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

**Authors:** Saulo Queiroz

In this letter we discuss the inconsistencies of $0/0\cdot x=y$, $x,y\in\real_{\ne 0}$
from the perspective of the zero property multiplication (ZPM) $x\cdot 0 = y\cdot 0$ on $\real$.
We axiomatize $x\cdot 0$ as a number $\inull(x)$ that has a real part $\Re(\inull(x))=0$
but indeed is not real. From this we define the set of null imaginary numbers
$\nullset$ as $\{\inull(x)|\forall x\in\real_{\ne 0}\} \cup \{0\}$.
We present the elementary algebra on $\nullset$ based on the definitions of
uniqueness (i.e., if $x\ne y\Leftrightarrow \inull(x)\ne \inull(y)$) and
the null division (i.e., $\inull(x)/0=x\ne 0$). Also, \emph{under the condition of existence
of $\nullset$}, we show that $0/0=\inull(0)/\inull(0)=1$ does not cause the logic trivialism
of the real mathematic.

**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

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