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Main form: normal, adjective. Used both descriptively (adding detail) and restrictively (restricting the subject). In sciences, typically used in a binary sense (something is either normal or is not normal), whereas in the social sciences and daily parlance, typically used to describe position on a wider scale.

Related forms: normality (how normal something is), normalcy (the condition of being normal), normalize/normalise (make normal)

Same roots: norm

Typical use:

Opposite words: abnormal, paranormal, strange

Derived words: conormal, abnormal, subnormal, supernormal, paranormal, contranormal, malnormal, pseudonormal, orthonormal, seminormal, quasinormal

Learn more about normal

Dictionary definitions: Oxford English Dictionary, Merriam Webster, Wiktionary, The Free Dictionary

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The term normal in mathematics is used in the following broad senses:

  • To denote something upright or perpendicular
  • To denote something that is as it should be. In this sense, normal means good or desirable rather than typical

It is typically used as a limiting adjective, and in a binary sense. In other words, an object in a certain class is normal if it satisfies certain conditions, and every object in that class is either normal or not normal. The noun form is normality. Thus, normality can be viewed as a property in various contexts.

In group theory

Normal subgroup: A subgroup of a group that occurs as the kernel of a homomorphism, or equivalently, such that every left coset and right coset are equal.

Related terms: Normality (the property of a subgroup being normal), normal core (the largest normal subgroup contained in a given subgroup), normal closure (the smallest normal subgroup containing a given subgroup), normalizer (the largest subgroup containing a given subgroup, in which it is normal), normal automorphism (an automorphism that restricts to an automorphism on every normal subgroup).

Term variations: subnormal subgroup, abnormal subgroup, quasinormal subgroup, and others. See Groupprops:Category:Variations of normality, Groupprops:Category:Opposites of normality, and Groupprops:Category:Analogues of normality.

Primary subject wiki entry: Groupprops:Normal subgroup. See also Groupprops:Questions:Normal subgroup.

Other subject wiki entries: Diffgeom:Normal subgroup

Coverage in guided tours: Groupprops guided tour for beginners; section not yet prepared.

Also located at: Wikipedia:Normal subgroup, Planetmath:NormalSubgroup, Mathworld:NormalSubgroup, Springer Online Reference Works, Citizendium:Normal subgroup

In topology

Normal space: A topological space is termed normal if all points are closed sets (the assumption), and any two disjoint closed subsets can be separated by disjoint open subsets.

In some definitions, the assumption is skipped.

Related terms: normality (the property of a topological space being normal)

Term variations: Variations of normality offers a list.

Primary subject wiki entry: Topospaces:Normal space

Also located at: Wikipedia:Normal space, Planetmath:NormalSpace, Mathworld:NormalSpace

In linear algebra

Normal form: also called canonical form, is a standard form in which to put a matrix (typically, upto conjugation). Two normal forms commonly used are the Jordan normal form and the rational normal form.

Normal matrix: A matrix over the complex numbers, that commutes with its conjugate-transpose.

No primary subject wiki entry.

Also located at: Wikipedia:Normal matrix, Mathworld:NormalMatrix, Planetmath:NormalMatrix

Normal operator

In field theory

Normal field extension: A field extension is termed normal if the fixed field under the automorphism group of the extension, is precisely the base field.

Primary subject wiki entry: Galois:Normal extension

Also located at: Planetmath:NormalExtension, Wikipedia:Normal extension, Mathworld:NormalExtension

In commutative algebra

Normal domain: An integral domain is termed normal if it is integrally closed in its field of fractions.

Primary subject wiki entry: Commalg:Normal domain

Normal ring: A commutative unital ring is termed normal if it is integrally closed in its total quotient ring.

Primary subject wiki entry: Commalg:Normal ring

In differential geometry

Normal to a curve in the plane: The line perpendicular to the tangent to the curve at a point.

Normal bundle to a submanifold, or immersed manifold: The quotient of the tangent bundle to the whole manifold (restricted to the submanifold), by the tangent bundle to the submanifold, or immersed manifold. When the manifolds are Riemannian, the normal bundle can be viewed as the orthogonal complement to the tangent bundle of the submanifold, in the tangent bundle to the whole manifold.

Vectors in the normal bundle are termed normal vectors.

Main subject wiki entry: Diffgeom:Normal bundle

Also located at: Wikipedia:Normal bundle, Mathworld:NormalBundle, Planetmath:NormalBundle

Normal coordinate system

In category theory

Normal monomorphism: A monomorphism in a preadditive category, or more generally, in a category enriched over pointed sets, that arises as the kernel of an epimorphism.

Primary subject wiki entry: Cattheory:Normal monomorphism

Normal epimorphism: An epimorphism in a preadditive category, or more generally a category enriched over pointed sets, that arises as the cokernel of a monomorphism.

Primary subject wiki entry: Cattheory:Normal epimorphism

Normal category: A preadditive category, or more generally a category enriched over pointed sets, in which every monomorphism is a normal monomorphism.

Primary subject wiki entry: Cattheory:Normal category

In probability/statistics

Normal distribution

In axiomatic set theory

Normal function

Normal measure


Normal number

Computer science

In formal language theory

Normal form (formal language theory)

In database theory

Normal form (database theory)

In game theory

Normal form (game theory)


In classical mechanics

Normal force: The component of contact force between two bodies, that acts in the direction perpendicular to the plane of contact. It manifests a tendency of the bodies to avoid moving into each other, and acts outwards on both bodies.

Primary subject wiki entry: Mech:Normal force

Also located at: Wikipedia:Normal force, Citizendium:Normal force


In organic chemistry nomenclature

Normal alkane: A straight-chain alkane. For instance, normal pentane is the straight-chain pentane. Also written by prefix , for instance, -pentane.

The prefix normal is used more generally for organic compounds, as a prefix denoting a straight-chain structure, with functional groups at one end.

In stoichiometry

Normality of aqueous solution: Number of gram equivalents of active reagent per liter (litre) of solution. For instance, an aqueous solution with 5 gram equivalents per solution is termed .


Normal good: A good for which demand increases with an increase in (real) income (other things remaining the same).

Primary subject wiki entry: Market:Normal good

Also located at Wikipedia:Normal good