quaternäärisiä

Quaternäärisiä, often referred to as quaternions, are a number system that extends complex numbers. They were first described by the Irish mathematician William Rowan Hamilton in 1843. A quaternion is typically expressed in the form $a + bi + cj + dk$, where $a, b, c, and d$ are real numbers, and $i, j, and k$ are fundamental quaternion units. These units follow specific multiplication rules: $i^2 = j^2 = k^2 = ijk = -1$. Additionally, the products of distinct units are $ij = k$, $ji = -k$, $jk = i$, $kj = -i$, $ki = j$, and $ik = -j$.

The introduction of quaternions was motivated by the desire to extend complex numbers to three dimensions. However, Hamilton discovered that a four-dimensional system was necessary to maintain certain desirable algebraic properties, such as associativity. Quaternions form a non-commutative division algebra over the real numbers, meaning that multiplication is not commutative ($ij \neq ji$) but division is well-defined.

Geometrically, quaternions are powerful tools for representing rotations in three-dimensional space. A unit quaternion can uniquely represent any rotation around an axis. This property makes them extremely useful in computer graphics, robotics, aerospace engineering, and physics, where precise manipulation of 3D orientation is crucial. Compared to other rotation representations like Euler angles or rotation matrices, quaternions often offer advantages in terms of computational efficiency and avoiding issues like gimbal lock. They are also used in abstract algebra and number theory.