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arXiv · 1705.06822

The Cayley-Dickson Construction in ACL2

John Cowles·Ruben Gamboa
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Abstract

The Cayley-Dickson Construction is a generalization of the familiar construction of the complex numbers from pairs of real numbers. The complex numbers can be viewed as two-dimensional vectors equipped with a multiplication. The construction can be used to construct, not only the two-dimensional Complex Numbers, but also the four-dimensional Quaternions and the eight-dimensional Octonions. Each of these vector spaces has a vector multiplication, v_1*v_2, that satisfies: 1. Each nonzero vector has a multiplicative inverse. 2. For the Euclidean length of a vector |v|, |v_1 * v_2| = |v_1| |v2|. Real numbers can also be viewed as (one-dimensional) vectors with the above two properties. ACL2(r) is used to explore this question: Given a vector space, equipped with a multiplication, satisfying the Euclidean length condition 2, given above. Make pairs of vectors into "new" vectors with a multiplication. When do the newly constructed vectors also satisfy condition 2?

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John Cowles, Ruben Gamboa. 2017-05-03. The Cayley-Dickson Construction in ACL2. https://doi.org/10.4204/eptcs.249.2

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