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Proof that e is irrational
Proof that e is irrational: the demonstration, first given by Euler in the 1730s, that Euler's number e cannot be written as a ratio of two integers.
Proof that π is irrational
Proof that π is irrational, first given by Johann Heinrich Lambert in 1761, established that π cannot be written as a ratio of two integers.
Proper morphism
In algebraic geometry, a proper morphism is a morphism of schemes that is separated, of finite type, and universally closed; projective space is proper, while affine space is not.
Proportionality (mathematics)
Proportionality is the mathematical relation between two varying quantities whose ratio or product is constant: a fixed ratio gives direct proportion and a fixed product gives inverse proportion.
Pseudoscalar
A pseudoscalar in geometric algebra (Clifford algebra) is the product of all orthonormal basis vectors, representing a unit oriented volume and fixing the space's orientation.
Puiseux series
A Puiseux series is a power series in fractional powers of a variable that expresses the local behavior of algebraic functions; the method goes back to Newton.
Pythagorean triple
A Pythagorean triple is a set of three positive integers (a, b, c) with a² + b² = c², such as (3, 4, 5), the side lengths of a right triangle.
Q (number format)
Q is a notation for fixed-point binary numbers, devised by Texas Instruments, in which Qm.n names m integer bits and n fractional bits, fixing an implicit scaling factor.
QR algorithm
The QR algorithm, or QR method, is an iterative numerical method for computing the eigenvalues of a square matrix, devised by J. G. F. Francis in 1961.
QR decomposition
The QR decomposition, also called QR factorization, expresses a matrix A as A = QR with Q orthogonal and R upper triangular, and underlies least-squares fitting and eigenvalue computation.
Quadratic equation
A quadratic equation, also called a second degree equation, is a polynomial equation of degree two, written ax² + bx + c = 0 and solved by the quadratic formula.
Quadratic field
A quadratic field is a number field of degree 2 over the rationals, of the form Q(√d) for a unique squarefree integer d; real when d > 0, imaginary when d < 0.
Quadratic form
A quadratic form is a homogeneous polynomial of degree 2 in several variables, equivalent to a symmetric matrix, and first studied systematically by Gauss in his 1801 Disquisitiones Arithmeticae.
Quadratic formula
The quadratic formula, also called Sridharacharya's formula, is an algebraic formula giving the roots of any quadratic equation directly from its coefficients, derived by completing the square.
Quadratic function
A quadratic function, also called a quadratic trinomial or second-order polynomial, is a polynomial of degree two, written f(x) = ax² + bx + c, whose graph is a parabola.
Quadratic reciprocity
Quadratic reciprocity is a number theory theorem relating the solvability of x² ≡ q (mod p) and x² ≡ p (mod q) for distinct odd primes.
Quadratic residue
Quadratic residue, also called quadratic residuosity, describes an integer a for which x² ≡ a (mod m) is solvable; the concept underlies quadratic reciprocity and cryptosystems like Goldwasser–Micali.
Quadratic sieve
The quadratic sieve is a general-purpose integer factorization algorithm invented by Carl Pomerance in 1981, fastest known until 1996 and still used below roughly 100 digits.
Quadric
A quadric, also known as a quadric surface or quadric hypersurface, is a geometric set defined by a degree-two polynomial equation; ellipsoids, hyperboloids, cones, cylinders, and paraboloids are familiar quadric surfaces.
Quadrivium
The quadrivium is the group of four mathematical arts of the classical seven liberal arts: arithmetic, geometry, music, and astronomy. It followed the trivium in medieval university education.
Quandle
A quandle is an algebraic structure, introduced into knot theory independently by David Joyce and Sergei Matveev in 1982, whose axioms capture conjugation in a group.
Quantale
A quantale is a complete lattice equipped with an associative binary operation distributing over arbitrary joins in each variable, introduced by C. J. Mulvey to extend locales non-commutatively for quantum logic.
Quantity
A quantity is an attribute of a phenomenon, body, or substance that can be determined quantitatively, such as length, mass, or electric charge, with values expressed as a number and a unit.
Quantized enveloping algebra
A quantized enveloping algebra is a one-parameter Hopf algebra deformation of the universal enveloping algebra of a semisimple Lie algebra, introduced by Drinfeld and Jimbo around 1985.
Quantum affine algebra
A quantum affine algebra, or affine quantum group, is a Hopf algebra that q-deforms the enveloping algebra of an affine Kac–Moody algebra, introduced by Drinfeld and Jimbo around 1985.
Quantum group
A quantum group is a noncommutative Hopf algebra that deforms the structure of a group or Lie algebra, introduced by Drinfeld and Jimbo in the 1980s.
Quantum plane
The quantum plane, also called the Manin plane, is a noncommutative algebra generated by x and y with xy = qyx, introduced by Yuri Manin in 1988.
Quantum trace
The quantum trace is a categorical generalization of the trace of a linear map, defined in ribbon categories and used to build knot invariants from braid representations.
Quartic function
A quartic function is a polynomial function of degree four, solved generally by Lodovico Ferrari's method in 1540. It is the highest degree solvable by radicals.
Quasigroup
A quasigroup is a set with a binary operation in which division is always possible in both directions, like a group without needing associativity or an identity.