Negative of Quotient of Real Numbers

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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

$\blacksquare$


Sources

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