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Quaternions: Extending Complex Numbers in 3D Space

Learn about quaternions, a number system that extends complex numbers, and their applications in mechanics and mathematical calculations involving three-dimensional rotations.

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Quaternions: Extending Complex Numbers in 3D Space

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  1. In mathematics, the quaternions are a number system that extends the complex numbers. They were first described by Irish mathematician Sir William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. A striking feature of quaternions is that the product of two quaternions is noncommutative, meaning that the product of two quaternions depends on which factor is to the left of the multiplication sign and which factor is to the right.

  2. Hamilton defined a quaternion as the quotient of two directed lines in a three-dimensional space[1] or equivalently as the quotient of two vectors.[2] Quaternions can also be represented as the sum of a scalar and a vector. Quaternions find uses in both theoretical and applied mathematics, in particular for calculations involving three-dimensional rotations such as in three-dimensional computer graphics and epipolar geometry. .

  3. They have been superseded in many applications by vectors and matrices.

  4. Quaternion plaque on Brougham (Broom) Bridge, Dublin, which says: Here as he walked by on the 16th of October 1843Sir William Rowan Hamilton in a flash of genius discovered the fundamental formula for quaternion multiplication i² = j² = k² = i j k = −1 & cut it on a stone of this bridge.

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