id: 06617839
dt: j
an: 2016e.00599
au: Sporn, Howard
ti: Split-quaternions and pseudo-Pythagorean quintuples.
so: Math. Comput. Educ. 50, No. 2, 94-105 (2016).
py: 2016
pu: MATYC Journal, Old Bethpage, NY
la: EN
cc: F50 F60 H40 H60
ut: complex numbers; Pythagorean triples; quaternions; hyperbolic unit;
matrices; split-quaternions; pseudo-Pythagorean quintuples;
hyperboloidal numbers; Pythagorean quadruples; pseudo-Pythagorean
quadruples
ci:
li:
ab: 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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