ThmDex – An index of mathematical definitions, results, and conjectures.

Result R4483 on D282: Set of real numbers

Real number is zero iff equal to its reflection

Formulation 0

Let $x \in \mathbb{R}$ be a D993: Real number.

Then \begin{equation} x = - x \quad \iff \quad x = 0 \end{equation}

Proofs

Proof 0

Let $x \in \mathbb{R}$ be a D993: Real number.

If $x = - x$, then adding $x$ to each side, we have \begin{equation} 2 x = x + x = - x + x = 0 \end{equation} Dividing each side by $2$, we then get $x = 0 / 2 = 0$. Conversely, if $x = 0$, then $x = 0 = - 0 = - x$. $\square$