# Zero is both Positive and Negative

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## 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:

- $x = 0$

$\blacksquare$

## Also defined as

In $\mathsf{Pr} \infty \mathsf{fWiki}$, $0$ is considered to be included in both the set of positive real numbers and the set of negative real numbers.

However, many sources consider $0$ to be neither positive nor negative.

Hence under that convention this result is no longer true.

## Sources

- 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): Chapter $\text I$: Algebraic Structures: $\S 1$: The Language of Set Theory