Zero is both Positive and Negative

Theorem

The number $0$ (zero) is the only (real) number which is both:

a positive (real) number

and

a negative (real) number.

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