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Binary Golay code
The binary Golay code is a perfect error-correcting code of length 23 that corrects any three bit errors, published by Marcel Golay in 1949 and used on NASA's Voyager missions.
Binary logarithm
The binary logarithm, also called log2 or the base-2 logarithm, is the logarithm to base 2, the exponent y such that 2 to the power y equals x.
Binary number
A binary number, also called a base-2 number, is written using only the digits 0 and 1, and digital computers use it to store and manipulate all data.
Binary operation
A binary operation, also called a binary operator or dyadic operation, is a function combining two elements of a set to yield an element of the same set.
Binary-coded decimal
Binary-coded decimal (BCD) is a way of representing decimal digits in digital systems, storing each digit 0 through 9 in its own 4-bit group of bits.
Binet's formula
Binet's formula is a closed-form expression computing the nth Fibonacci number directly from the golden ratio, published by Jacques Binet in 1843 though Euler had it in 1765.
Binomial coefficient
The binomial coefficient, or choose function, counts the ways to choose k objects from n distinct objects without order, and gives the coefficients in the expansion of (a+b)^n.
Binomial series
The binomial series, also known as Newton's binomial series, is the power series for (1+z)^α, converging for |z| < 1 and probably first stated by Newton in 1664–1665.
Binomial theorem
The binomial theorem, or binomial formula, expands powers of a sum using binomial coefficients from Pascal's triangle; first attributed to al-Karaji around 1029 and extended to fractional powers by Newton.
Biquaternion
A biquaternion is a quaternion whose four coefficients are complex rather than real numbers, introduced by Hamilton by 1850 and generalized by Clifford in 1871, used in physics.
Birational geometry
Birational geometry, also known as birational algebraic geometry, is the branch of algebraic geometry that studies varieties up to birational equivalence, meaning rational maps invertible on dense open subsets.
Birch and Swinnerton-Dyer conjecture
The Birch and Swinnerton-Dyer conjecture, a Clay Millennium Prize Problem in number theory, states that an elliptic curve's L-function vanishes at s = 1 to order equal to its rank of rational points.
Bit numbering
Bit numbering assigns index numbers to the individual bits of a value. Conventions include LSB 0, where 0 is least significant, and MSB 0, where 0 is most significant.
Bitwise operation
A bitwise operation, also known as a bit operation, acts on the individual bits of binary numerals rather than on the numbers those bits represent, using operators like AND, OR, XOR, and shifts.
Bitwise operations in C
Bitwise operations in C manipulate individual bits of integer values with operators such as AND, OR, XOR, complement, and shifts, letting C work close to the hardware.
Bivariate bicycle code
The bivariate bicycle code is a quantum low-density parity-check code introduced by IBM researchers in 2024, whose gross code encodes 12 logical qubits into 144 data qubits.
Bivector
A bivector is an oriented quantity of grade 2 in exterior algebra, formed as the outer product of two vectors and representing an oriented plane segment.
Black box group
A black box group is a finite group whose elements are encoded by bit strings, with operations available only through oracles; it was introduced by Babai and Szemerédi in 1984.
Blackboard bold
Blackboard bold, also called double-struck or chalkboard bold, is a mathematical typeface with doubled strokes imitating bold, used chiefly for number systems like ℤ, ℝ, and ℂ.
Bloch–Kato conjecture
The Bloch–Kato conjecture, now the norm residue isomorphism theorem, identifies Milnor K-theory modulo ℓ with étale cohomology; proved by Rost, Voevodsky, and Weibel.
Bloch's higher Chow group
Bloch's higher Chow groups are invariants CH^p(X, n) of algebraic varieties, defined by Spencer Bloch in 1986, recovering the classical Chow groups at n = 0.
Block matrix
A block matrix, also called a partitioned matrix, is a matrix divided into smaller rectangular submatrices called blocks, allowing sums, products, and inverses to be computed block by block.
Blow-up (mathematics)
A blow-up in algebraic geometry is a birational transformation that replaces a closed subscheme with the tangent directions out of it; by Hironaka's 1964 theorem, resolutions of singularities are built from sequences of blowups.
Blum–Shub–Smale machine
The Blum–Shub–Smale machine (BSS machine) is a model of computation over the real numbers, treating each arithmetic operation as a single step, introduced by Blum, Shub, and Smale in 1989.
Bombieri–Vinogradov theorem
A 1965 theorem of Enrico Bombieri and A. I. Vinogradov showing primes are well distributed in arithmetic progressions on average over moduli up to about the square root.
Boolean algebra
Boolean algebra, also known as Boolean logic or switching algebra, is the algebra of two-valued logic and of sets under union and complementation, introduced by George Boole in 1847.
Boolean lattice
A Boolean lattice is a distributive lattice in which every element has a unique complement, exemplified by the power set of a set ordered by inclusion, where meet is intersection and join is union.
Boolean ring
A Boolean ring is a ring in which every element satisfies x² = x; such rings are automatically commutative of characteristic 2 and are definitionally equivalent to Boolean algebras.
Booth's multiplication algorithm
Booth's multiplication algorithm, also known as Booth's algorithm, is a signed binary multiplication technique published by Andrew D. Booth in 1951 that handles two's complement numbers of either sign without separate correction.
Borel subalgebra
A Borel subalgebra of a Lie algebra is a maximal solvable subalgebra, named for Armand Borel. It organizes the structure and representation theory of semisimple Lie algebras.