# 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:

$x = 0$

$\blacksquare$

## Note

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.