Kvaternionteja

Kvaternionteja, also known as quaternions, are a number system that extends the complex numbers. They were first described by the Irish mathematician William Rowan Hamilton in 1843. A quaternion is defined as 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 satisfy the following multiplication rules: i^2 = j^2 = k^2 = ijk = -1, and ij = k, ji = -k, jk = i, kj = -i, ki = j, and ik = -j. Quaternions have applications in various fields, including computer graphics, robotics, and physics. They are particularly useful in representing rotations in three-dimensional space, as a single quaternion can describe a rotation without the need for trigonometric functions or gimbal lock, which can occur with Euler angles. Quaternions also have a norm, which is the square root of the sum of the squares of their components, and a conjugate, which is obtained by changing the sign of the imaginary parts. The set of all quaternions forms a non-commutative division algebra over the real numbers.