Quaternion | Wikipedia audio article
Channel: wikipedia tts
Date: 07/05/2019

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Quaternion


00:03:59 1 History
00:11:47 1.1 Quaternions in physics
00:14:34 2 Definition
00:25:22 2.1 Multiplication of basis elements
00:29:53 2.2 Center
00:30:09 2.3 Hamilton product
00:31:38 2.4 Scalar and vector parts
00:31:55 3 Conjugation, the norm, and reciprocal
00:32:43 3.1 Unit quaternion
00:33:07 4 Algebraic properties
00:33:21 5 Quaternions and the geometry of Rsup3/sup
00:43:26 6 Matrix representations
00:48:55 7 Lagrange’s four square theorem
00:57:46 8 Quaternions as pairs of complex numbers
01:00:33 9 Square roots of &#8722;1
01:03:50 9.1 H as a union of complex planes
01:05:20 9.2 Commutative subrings
01:15:09 10 Functions of a quaternion variable
01:16:50 10.1 Exponential, logarithm, and power functions
01:17:08 10.2 Geodesic norm
01:18:49 11 Three-dimensional and four-dimensional rotation groups
01:19:00 12 Quaternion algebras
01:30:17 13 Quaternions as the even part of Cℓsub3,0/sub(R)
01:32:35 14 Brauer group
01:33:32 15 Quotations
01:34:58 16 See also
01:36:23 17 Notes
01:38:39 18 References
01:39:05 19 External articles and resources
01:40:35 19.1 Books and publications
01:45:25 19.2 Links and monographs
01:45:55 qp) only if they lie in the same complex subplane of H, the profile of H as a union of complex planes arises when one seeks to find all commutative subrings of the quaternion ring. This method of commutative subrings is also used to profile the split-quaternions, which as an algebra over the reals are isomorphic to 2 × 2 real matrices.
01:46:29 Functions of a quaternion variable
01:46:41 Like functions of a complex variable, functions of a quaternion variable suggest useful physical models. For example, the original electric and magnetic fields described by Maxwell were functions of a quaternion variable. Examples of other functions include the extension of the Mandelbrot set and Julia sets into 4 dimensional space.
01:47:17 Exponential, logarithm, and power functions
01:53:44 Geodesic norm
01:55:16 Three-dimensional and four-dimensional rotation groups
01:59:41 Quaternion algebras
02:00:18 a, j2
02:00:30 −ji (so (ij)2
02:01:02 Quaternions as the even part of Cℓ3,0(R)
02:10:39 Brauer group
02:12:29 Quotations
02:17:00 See also



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SUMMARY
=======
In mathematics, the quaternions are a number system that extends the complex numbers. They were first described by Irish mathematician William Rowan Hamilton in 1843  and applied to mechanics in three-dimensional space. A feature of quaternions is that multiplication of two quaternions is noncommutative. Hamilton defined a quaternion as the quotient of two directed lines in a three-dimensional space or equivalently as the quotient of two vectors.Quaternions are generally represented in the form:

  
    
      
        a
        +
        b
         
        
          i
        
        +
        c
         
        
          j
        
        +
        d
         
        
          k
        
      
    
    {\displaystyle a+b\ \mathbf {i} +c\ \mathbf {j} +d\ \mathbf {k} }
  where a, b, c, and d are real numbers, and  i, j, and k are the fundamental quaternion units.
Quaternions find uses in both pure and applied mathematics, in particular for calculations involving three-dimensional rotations such as in three-dimensional computer graphics, computer vision, and crystallographic texture analysis. In practical applications, they can be used alongside other methods, such as Euler angles and rotation matrices, or as an alternative to them, depending on the application.
In modern mathematical language, quaternions form a four-dimensional associative normed division algebra over the real numbers, and therefore also a domain. In fact, the quaternions were the first noncommutative division algebra to be discovered. The algebra of ...