Zero is both Positive and Negative
Theorem
The number $0$ (zero) is the only (real) number which is both:
and
Proof
Let $x$ be a real number which is both positive and negative.
Thus:
- $x \in \set {x \in \R: x \ge 0}$
and:
- $x \in \set {x \in \R: x \le 0}$
and so:
- $0 \le x \le 0$
from which:
- $x = 0$
$\blacksquare$
Note
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Move this note, appropriately titled. into an "also defined as" section in a separate page, which can be transcluded into whatever pages define positiveness and negativeness.
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In $\mathsf{Pr} \infty \mathsf{fWiki}$, we include $0$ in both the positive real numbers set and negative real numbers set.
However, many sources consider $0$ to be neither positive nor negative, so this theorem is no longer true if we consider their convention.
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text I$: Algebraic Structures: $\S 1$: The Language of Set Theory