2013, J. A. Sparenberg, Hydrodynamic Propulsion and Its Optimization, Springer, page 183,
In Section 4.3 weintroduced the imaginary unit with , in the complex space domain. In section 4.4 we introduced the complex unit with , in connection with the time dependency of the shape (4.4.16) of the swimming profile. These different imaginary units have no interaction, we leave unaltered the product .
2014, Dennis G. Zill, Warren S. Wright, Advanced Engineering Mathematics, Ascend Learning (Jones & Bartlett Learning), 5th Edition, page 793,
Wenowsimplysay that is the imaginary unit and define it by the property . Using the imaginary unit, we build a general complex number out of two real numbers.
2021, Bruce Hunt, Locally Mixed Symmetric Spaces, Springer, page 123,
Considernow inaddition to the given, one of the totally imaginary units not contained in , and note that acts transitively on the set of totally imaginary units of a quaternion algebra; as we are assuming , we may in fact assume that with .