kvaterniók

Quaternions are a number system that extends complex numbers. They were introduced by the mathematician William Rowan Hamilton in 1843, who discovered that a four‑dimensional algebra over the real numbers could be defined with elements of the form a + bi + cj + dk, where a, b, c, d are real numbers and i, j, k are imaginary units. The fundamental multiplication rules are i² = j² = k² = ijk = –1 and the units satisfy non‑commutative multiplication such as ij = k, ji = –k.

The algebra of quaternions, denoted ℍ, forms a non‑commutative, associative division ring. Every non‑zero quaternion has an inverse, given by its conjugate over its norm. Quaternions can be represented as 4‑tuples or as 2×2 complex matrices, providing practical computational approaches.

Mathematically, quaternions are useful for representing rotations in three‑dimensional space. A unit quaternion can be used to rotate a vector by a given angle around a specified axis, avoiding singularities such as gimbal lock that can occur with Euler angles. This property makes them valuable in computer graphics, robotics, aerospace, and physics for attitude representation and interpolation (slerp).

Beyond rotations, quaternions appear in the study of spinors, quantum mechanics, and relativity. They also form the basis of higher dimensional number systems such as octonions and are used in constructing Clifford algebras. Qubit‑based quantum computing and the mathematical formulation of quaternionic quantum mechanics explore further applications. The quaternion algebra remains a fundamental object in modern algebra, providing insights into geometry, group theory, and applied mathematics.