The tessarines are a mathematical idea introduced by James Cockle in 1848. The concept includes both ordinary complex numbers and split-complex numbers. A tessarine t may be described as a 2 x 2 complex matrix In mathematics, the complex numbers are an extension of the real numbers by the inclusion of the imaginary unit , satisfying . ... In mathematics, the split-complex numbers are an extension of the real numbers defined analogously to the complex numbers. ...
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When w and z are both real numbers, then t amounts to a split-complex number. The particular tessarine The text or formatting below is generated by a template which has been proposed for deletion. ...
has the property that its matrix product square is the identity matrix. This property lead Cockle to call the tessarine a "new imaginary in algebra". The significance of the commutative and associativering of all tessarines seems to have been less than the significance of this particular tessarine and the plane it generates beyond the real line. In mathematics, an imaginary number (or purely imaginary number) is a complex number whose square is negative or zero. ... In mathematics, especially abstract algebra, a binary operation * on a set S is commutative if x * y = y * x for all x and y in S. Otherwise * is noncommutative. ... In mathematics, associativity is a property that a binary operation can have. ... A ring is usually anything resembling a circle, or a noise that cycles rapidly. ...
While for Cayley-Dickson constructs, split-complex algebra, and Clifford algebra all non-real bases are anti-commutative, use of a commutativeimaginary base leads to four dimensional Tessarines, eight dimensional biquaternions, and 16 dimensional conic sedenions.
A special case are the bicomplex numbers which are isomorphic to tessarines, conic quaternions (from Musès' hypernumbers), and are also contained in the 'hypercomplex number' definition by Kantor and Solodovnikov.
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