Multiplication by Negative Real Number
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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
| \(\ds x \times \paren {\paren {-y} + y}\) | \(=\) | \(\ds x \times 0\) | Real Number Axiom $\R \text A4$: Inverses for Addition | |||||||||||
| \(\ds \) | \(=\) | \(\ds 0\) | Real Zero is Zero Element | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \paren {x \times \paren {-y} } + \paren {x \times y}\) | \(=\) | \(\ds 0\) | Real Number Axiom $\R \text D$: Distributivity of Multiplication over Addition | ||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \paren {\paren {x \times \paren {-y} } + \paren {x \times y} } + \paren {-\paren {x \times y} }\) | \(=\) | \(\ds 0 + \paren {-\paren {x \times y} }\) | adding $-\paren {x \times y}$ to both sides | ||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \paren {x \times \paren {-y} } + \paren {\paren {x \times y} + \paren {-\paren {x \times y} } }\) | \(=\) | \(\ds 0 + \paren {-\paren {x \times y} }\) | Real Number Axiom $\R \text A1$: Associativity of Addition | ||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \paren {x \times \paren {-y} } + 0\) | \(=\) | \(\ds 0 + \paren {-\paren {x \times y} }\) | Real Number Axiom $\R \text A4$: Inverses for Addition | ||||||||||
| \(\ds \leadsto \ \ \) | \(\ds x \times \paren {-y}\) | \(=\) | \(\ds -\paren {x \times y}\) | Real Number Axiom $\R \text A3$: Identity for Addition |
Similarly:
| \(\ds \paren {x + \paren {-x} } \times y\) | \(=\) | \(\ds 0 \times y\) | Real Number Axiom $\R \text A4$: Inverses for Addition | |||||||||||
| \(\ds \) | \(=\) | \(\ds 0\) | Real Zero is Zero Element | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \paren {x \times y} + \paren {\paren {-x} \times y}\) | \(=\) | \(\ds 0\) | Real Number Axiom $\R \text D$: Distributivity of Multiplication over Addition | ||||||||||
| \(\ds \leadsto \ \ \) | \(\ds -\paren {x \times y} + \paren {\paren {x \times y} + \paren {\paren {-x} \times y} }\) | \(=\) | \(\ds -\left({x \times y}\right) + 0\) | adding $-\paren {x \times y}$ to both sides | ||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \paren {-\paren {x \times y} + \paren {x \times y} } + \paren {\paren {-x} \times y}\) | \(=\) | \(\ds -\paren {x \times y} + 0\) | Real Number Axiom $\R \text A1$: Associativity of Addition | ||||||||||
| \(\ds \leadsto \ \ \) | \(\ds 0 + \paren {\paren {-x} \times y}\) | \(=\) | \(\ds -\paren {x \times y} + 0\) | Real Number Axiom $\R \text A4$: Inverses for Addition | ||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \paren {-x} \times y\) | \(=\) | \(\ds -\paren {x \times y}\) | 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 $1 \ \text{(e)}$