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Therefore, nonzero, non-scalar quaternions, or positive scalar quaternions, have exactly two roots, while 0 has exactly one root (0), and negative scalar quaternions have infinitely many roots, which are the vector quaternions located on {} (), i.e., where the scalar part is zero and the vector part is located on the 2-sphere with radius .
3D visualization of a sphere and a rotation about an Euler axis (^) by an angle of In 3-dimensional space, according to Euler's rotation theorem, any rotation or sequence of rotations of a rigid body or coordinate system about a fixed point is equivalent to a single rotation by a given angle about a fixed axis (called the Euler axis) that runs through the fixed point. [6]
In mathematics, quaternionic analysis is the study of functions with quaternions as the domain and/or range. Such functions can be called functions of a quaternion variable just as functions of a real variable or a complex variable are called.
Spatial rotations in three dimensions can be parametrized using both Euler angles and unit quaternions. This article explains how to convert between the two representations. Actually this simple use of "quaternions" was first presented by Euler some seventy years earlier than Hamilton to solve the problem of magic squares. For this reason the ...
Quaternions continued to be a well-studied mathematical structure in the twentieth century, as the third term in the Cayley–Dickson construction of hypercomplex number systems over the reals, followed by the octonions, the sedenions, the trigintaduonions; they are also a useful tool in number theory, particularly in the study of the ...
This article describes Hamilton's original treatment of quaternions, using his notation and terms. Hamilton's treatment is more geometric than the modern approach, which emphasizes quaternions' algebraic properties. Mathematically, quaternions discussed differ from the modern definition only by the terminology which is used.
It can also be realized as the subgroup of unit quaternions generated by [10] = / and =. The generalized quaternion groups have the property that every abelian subgroup is cyclic. [ 11 ] It can be shown that a finite p -group with this property (every abelian subgroup is cyclic) is either cyclic or a generalized quaternion group as defined ...
The quaternions are a non-commutative extension of the complex numbers which have numerous applications in mathematics, physics, and computer graphics The main article for this category is Quaternion .