# Negative of Quotient of Real Numbers

## Theorem

$\forall x \in \R, y \in \R_{\ne 0}: \dfrac {-x} y = -\dfrac x y = \dfrac x {-y}$

## Proof

 $\ds \frac {-x} y$ $=$ $\ds \frac {\paren {-1} \times x} y$ Multiplication by Negative Real Number: Corollary $\ds$ $=$ $\ds \paren {-1} \times \frac x y$ Product of Real Number with Quotient $\ds$ $=$ $\ds -\frac x y$ Multiplication by Negative Real Number: Corollary $\ds$ $=$ $\ds \paren {-1} \times \frac x y$ Multiplication by Negative Real Number: Corollary $\ds$ $=$ $\ds \paren {-1} \times 1 \times \frac x y$ Real Number Axiom $\R \text M3$: Identity Element for Multiplication $\ds$ $=$ $\ds \paren {-1} \times \frac {-1} {-1} \times \frac x y$ Real Number Divided by Itself $\ds$ $=$ $\ds \frac {-1 \times -1} {-1} \times \frac x y$ Product of Real Number with Quotient $\ds$ $=$ $\ds \frac {-\paren {-1} } {-1} \times \frac x y$ Multiplication by Negative Real Number: Corollary $\ds$ $=$ $\ds \frac 1 {-1} \times \frac x y$ Negative of Negative Real Number $\ds$ $=$ $\ds \frac {1 \times x} {\paren {-1} \times y}$ Product of Quotients of Real Numbers $\ds$ $=$ $\ds \frac {1 \times x} {-y}$ Multiplication by Negative Real Number: Corollary $\ds$ $=$ $\ds \frac x {-y}$ Real Number Axiom $\R \text M3$: Identity Element for Multiplication

$\blacksquare$