For mathematicians, the binary numeral system is a common system. It is a base-2 number system, a numeral system that represents numeric values using two symbols, usually 0 and 1. More specifically, the usual base-2 system is a positional notation with a radix of 2. Owing to its straightforward implementation in electronic circuitry, the binary system is used internally by virtually all modern computers.
According to wikipedia.com, the ancient Indian mathematician Pingala presented the first known description of a binary numeral system around 800 BC written in Hindu numerals. The numeration system was based on the Eye of Horus Old Kingdom numeration system.
A full set of 8 trigrams and 64 hexagrams, analogous to the 3-bit and 6-bit binary numerals, were known to the ancient Chinese in the classic text I Ching. Similar sets of binary combinations have also been used in traditional African divination systems such as Ifá as well as in medieval Western geomancy.
An ordered binary arrangement of the hexagrams of the I Ching, representing the decimal sequence from 0 to 63, and a method for generating the same, was developed by the Chinese scholar and philosopher Shao Yong in the 11th century. However, there is no evidence that Shao understood binary computation.
In 1605 Francis Bacon discussed a system by which 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. 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 onely; as by Bells, by Trumpets, by Lights and Torches, by the report of Muskets, and any instruments of like nature.”[2] (See Bacon’s cipher.)
The modern binary number system was fully documented by Gottfried Leibniz in the 17th century in his article Explication de l’Arithmétique Binaire. Leibniz’s system used 0 and 1, like the modern binary numeral system.
In 1854, British mathematician George Boole published a landmark paper detailing a system of logic that would become known as Boolean algebra. His logical system proved instrumental in the development of the binary system, particularly in its implementation in electronic circuitry.
In 1937, Claude Shannon produced his master’s thesis at MIT that implemented Boolean algebra and binary arithmetic using electronic relays and switches for the first time in history. Entitled A Symbolic Analysis of Relay and Switching Circuits, Shannon’s thesis essentially founded practical digital circuit design.
In November of 1937, George Stibitz, then working at Bell Labs, completed a relay-based computer he dubbed the “Model K” (for “Kitchen”, where he had assembled it), which calculated using binary addition. Bell Labs thus authorized a full research program in late 1938 with Stibitz at the helm. Their Complex Number Computer, completed January 8, 1940, was able to calculate complex numbers. In a demonstration to the American Mathematical Society conference at Dartmouth College on September 11, 1940, Stibitz was able to send the Complex Number Calculator remote commands over telephone lines by a teletype. It was the first computing machine ever used remotely over a phone line. Some participants of the conference who witnessed the demonstration were John Von Neumann, John Mauchly, and Norbert Wiener, who wrote about it in his memoirs.
Representation
A binary number can be represented by any sequence of bits (binary digits), which in turn may be represented by any mechanism capable of being in two mutually exclusive states. The following sequences of symbols could all be interpreted as the same binary numeric value of 667:
1 0 1 0 0 1 1 0 1 1
| - | - - | | - | |
x o x o o x x o x x
y n y n n y y n y y
A binary clock might use LEDs to express binary values. In this clock, each column of LEDs shows a binary-coded decimal numeral of the traditional sexagesimal time.
The numeric value represented in each case is dependent upon the value assigned to each symbol. In a computer, the numeric values may be represented by two different voltages; on a magnetic disk, magnetic polarities may be used. A “positive”, “yes”, or “on” state is not necessarily equivalent to the numerical value of one; it depends on the architecture in use.
In keeping with customary representation of numerals using Arabic numerals, binary numbers are commonly written using the symbols 0 and 1. When written, binary numerals are often subscripted, prefixed or suffixed in order to indicate their base, or radix. The following notations are equivalent:
- 100101 binary (explicit statement of format)
- 100101b (a suffix indicating binary format)
- 100101B (a suffix indicating binary format)
- bin 100101 (a prefix indicating binary format)
- 1001012 (a subscript indicating base-2 (binary) notation)
- %100101 (a prefix indicating binary format)
- 0b100101 (a prefix indicating binary format, common in programming languages)
When spoken, binary numerals are usually pronounced by pronouncing each individual digit, in order to distinguish them from decimal numbers. For example, the binary numeral “100″ is pronounced “one zero zero”, rather than “one hundred”, to make its binary nature explicit, and for purposes of correctness. Since the binary numeral “100″ is equal to the decimal value four, it would be confusing, and numerically incorrect, to refer to the numeral as “one hundred” so to speak.
