Difference between revisions of "Forward circle plus"
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Revision as of 20:02, 29 December 2015
Let $h>0$ and $z_1,z_2 \in \mathbb{C}_h$, the Hilger complex plane. Then we define the $\oplus_h$ operation by $$z_1 \oplus_h z_2 = z_1+z_2+z_1 z_2h.$$
Contents
Properties
Theorem: The structure $(\mathbb{C}_h,\oplus_h)$ is an Abelian group.
Proof: █
Theorem
The following formula holds: $$z = \mathrm{Re}_h(z) \oplus_h \mathring{\iota} \mathrm{Im}_h(z),$$ where $\mathrm{Re}_h$ denotes the Hilger real part of $z$, $\mathrm{Im}_h$ denotes the Hilger imaginary part of $z$, $\oplus_h$ denotes the circle plus operation, and $\mathring{\iota}$ denotes the Hilger pure imaginary.
