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

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Result R4483 on D282: Set of real numbers
Subresult of R5244: Complex number is zero iff equal to its reflection

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.

$x = - x$ , 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

$x = 0 = - 0 = - x$ .

$\square$