# Multiplication by Negative Real Number

## Theorem

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

### Corollary

$\forall x \in \R: \paren {-1} \times x = -x$

## Proof

 $\displaystyle x \times \paren {\paren {-y} + y}$ $=$ $\displaystyle x \times 0$ Real Number Axioms: $\R A 4$: Inverse for Addition $\displaystyle$ $=$ $\displaystyle 0$ Real Zero is Zero Element $\displaystyle \leadsto \ \$ $\displaystyle \paren {x \times \paren {-y} } + \paren {x \times y}$ $=$ $\displaystyle 0$ Real Number Axioms: $\R D$: Distributivity $\displaystyle \leadsto \ \$ $\displaystyle \paren {\paren {x \times \paren {-y} } + \paren {x \times y} } + \paren {-\paren {x \times y} }$ $=$ $\displaystyle 0 + \paren {-\paren {x \times y} }$ adding $-\paren {x \times y}$ to both sides $\displaystyle \leadsto \ \$ $\displaystyle \paren {x \times \paren {-y} } + \paren {\paren {x \times y} + \paren {-\paren {x \times y} } }$ $=$ $\displaystyle 0 + \paren {-\paren {x \times y} }$ Real Number Axioms: $\R A 1$: Associativity $\displaystyle \leadsto \ \$ $\displaystyle \paren {x \times \paren {-y} } + 0$ $=$ $\displaystyle 0 + \paren {-\paren {x \times y} }$ Real Number Axioms: $\R A 4$: Inverse for Addition $\displaystyle \leadsto \ \$ $\displaystyle x \times \paren {-y}$ $=$ $\displaystyle -\paren {x \times y}$ Real Number Axioms: $\R A 3$: Identity for Addition

Similarly:

 $\displaystyle \paren {x + \paren {-x} } \times y$ $=$ $\displaystyle 0 \times y$ Real Number Axioms: $\R A 4$: Inverse for Addition $\displaystyle$ $=$ $\displaystyle 0$ Real Zero is Zero Element $\displaystyle \leadsto \ \$ $\displaystyle \paren {x \times y} + \paren {\paren {-x} \times y}$ $=$ $\displaystyle 0$ Real Number Axioms: $\R D$: Distributivity $\displaystyle \leadsto \ \$ $\displaystyle -\paren {x \times y} + \paren {\paren {x \times y} + \paren {\paren {-x} \times y} }$ $=$ $\displaystyle -\left({x \times y}\right) + 0$ adding $-\left({x \times y}\right)$ to both sides $\displaystyle \leadsto \ \$ $\displaystyle \paren {-\paren {x \times y} + \paren {x \times y} } + \paren {\paren {-x} \times y}$ $=$ $\displaystyle -\paren {x \times y} + 0$ Real Number Axioms: $\R A 1$: Associativity $\displaystyle \leadsto \ \$ $\displaystyle 0 + \paren {\paren {-x} \times y}$ $=$ $\displaystyle -\paren {x \times y} + 0$ Real Number Axioms: $\R A 4$: Inverse for Addition $\displaystyle \leadsto \ \$ $\displaystyle \paren {-x} \times y$ $=$ $\displaystyle -\paren {x \times y}$ Real Number Axioms: $\R A 3$: Identity for Addition

$\blacksquare$