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Split-quaternions and pseudo-Pythagorean quintuples. (English)
Math. Comput. Educ. 50, No. 2, 94-105 (2016).
From the text: This article deals with a new number system, the split-quaternions. This is an extension of the concept, familiar to students, of taking the real numbers and using them to build a new number system, the complex numbers. In addition, students are familiar with Pythagorean triples, and this paper deals with several higher-dimensional extensions of them. This material may be appropriate for an undergraduate mathematics major or a college mathematics club project. In a previous paper, it was shown that Gaussian integers (complex numbers with integer components) can be used to generate Pythagorean triples. It was further shown that quaternions (a four-dimensional analog of the complex numbers) can be used to generate Pythagorean quintuples (a five-dimensional analog of Pythagorean triples). It seemed like the next natural step would be to consider the split-quaternions, a number system related to, but different from, the quaternions, and their analogous connection to Pythagorean $n$-tuples, if any. It will be shown that these split-quaternions can be used to generate pseudo-Pythagorean quintuples, which are related to but different from Pythagorean quintuples. The split-quaternions are isomorphic to the real-valued $2\times 2$ matrices under addition and multiplication. The split-quaternions, and that set of matrices, are 4-dimensional algebras with zero-divisors. Split-quaternions also have applications in relativistic physics. After a review of the use of complex numbers to generate Pythagorean triples, we’ll look at the split-quaternions and use them to generate pseudo-Pythagorean quintuples and numbers satisfying Diophantine equations of the forms $a^2+b^2+c^2=d^2+e^2$, $a^2+b^2+c ^2=d^2$, and $a^2+b^2=c^2+d^2$.
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