Kvaternäärisiä

Kvaternäärisiä, or quaternions in English, are a number system that extends complex numbers. They were first described by Irish mathematician William Rowan Hamilton in 1843. Quaternions are often represented in the form a + bi + cj + dk, where a, b, c, and d are real numbers, and i, j, and k are the fundamental quaternion units. These units have the property that i^2 = j^2 = k^2 = ijk = -1.

The multiplication of quaternions is not commutative, meaning that the order of multiplication matters (e.g., ij is not necessarily equal to ji). This non-commutativity is a key characteristic that distinguishes them from complex numbers. Despite this, quaternion multiplication is associative.

Quaternions have found significant applications in various fields, particularly in computer graphics and robotics. They are widely used for representing rotations in three-dimensional space. Compared to other methods like Euler angles, quaternions offer advantages such as avoiding gimbal lock and providing smoother interpolations. They are also used in physics, engineering, and theoretical mathematics. The study of quaternions and their properties is part of abstract algebra.