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| Like the sedenions, which they contain as a sub-algebra, trigintaduonions are neither ], ] nor ]. They do preserve the property of being ].<ref name=cawgas/> | Like the sedenions, which they contain as a sub-algebra, trigintaduonions are neither ], ] nor ]. They do preserve the property of being ].<ref name=cawgas/> | ||
| -'''Intro''' | |||
| A trigintaduonion is of the form a<sub>0</sub> + ia<sub>1</sub> + ja<sub>2</sub> + ka<sub>3</sub> + e'a<sub>4</sub> + i'a<sub>5</sub> +...+ k<sup>(7)</sup>a<sub>31</sub> where each an is a real numbers. The symbols i, j, k, e', i', j', k', e'', i'', j'', k'', e''', i''', j''', k''', e<sup>(4)</sup>, i<sup>(4)</sup>, j<sup>(4)</sup>, k<sup>(4)</sup>, e<sup>(5)</sup>, i<sup>(5)</sup>, j<sup>(5)</sup>, k<sup>(5)</sup>, e<sup>(6)</sup>, i<sup>(6)</sup>, j<sup>(6)</sup>, k<sup>(6)</sup>, e<sup>(7)</sup>, i<sup>(7)</sup>, j<sup>(7)</sup>, k<sup>(7)</sup> commute with real numbers and obey the multiplication rules i<sup>2</sup> = j<sup>2</sup> = k<sup>2</sup> = e'<sup>2</sup> = i'<sup>2</sup> =...= k<sup>(7)</sup><sup>2</sup> = -1, ij = k, ie' = i', je' = j', ke' = k', ie'' = i'', je'' = j'', ke'' = k'', ie''' = i''', je''' = j''', and ke''' = k'''. | |||
| In the description of each non-static method, this trigintaduonion is represented by s = a<sub>0</sub> + ia<sub>1</sub> + ja<sub>2</sub> + ka<sub>3</sub> + e'a<sub>4</sub> + i'a<sub>5</sub> +...+ k<sup>(7)</sup>a<sub>31</sub>. a<sub>0</sub> and ia<sub>1</sub> + ja<sub>2</sub> + ka<sub>3</sub> + e'a<sub>4</sub> + i'a<sub>5</sub> +...+ k<sup>(7)</sup>a<sub>31</sub> are the real and imaginary parts of s, respectively. The real numbers an are the coefficients of the trigintaduonion s. The components a<sub>0</sub>, ia<sub>1</sub>, ..., k<sup>(7)</sup>a<sub>31</sub> will be referred to as the 0th, 1st, ..., and 17th terms of the trigintaduonion s, respectively.\ | |||
| Any other trigintaduonion is represented by t = b<sub>0</sub> + ib<sub>1</sub> + jb<sub>2</sub> + kb<sub>3</sub> + e'b<sub>4</sub> + i'b<sub>5</sub> +...+ k<sup>(7)</sup>b<sub>31</sub>. The variables x and n are used to represent real numbers and integers, respectively. | |||
| In reference to methods which must deal with branch cuts, the symbol u refers to any unit imaginary trigintaduonion (real(u) = 0 and |u| = 1). | |||
| Instances of trigintaduonions are immutable.<ref>http://www.ece.uwaterloo.ca/~dwharder/Java/doc/ca/uwaterloo/alumni/dwharder/Numbers/Trigintaduonion.html</ref> | |||
| ==References== | ==References== | ||
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In abstract algebra, the trigintaduonions (from the Latin trigintaduo meaning 32) form a 32-dimensional algebra over the reals. They are constructed from the sedenions by applying the Cayley–Dickson construction.
Like the sedenions, which they contain as a sub-algebra, trigintaduonions are neither commutative, associative nor alternative. They do preserve the property of being power associative.
-Intro A trigintaduonion is of the form a0 + ia1 + ja2 + ka3 + e'a4 + i'a5 +...+ ka31 where each an is a real numbers. The symbols i, j, k, e', i', j', k', e, i, j, k, e, i, j, k, e, i, j, k, e, i, j, k, e, i, j, k, e, i, j, k commute with real numbers and obey the multiplication rules i = j = k = e' = i' =...= k = -1, ij = k, ie' = i', je' = j', ke' = k', ie = i, je = j, ke = k, ie = i, je = j, and ke = k. In the description of each non-static method, this trigintaduonion is represented by s = a0 + ia1 + ja2 + ka3 + e'a4 + i'a5 +...+ ka31. a0 and ia1 + ja2 + ka3 + e'a4 + i'a5 +...+ ka31 are the real and imaginary parts of s, respectively. The real numbers an are the coefficients of the trigintaduonion s. The components a0, ia1, ..., ka31 will be referred to as the 0th, 1st, ..., and 17th terms of the trigintaduonion s, respectively.\ Any other trigintaduonion is represented by t = b0 + ib1 + jb2 + kb3 + e'b4 + i'b5 +...+ kb31. The variables x and n are used to represent real numbers and integers, respectively. In reference to methods which must deal with branch cuts, the symbol u refers to any unit imaginary trigintaduonion (real(u) = 0 and |u| = 1). Instances of trigintaduonions are immutable.
References
- C++ Complex Numbers, Quaternions, Octonions, Sedenions, etc., University of Waterloo
- ^ The Basic Subalgebra Structure of the Cayley-Dickson Algebra of Dimension 32 (Trigintaduonions), Raoul E. Cawgas and co-authors
- http://www.ece.uwaterloo.ca/~dwharder/Java/doc/ca/uwaterloo/alumni/dwharder/Numbers/Trigintaduonion.html
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