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Quaternions, much like octonions which I will not discuss here yet, can be understood as a sort of extension of complex numbers. This is the reason why I recalled the essential characteristics of the latter on one of the previous pages of the site.
Quaternions (a set traditionally denoted by the letter H) were introduced in 1840 by O. Rodrigues [01] and in 1843 by W. R. Hamilton [02].
There were some precursor elements of data appearing in the study of quaternions; notably (i) their modulus, which is a sum of four squares, and (ii) their purely imaginary part, so practical for describing rotations in a three-dimensional space.
Regarding their modulus, it is worth recalling Euler's four-square identity (May 4, 1748) and the formula on the sum of four squares (Lagrange's theorem proving in 1770 the conjecture of Claude Gaspard Bachet de Mériziac formulated in 1621; nota bene: there is a concrete example here of a recommendation made to future researchers on the "amateur and researcher" page regarding the choice of a thesis topic).
As for the description of rotations in space, it is fair to remember the efforts of Olinde Rodrigues (1795 in Bordeaux – 1851 in Paris) seeking to express the combination of two rotations using trigonometric elements of spherical geometry published in the Annales de Gergonne in 1840, three years before W. R. Hamilton.
These numbers are now proving very useful for describing rotations in space, for orientation, and for satellite guidance [03]. They have recently been integrated into the theory of relativity [04].
The quasi-vectorial representation of complex numbers and its striking analogy with Euclidean geometry suggests an idea:
"Why not imagine the existence of three purely imaginary numbers whose squares equal negative one; one number per axis of a three-dimensional imaginary space?"
1) By deciding to name these purely imaginary numbers: i, j, k or i1, i2, i3;
2) By assuming that quaternions also have a real part generated by the integer 1 with which each of the purely imaginary numbers commutes: 1.i = i.1, 1.j = j.1, 1.k = k.1;
3) By starting from the principle that the multiplication of the generators of the set H is determined by the following multiplication table, traces of which can be found in various references [05]:
. 1 i j k
1 1 i j k
i i -1 k -j
j j -k -1 i
k k j -i -1
Quaternions are based on (1, i, j, k) or, in a more concise notation: (i0, i1, i2, i3).
With that being said, quaternions can be written in various ways, depending on (i) the authors and references consulted, and also (ii) to facilitate the rendering of their properties. The few examples below illustrate the multitude of possible notations:
1) Perhaps the simplest and most well-known [05]:
q = a + b.i + c.j + d.k
2) A variant represents them as quadruplets of real numbers (a, b, c, d) having very specific properties regarding multiplication [03; § 1.1.1].
3) I prefer to use the notation inspired by reference [06], which divides H into a Cartesian product R x ImH, where R represents the commutative field of real numbers, while ImH symbolizes the vector space of purely imaginary quaternions:
q = [a, (h, q)]
Here, a is a real number, h is the vector (i, j, k), q is the vector having (b, c, d) as real components in R^3, and (..., ...) is a dot product.
It is easy to show that if q and q' are two elements of ImH, then:
q.q’ = [0, (h, q)]. [0, (h, q')] = [- (q, q'), (h, q x q’)]
The product of two purely imaginary quaternions has both a real part and an imaginary part.
The square of a purely imaginary quaternion q is purely real and is equal to minus the square of the Euclidean norm of the vector q:
q^2 = - (q, q) = - ||q||^2
The properties of purely imaginary quaternion products suggested an intuition to me regarding certain physical situations, which I detail on the following page.
[01] Rodrigues, O. : des lois qui régissent les déplacements d’un système solide dans l’espace, et de la variation des coordonnées provenant de ces déplacements considérés indépendamment des causes qui peuvent les produire ; Journal de Mathématiques Pures et Appliquées (1840) : pages 380-440. En ligne sur le site non sécurisé <eudml.org/doc/234443> [consulté le 30 juillet 2026].
[02] Hamilton, W. R. : Elements of quaternions ; édité par son fils chez Longmans, Green and Cie, Londres, 1866, 866 pages au total (introduction, table des matières, les trois livres et la liste des œuvres publiées à l'époque par la maison d'édition).
[03] Florian Monteghetti. Quaternions, orientation et mouvement. [Rapport de recherche] ISAE-SUPAERO. 2012. ⟨hal-01618257⟩.
[04] Girard, P. : Quaternions, algèbre de Clifford et physique relativiste, Presses Polytechniques Romandes, 2004.
[05] Bordas encyclopédie, 50/51 mathématiques, © Bordas-Éditeurs, 1972, Paris, 184 pages.
[06] Quaternions réels ; Université Pierre et Marie Curie, année 2005-2006. Agrégation externe de Mathématiques.