Kvaternionin

Kvaternionin, also known 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 represented 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: $i^2 = j^2 = k^2 = ijk = -1$. Furthermore, $ij = k$, $ji = -k$, $jk = i$, $kj = -i$, $ki = j$, and $ik = -j$. This non-commutative multiplication is a key characteristic of quaternions.

Quaternions have found significant applications in various fields, particularly in three-dimensional computer graphics and robotics. Their ability to represent rotations in 3D space without the Gimbal lock phenomenon, which can occur with Euler angles, makes them a preferred choice for animation and navigation systems. They are also used in physics, especially in theoretical physics, and in some areas of engineering. The algebraic structure of quaternions forms a division algebra over the real numbers, meaning that every non-zero quaternion has a multiplicative inverse.