Binary numbers

Counting with two symbols is far older than the machines that made it unavoidable. Pingala's Chandahsastra, a Sanskrit treatise on poetic metre usually dated to the third or second century BCE, classified lines of verse by their patterns of short and long syllables, handling two-valued sequences systematically; later commentators pushed the scheme close to a binary numbering of metres. In 1605 Francis Bacon described a "biliteral" alphabet in which each letter became five places of two symbols — a five-bit code, built for concealment rather than calculation, and one he noted would work with any objects "capable of a twofold difference only". Thomas Harriot investigated binary along with several other positional systems, but published none of his results; they were found later among his papers.

The first widely read publication was Gottfried Wilhelm Leibniz's Explication de l'Arithmétique Binaire, in 1703. Leibniz had been working on base two well before that, which matters because the familiar story runs backwards: he did not take the idea from the I Ching. The Jesuit missionary Joachim Bouvet corresponded with him about the 64 hexagrams in 1701, and the letters established the I Ching as an independent, parallel invention of binary notation — Leibniz read it as an ancient tradition confirming an arithmetic he already possessed, and he liked the theology of a system that builds everything out of nothing and one.

Binary became a technology in two steps. George Boole's An Investigation of the Laws of Thought (1854) gave two-valued logic an algebra. Then Claude Shannon's 1937 master's thesis at MIT, A Symbolic Analysis of Relay and Switching Circuits, showed that Boole's algebra described exactly what networks of switches do — the hinge on which electronic computing turns. Konrad Zuse's Z3, finished in Berlin in 1941, already calculated in binary floating point, while the American ENIAC of 1945 was built as a decimal machine. The word "bit", a contraction of binary digit, was coined by John W. Tukey in a Bell Labs memo of 9 January 1947; Shannon put it into print the following year and credited Tukey for it.

Key properties

• A positive integer n needs floor(log2 n) + 1 binary digits, so 1,000,000 fits in 20 bits.
• 2^k is a 1 followed by k zeros, and 2^k - 1 is a run of k ones.
• The last binary digit is the parity: even numbers end in 0, odd numbers end in 1.
• Doubling shifts every digit one place left; halving an even number shifts one place right.
• n is a power of two exactly when n > 0 and n AND (n - 1) equals zero.
• The count of 1s in a binary numeral is its Hamming weight, or popcount; for 2^k - 1 that count is k.
• Eight bits give 2^8 = 256 distinct values: 0 to 255 unsigned, or -128 to 127 in two’s complement.
• Every positive integer has exactly one binary representation without leading zeros.

Where they turn up

• Digital logic: processors, memory cells and bus lines all hold data as two voltage states, which is why computing settled on base 2 rather than the decimal design ENIAC used.
• A Braille cell is six positions, each raised or flat — a 6-bit code with 64 patterns, devised by Louis Braille in the 1820s.
• Unix file permissions such as chmod 755 are three groups of three bits (read, write, execute) written in octal.
• IPv4 subnet masks and CIDR suffixes like /24 simply count the leading 1 bits of a 32-bit address.
• The 64 hexagrams of the I Ching are six two-state lines apiece; reading them as the binary numbers 0 to 63 is a later interpretation, one Leibniz himself advanced, rather than part of the original text.