Negative of Negative Real Number

Theorem

$\forall x \in \R: -\paren {-x} = x$

Proof

\(\ds 0\)

\(=\)

\(\ds \paren {-x} + x\)

Real Number Axiom $\R \text A4$: Inverses for Addition

\(\ds \leadsto \ \ \)

\(\ds -\paren {-x} + 0\)

\(=\)

\(\ds -\paren {-x} + \paren {-x} + x\)

adding $-\paren {-x}$ to both sides

\(\ds \leadsto \ \ \)

\(\ds -\paren {-x} + 0\)

\(=\)

\(\ds \paren {-\paren {-x} + \paren {-x} } + x\)

Real Number Axiom $\R \text A1$: Associativity of Addition

\(\ds \leadsto \ \ \)

\(\ds -\paren {-x} + 0\)

\(=\)

\(\ds 0 + x\)

Real Number Axiom $\R \text A4$: Inverses for Addition

\(\ds \leadsto \ \ \)

\(\ds -\paren {-x}\)

\(=\)

\(\ds x\)

Real Number Axiom $\R \text A3$: Identity for Addition

$\blacksquare$

Sources

• 2000: James R. Munkres: Topology (2nd ed.) ... (previous) ... (next): $1$: Set Theory and Logic: $\S 4$: The Integers and the Real Numbers: Exercise $1 \ \text{(d)}$