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A quaternion is a number system that extends the complex numbers. It is a four-dimensional structure, initially introduced by Irish mathematician William Rowan Hamilton in 1843. The system is often used in three-dimensional rotation representations, particularly in applications like computer graphics, robotics, and aerospace.

A quaternion can be written as q = 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 their squares are -1 (i² = j² = k² = -1), and their products are non-commutative, meaning that the order of multiplication affects the result. Specifically, ij = k, jk = i, ki = j, and ji = -k, kj = -i, ik = -j.

Quaternions can be used to represent rotations in three-dimensional space. Unlike rotation matrices, quaternions do not suffer from gimbal lock, a problem that can occur with other rotation representations. This makes quaternions particularly useful in applications where smooth and continuous rotations are required.

In addition to their use in rotations, quaternions have applications in various fields of mathematics and physics. They form a non-commutative, associative, and normed division algebra over the real numbers. This means that they have properties similar to those of complex numbers but with additional structure due to their four-dimensional nature.

Quaternions are also used in the study of rotations and orientations in three-dimensional space, as well as in the representation of spatial transformations. They are a powerful tool in many areas of mathematics and engineering, providing a compact and efficient way to represent and manipulate rotations and other transformations.