The quaternions extend the complex numbers and are used in physics and engineering. Division of quaternions by zero is not defined, which limits physical theories and engineering applications. We now introduce transquaternions, which totalise the arithmetical operations of quaternion addition, subtraction, multiplication, and both left and right division. In particular, division of quaternions by zero is allowed.
The transquaternions are homeomorphic to the unit hypersphere or glome, including its interior, together with an isolated point. The 4D interior of the hypersphere is made up of the ordinary quaternions. The 3D surface of the hypersphere is made up of the infinite transquaternions, which are produced by dividing non-zero quaternions by zero. The isolated point, that lies outside the 4D space containing the hypersphere, is the transquaternion nullity, which is produced by dividing zero by zero.
Transquaternions are a separable compact complete metric topological space.
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Copyright (c) 2022 Tiago Soares dos Reis, James A.D.W. Anderson
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