## On The Non-Real Nature of x.0 (x in R*): The Set of Null Imaginary Numbers

**Authors:** Saulo Jorge

In this work I axiomatize the result of $x \cdot 0$ ($x\in\real_{\ne 0}$) as a number $\inull(x)$ that has a null real part (denoted as $\Re(\inull(x))=0$) but that is not real. This implies that $y+\Re(\inull(x)) = y$ but $y+\inull(x) = y + x\cdot 0 \not=y$, $y\in\real_{\ne 0}$. From this I define the set of null imaginary numbers $\nullset=\{\inull(x)=x\cdot 0|\forall x\in\real_{\ne 0}\}$
and present its elementary algebra taking the axiom of uniqueness as base (i.e., if $x\ne y\Leftrightarrow \inull(x)\ne \inull(y)$).
Under the condition of existence of $\nullset$ I show that division by zero can be defined without causing inconsistencies in elementary algebra.

**Comments:** 7 Pages.

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### Submission history

[v1] 2018-09-23 23:17:37

[v2] 2018-09-24 10:18:41

[v3] 2018-10-01 16:23:37

[v4] 2018-10-23 08:04:46

[v5] 2019-06-12 14:05:41

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