye fractions are a binary numbering system for fractional quantities of grain, liquids, or other measures, in which a fraction of a hekat is expressed as a sum of the binary fractions , , , , , and . Early forms of this system can be found in documents from the Fifth Dynasty of Egypt, approximately 2400 BC, and its fully developed hieroglyphic form dates to the Nineteenth Dynasty of Egypt, approximately 1200 BC.

The method used for ancient Egyptian multiplication is also closely related to binary numbers. In this method, multiplying one number by a second is performed by a sequence of steps in which a value (initially the first of the two numbers) is either doubled or has the first number added back into it; the order in which these steps are to be performed is given by the binary representation of the second number. This method can be seen in use, for instance, in the Rhind Mathematical Papyrus, which dates to around 1650 BC.

China

thumb|Daoist Bagua

The I Ching dates from the 9th century BC in China. The binary notation in the I Ching is used to interpret its quaternary technique of divination. Eight trigrams (Bagua) and a set of 64 hexagrams ("sixty-four" gua), analogous to the three-bit and six-bit binary numerals, were in use at least as early as the Zhou dynasty of ancient China. Viewing the least significant bit on top of single hexagrams in Shao Yong's square

and reading along rows either from bottom right to top left with solid lines as 0 and broken lines as 1 or from top left to bottom right with solid lines as 1 and broken lines as 0 hexagrams can be interpreted as sequence from 0 to 63.

Classical antiquity

Etruscans divided the outer edge of divination livers into sixteen parts, each inscribed with the name of a divinity and its region of the sky. Each liver region produced a binary reading which was combined into a final binary for divination.

Divination at Ancient Greek Dodona oracle worked by drawing from separate jars, questions tablets and "yes" and "no" pellets. The result was then combined to make a final prophecy.

India

The Indian scholar Pingala (c. 2nd century BC) developed a binary system for describing prosody. He described meters in the form of short and long syllables (the latter equal in length to two short syllables). They were known as laghu (light) and guru (heavy) syllables.

Pingala's Hindu classic titled Chandaḥśāstra (8.23) describes the formation of a matrix in order to give a unique value to each meter. "Chandaḥśāstra" literally translates to science of meters in Sanskrit. The binary representations in Pingala's system increases towards the right, and not to the left like in the binary numbers of the modern positional notation. In Pingala's system, the numbers start from number one, and not zero. Four short syllables "0000" is the first pattern and corresponds to the value one. The numerical value is obtained by adding one to the sum of place values.

West Africa

The Ifá is a West African divination system popular among the Yoruba tribe of the Old Oyo Empire. Similar to the I Ching, but has up to 256 binary signs, unlike the I Ching which has 64. The number comes from squaring 16 which also matches the total possibilities in an 8-bit sequence. In Ifá divination, this reflects the possible outcomes called Odú. These Odú are determined using an Ọpẹlẹ chain, which has 8 seeds. Each seed can land in one of two positions (open or closed) creating all the possible combinations. The Ifá originated in 15th century West Africa among Yoruba people. In 2008, UNESCO added Ifá to its list of the "Masterpieces of the Oral and Intangible Heritage of Humanity".

Other cultures

The residents of the island of Mangareva in French Polynesia were using a hybrid binary-decimal system before 1450. Slit drums with binary tones are used to encode messages across Africa and Asia.

Western predecessors to Leibniz

In the late 13th century Ramon Llull had the ambition to account for all wisdom in every branch of human knowledge of the time. For that purpose he developed a general method or "Ars generalis" based on binary combinations of a number of simple basic principles or categories, for which he has been considered a predecessor of computing science and artificial intelligence.

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.

In 1617, John Napier described a system he called location arithmetic for doing binary calculations using a non-positional representation by letters.

Thomas Harriot investigated several positional numbering systems, including binary, but did not publish his results; they were found later among his papers.

Possibly the first publication of the system in Europe was by Juan Caramuel y Lobkowitz, in 1700.

Leibniz

thumb|upright|Gottfried Leibniz

Leibniz wrote in excess of a hundred manuscripts on binary, most of them remaining unpublished. Before his first dedicated work in 1679, numerous manuscripts feature early attempts to explore binary concepts, including tables of numbers and basic calculations, often scribbled in the margins of works unrelated to mathematics. Leibniz's system uses 0 and 1, like the modern binary numeral system. An example of Leibniz's binary numeral system is as follows: Of this parallel invention, Leibniz wrote in his "Explanation Of Binary Arithmetic" that "this restitution of their meaning, after such a great interval of time, will seem all the more curious."

The relation was a central idea to his universal concept of a language or characteristica universalis, a popular idea that would be followed closely by his successors such as Gottlob Frege and George Boole in forming modern symbolic logic.

Leibniz was first introduced to the I Ching through his contact with the French Jesuit Joachim Bouvet, who visited China in 1685 as a missionary. Leibniz saw the I Ching hexagrams as an affirmation of the universality of his own religious beliefs as a Christian. Binary numerals were central to Leibniz's theology. He believed that binary numbers were symbolic of the Christian idea of creatio ex nihilo or creation out of nothing.

{3_{10 = \frac{1_2}{11_2} = 0.01010101\overline{01}\ldots\,_2 </math>

<math display="block">\frac{12_{10{17_{10 = \frac{1100_2}{10001_2} = 0.10110100 10110100\overline{10110100}\ldots\,_2 </math>

The phenomenon that the binary representation of any rational is either terminating or recurring also occurs in other radix-based numeral systems. See, for instance, the explanation in decimal. Another similarity is the existence of alternative representations for any terminating representation, relying on the fact that 0.111111... is the sum of the geometric series 2<sup>−1</sup> + 2<sup>−2</sup> + 2<sup>−3</sup> + ... which is 1.

Binary numerals that neither terminate nor recur represent irrational numbers. For instance,

  • 0.10100100010000100000100... does have a pattern, but it is not a fixed-length recurring pattern, so the number is irrational
  • 1.0110101000001001111001100110011111110... is the binary representation of <math>\sqrt{2}</math>, the square root of 2, another irrational. It has no discernible pattern.

See also

  • ASCII
  • Balanced ternary
  • Bitwise operation
  • Binary code
  • Binary-coded decimal
  • Binary non positional code
  • Finger binary
  • Gray code
  • IEEE 754
  • Linear-feedback shift register
  • Offset binary
  • Quibinary
  • Reduction of summands
  • Redundant binary representation
  • Repeating decimal
  • Two's complement
  • Unicode

References

Sources

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  • Binary System at cut-the-knot
  • Conversion of Fractions at cut-the-knot
  • Sir Francis Bacon's BiLiteral Cypher system , predates binary number system.
  • An analysis of binary as an efficient base