Algebraic properties

The set H of all quaternions is a agent amplitude over the absolute numbers with ambit 4. (In comparison, the absolute numbers accept ambit 1, the circuitous numbers accept ambit 2, and the octonions accept ambit 8.) The quaternions accept a multiplication that is akin and that distributes over agent addition, but which is not commutative. Therefore the quaternions H are a non-commutative akin algebra over the absolute numbers. Even admitting H contains copies of the circuitous numbers, it is not an akin algebra over the circuitous numbers.

Because it is accessible to bisect quaternions, they anatomy a analysis algebra. This is a anatomy agnate to a acreage except for the commutativity of multiplication. Finite-dimensional akin analysis algebras over the absolute numbers are actual rare. The Frobenius assumption states that there are absolutely three: R, C, and H. The barometer makes the quaternions into a normed algebra, and normed analysis algebras over the reals are aswell actual rare: Hurwitz's assumption says that there are alone four: R, C, H, and O (the octonions). The quaternions are aswell an archetype of a agreement algebra and of a unital Banach algebra.

Because the artefact of any two base vectors is additional or bare addition base vector, the set {±1, ±i, ±j, ±k} forms a accumulation beneath multiplication. This accumulation is alleged the quaternion accumulation and is denoted Q8.17 The absolute accumulation ring of Q8 is a ring RQ8 which is aswell an eight-dimensional agent amplitude over R. It has one base agent for anniversary aspect of Q8. The quaternions are the caliber ring of RQ8 by the ideal generated by the elements 1 + (−1), i + (−i), j + (−j), and k + (−k). Here the aboriginal appellation in anniversary of the differences is one of the base elements 1, i, j, and k, and the additional appellation is one of base elements −1, −i, −j, and −k, not the accretion inverses of 1, i, j, and k.