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
- 2000: James R. Munkres: Topology (2nd ed.) ... (previous) ... (next): $1$: Set Theory and Logic: $\S 4$: The Integers and the Real Numbers: Exercise $1 \ \text{(t)}$