A member of a *noncommutative* Division Algebra first invented by
William Rowan Hamilton. The quaternions are sometimes also known as Hypercomplex Numbers and denoted . While the quaternions are not commutative, they are associative.

The quaternions can be represented using complex Matrices

(1) |

(2) |

(3) | |||

(4) | |||

(5) | |||

(6) |

(Note that here, is used to denote the Identity Matrix, not .) The matrices are closely related to the Pauli Spin Matrices , , , combined with the Identity Matrix. From the above definitions, it follows that

(7) | |||

(8) | |||

(9) |

Therefore , , and are three essentially different solutions of the matrix equation

(10) |

In , the basis of the quaternions can be given by

(11) | |||

(12) | |||

(13) | |||

(14) |

The quaternions satisfy the following identities, sometimes known as
Hamilton's Rules,

(15) |

(16) |

(17) |

(18) |

1 | ||||

1 | 1 | |||

The quaternions , , , and form a non-Abelian Group of order eight (with multiplication as the group operation) known as .

The quaternions can be written in the form

(19) |

(20) |

(21) |

(22) |

so the norm is

(23) |

Quaternions can be interpreted as a Scalar plus a Vector by writing

(24) |

(25) |

(26) |

(27) |

A rotation about the Unit Vector by an angle can be computed using the
quaternion

(28) |

(29) |

(30) |

**References**

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Arvo, J. *Graphics Gems II.* New York: Academic Press, pp. 351-354 and 377-380, 1994.

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Beeler, M.; Gosper, R. W.; and Schroeppel, R. *HAKMEM.* Cambridge, MA: MIT
Artificial Intelligence Laboratory, Memo AIM-239, Item 107, Feb. 1972.

Conway, J. H. and Guy, R. K. *The Book of Numbers.* New York: Springer-Verlag, pp. 230-234, 1996.

Crowe, M. J. *A History of Vector Analysis: The Evolution of the Idea of a Vectorial System.* New York: Dover, 1994.

Dickson, L. E. *Algebras and Their Arithmetics.* New York: Dover, 1960.

Du Val, P. *Homographies, Quaternions, and Rotations.* Oxford, England: Oxford University Press, 1964.

Ebbinghaus, H. D.; Hirzebruch, F.; Hermes, H.; Prestel, A; Koecher, M.; Mainzer, M.; and Remmert, R.
*Numbers.* New York: Springer-Verlag, 1990.

Goldstein, H. *Classical Mechanics, 2nd ed.* Reading, MA: Addison-Wesley, p. 151, 1980.

Hamilton, W. R. *Lectures on Quaternions: Containing a Systematic Statement of a New Mathematical Method.* Dublin: Hodges and Smith, 1853.

Hamilton, W. R. *Elements of Quaternions.* London: Longmans, Green, 1866.

Hamilton, W. R. *The Mathematical Papers of Sir William Rowan Hamilton.* Cambridge, England: Cambridge University Press, 1967.

Hardy, A. S. *Elements of Quaternions.* Boston, MA: Ginn, Heath, & Co., 1881.

Hardy, G. H. and Wright, E. M. *An Introduction to the Theory of Numbers, 5th ed.*
Cambridge, England: Clarendon Press, 1965.

Hearn, D. and Baker, M. P. *Computer Graphics: C Version, 2nd ed.* Englewood Cliffs, NJ:
Prentice-Hall, pp. 419-420 and 617-618, 1996.

Joly, C. J. *A Manual of Quaternions. * London: Macmillan, 1905.

Julstrom, B. A. ``Using Real Quaternions to Represent Rotations in Three Dimensions.''
*UMAP Modules in Undergraduate Mathematics and Its Applications, Module 652.* Lexington, MA: COMAP, Inc., 1992.

Kelland, P. and Tait, P. G. *Introduction to Quaternions, 3rd ed.* London: Macmillan, 1904.

Nicholson, W. K. *Introduction to Abstract Algebra.* Boston, MA: PWS-Kent, 1993.

Tait, P. G. *An Elementary Treatise on Quaternions, 3rd ed., enl.* Cambridge, England: Cambridge University Press, 1890.

Tait, P. G. ``Quaternions.'' *Encyclopædia Britannica, 9th ed.* ca. 1886.
ftp://ftp.netcom.com/pub/hb/hbaker/quaternion/tait/Encyc-Brit.ps.gz.

© 1996-9

1999-05-25