transfinite arithmetic, cardinal arithmetic, ordinal arithmetic
prime field, p-adic integer, p-adic rational number, p-adic complex number
arithmetic geometry, function field analogy
A quaternion or Hamilton number is a kind of number similar to the complex numbers but with three instead of one square root of adjoined, satisfying certain relations.
The quaternions form the largest associative normed division algebra, usually denoted after William Rowan Hamilton (since is taken for the rational numbers).
The structure of as an -algebra is given by a basis of the underlying vector space of , equipped with a multiplication table where is the identity element and otherwise uniquely specified by the equations
and extended by -linearity to all of . The norm on is given by
where given an -linear combination , we define the conjugate . A simple calculation yields
whence for , the multiplicative inverse is
In this way is a normed division algebra.
We have canonical left and right module structures on , but as is not commutative, if we want to talk about tensor products of modules, we need to consider bimodules. This also means that ordinary linear algebra as is used over a field is not quite the same when dealing with quaternions. For instance, one needs to distinguish between left and right eigenvalues of matrices in (using the left and right module structures on respectively), and only left eigenvalues relate to the spectrum of the associated linear operator.
Using the conjugation operation one can define an inner product on so that the corresponding orthogonal group is the compact symplectic group.
The automorphism group of the quaternions, as a real algebra, is SO(3), acting canonically on their imaginary part (in generalization of how the product of complex numbers respects the complex conjugation action)
See also at normed division algebra – automorphism
(e.g. Klimov-Zhuravlev, p. 85)
exceptional spinors and real normed division algebras
Originally due to:
William Rowan Hamilton: On Quaternions, or on a new System of Imaginaries in Algebra, Copy of a Letter from Sir William R. Hamilton to John T. Graves, Philosophical Magazine 25 (1844) 489-495 [pdf]
William Rowan Hamilton: On a new Species of Imaginary Quantities connected with a theory of Quaternions, Proceedings of the Royal Irish Academy 2 (1844) 424-434 [pdf]
William Rowan Hamilton: On Quaternions; or on a new System of Imaginaries in Algebra, Philosophical Magazine xxv (1844) - xxxvi (1850) [pdf]
Early review:
Dedicated monographs:
Andrew J. Hanson: Visualizing Quaternions, Series in Interactive 3D Technology, Morgan Kaufman (2006) [ISBN:978-0-12-088400-1, pdf]
Andrew J. Hanson: Visualizing More Quaternions, Morgan Kaufman (2024) [ISBN:978-0-323-99202-2]
João Pedro Morais, Svetlin Georgiev, Wolfgang Sprößig: An Introduction to Quaternions, chapter 1 in: Real Quaternionic Calculus Handbook, Birkhäuser (2014) 1-34 [doi:10.1007/978-3-0348-0622-0_1]
Further monographs:
D. M. Klimov, V. Ph. Zhuravlev, Group-Theoretic Methods in Mechanics and Applied Mathematics, Routledge (2004, 2020) [ISBN:9780367446987]
Ernst Binz, Sonja Pods, Ch 1 in: The geometry of Heisenberg groups — With Applications in Signal Theory, Optics, Quantization, and Field Quantization, Mathematical Surveys and Monographs 151, American Mathematical Society (2008) [ams:surv-151]
Tevian Dray, Corinne Manogue, Section 3.1 of: The Geometry of Octonions, World Scientific (2015) [doi:10.1142/8456]
See also:
Wikipedia, Quaternions
T. Y. Lam, Hamilton’s Quaternions (ps)
Last revised on June 21, 2025 at 17:17:49. See the history of this page for a list of all contributions to it.