Binary number

[1] The modern binary number system was studied in Europe in the 16th and 17th centuries by Thomas Harriot, and Gottfried Leibniz.

However, systems related to binary numbers have appeared earlier in multiple cultures including ancient Egypt, China, Europe and India.

[7] 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.

[5] The Song dynasty scholar Shao Yong (1011–1077) rearranged the hexagrams in a format that resembles modern binary numbers, although he did not intend his arrangement to be used mathematically.

[10] Divination at Ancient Greek Dodona oracle worked by drawing from separate jars, questions tablets and "yes" and "no" pellets.

Pingala's Hindu classic titled Chandaḥśāstra (8.23) describes the formation of a matrix in order to give a unique value to each meter.

[7] Sets of binary combinations similar to the I Ching have also been used in traditional African divination systems, such as Ifá among others, as well as in medieval Western geomancy.

[21] 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.

[22] 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.

Leibniz & Bouvet concluded that this mapping was evidence of major Chinese accomplishments in the sort of philosophical mathematics he admired.

[28] 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.

[30] 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.

[33] 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.

[34] In November 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.

In a demonstration to the American Mathematical Society conference at Dartmouth College on 11 September 1940, Stibitz was able to send the Complex Number Calculator remote commands over telephone lines by a teletype.

[36][37][38] The Z1 computer, which was designed and built by Konrad Zuse between 1935 and 1938, used Boolean logic and binary floating-point numbers.

This is similar to what happens in decimal when certain single-digit numbers are added together; if the result equals or exceeds the value of the radix (10), the digit to the left is incremented: This is known as carrying.

It is based on the simple premise that under the binary system, when given a stretch of digits composed entirely of n ones (where n is any integer length), adding 1 will result in the number 1 followed by a string of n zeros.

From that one finds that large binary numbers can be added using two simple steps, without excessive carry operations.

Using two's complement notation, subtraction can be summarized by the following formula: Multiplication in binary is similar to its decimal counterpart.

The procedure is the same as that of decimal long division; here, the divisor 1012 goes into the first three digits 1102 of the dividend one time, so a "1" is written on the top line.

Aside from long division, one can also devise the procedure so as to allow for over-subtracting from the partial remainder at each iteration, thereby leading to alternative methods which are less systematic, but more flexible as a result.

Though not directly related to the numerical interpretation of binary symbols, sequences of bits may be manipulated using Boolean logical operators.

For example, an arithmetic shift left of a binary number is the equivalent of multiplication by a (positive, integral) power of 2.

This suggests the algorithm: Repeatedly double the number to be converted, record if the result is at least 1, and then throw away the integer part.

A simple divide-and-conquer algorithm is more effective asymptotically: given a binary number, it is divided by 10k, where k is chosen so that the quotient roughly equals the remainder; then each of these pieces is converted to decimal and the two are concatenated.

Other rational numbers have binary representation, but instead of terminating, they recur, with a finite sequence of digits repeating indefinitely.