kvaternioon

Kvaternioon is a mathematical concept that extends complex numbers. While complex numbers have a real part and an imaginary part, kvaternioons have a real part and three imaginary parts. They are typically written in the form $a + bi + cj + dk$, where $a, b, c, d$ are real numbers and $i, j, k$ are imaginary units.

The multiplication of these imaginary units follows specific rules: $i^2 = j^2 = k^2 = ijk = -1$. This leads to non-commutative multiplication, meaning the order in which kvaternioons are multiplied matters, unlike with real or complex numbers. For example, $ij = k$ but $ji = -k$.

Kvaternioons were invented by William Rowan Hamilton in 1843. He was searching for a way to extend complex numbers to three dimensions, but he discovered that four dimensions were necessary to maintain certain algebraic properties, particularly the existence of a multiplicative inverse for every non-zero kvaternioon.

Applications of kvaternioons are found in various fields, including computer graphics for representing rotations in 3D space, robotics, and navigation systems. Their ability to concisely and efficiently represent rotations without gimbal lock makes them a valuable tool in these areas. Despite their abstract nature, kvaternioons provide elegant solutions to practical problems in physics and engineering.