kvaternió

Kvaternió is a mathematical concept that extends complex numbers. A kvaternió is an element of a four-dimensional algebra over the real numbers, typically denoted as $a + bi + cj + dk$, where $a, b, c, d$ are real numbers and $i, j, k$ are basis elements. These basis elements follow specific multiplication rules: $i^2 = j^2 = k^2 = ijk = -1$. From these fundamental rules, other relations can be derived, such as $ij = k$, $ji = -k$, $jk = i$, $kj = -i$, $ki = j$, and $ik = -j$. This non-commutative multiplication is a key characteristic of kvaternió algebra.

Kvaternió were first described by William Rowan Hamilton in 1843. He was looking for a way to extend complex numbers, which are two-dimensional, to a higher dimension that would still possess multiplicative properties useful in geometry. The development of kvaternió was a significant step in abstract algebra.

Kvaternió have found applications in various fields, particularly in computer graphics and robotics for representing rotations in three-dimensional space. Their ability to avoid the gimbal lock problem, which can occur with other rotation representations like Euler angles, makes them advantageous. While conceptually more complex than vectors or matrices, their efficiency in combining rotations and representing orientations has led to their widespread adoption in these areas. The study of kvaternió continues to be relevant in mathematics and its applications.