Hilger real part
From timescalewiki
Let $h>0$ and let $z \in \mathbb{C}_h$, the Hilger complex plane. The Hilger real part of $z$ is defined by $$\mathrm{Re}_h(z)=\dfrac{|zh+1|-1}{h}.$$
Properties
Theorem: The following inequality holds for $z \in \mathbb{C}_h$: $$-\dfrac{1}{h} < \mathrm{Re}_h(z) < \infty.$$
Proof: █
Limit of Hilger real and imag parts yields classical
Hilger real part oplus Hilger imaginary part equals z
