Explore Quaternion on MathWorld

A quaternion is a member of a four-dimensional noncommutative division algebra (i.e., a ring in which every nonzero element has a multiplicative inverse, but multiplication is not necessarily commutative) over the real numbers.

Quaternion is a college-level concept that would be first encountered in an abstract algebra course covering rings and fields.


Commutative: An operation * is said to be commutative if x*y = y*x for all x and y.
Complex Number: A complex number is a number consisting of a real part and an imaginary part. A complex number is an element of the complex plane.
Field: A field is a ring in which every nonzero element has a multiplicative inverse. The real numbers and the complex numbers are both fields.
Ring: In mathematics, a ring is an Abelian group together with a rule for multiplying its elements.
i: i is the symbol used to denote the principal square root of -1, also called the imaginary unit.

Classroom Articles on Rings and Fields

  • Algebra
  • Gaussian Integer
  • Algebraic Number
  • Ideal
  • Finite Field

  • Classroom Articles on Abstract Algebra (Up to College Level)

  • Abelian Group
  • Group Representation
  • Abstract Algebra
  • Group Theory
  • Boolean Algebra
  • Isomorphism
  • Cyclic Group
  • Normal Subgroup
  • Dihedral Group
  • Simple Group
  • Finite Group
  • Subgroup
  • Group
  • Symmetric Group
  • Group Action
  • Symmetry Group