Negative of Quotient of Real Numbers

From ProofWiki
Jump to navigation Jump to search

Theorem

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


Proof

\(\displaystyle \frac {-x} y\) \(=\) \(\displaystyle \frac {\paren {-1} \times x} y\) Multiplication by Negative Real Number: Corollary
\(\displaystyle \) \(=\) \(\displaystyle \paren {-1} \times \frac x y\) Product of Real Number with Quotient
\(\displaystyle \) \(=\) \(\displaystyle - \frac x y\) Multiplication by Negative Real Number: Corollary
\(\displaystyle \) \(=\) \(\displaystyle \paren {-1} \times \frac x y\) Multiplication by Negative Real Number: Corollary
\(\displaystyle \) \(=\) \(\displaystyle \paren {-1} \times 1 \times \frac x y\) Real Number Axioms: $\R M3$: Identity
\(\displaystyle \) \(=\) \(\displaystyle \paren {-1} \times \frac {-1} {-1} \times \frac x y\) Real Number Divided by Itself
\(\displaystyle \) \(=\) \(\displaystyle \frac {-1 \times -1} {-1} \times \frac x y\) Product of Real Number with Quotient
\(\displaystyle \) \(=\) \(\displaystyle \frac {-\paren {-1} } {-1} \times \frac x y\) Multiplication by Negative Real Number: Corollary
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 {-1} \times \frac x y\) Negative of Negative Real Number
\(\displaystyle \) \(=\) \(\displaystyle \frac {1 \times x} {\paren {-1} \times y}\) Product of Quotients of Real Numbers
\(\displaystyle \) \(=\) \(\displaystyle \frac {1 \times x} {- y}\) Multiplication by Negative Real Number: Corollary
\(\displaystyle \) \(=\) \(\displaystyle \frac x {- y}\) Real Number Axioms: $\R M3$: Identity

$\blacksquare$


Sources