Quotient of Quotients of Real Numbers

Theorem

$\forall x \in \R, y, w, z \in \R_{\ne 0}: \dfrac {x / y} {w / z} = \dfrac {x \times z} {y \times w}$


Proof

\(\ds \frac {x / y} {w / z}\) \(=\) \(\ds \frac x y \times \frac 1 {w / z}\) Definition of Real Division
\(\ds \) \(=\) \(\ds \frac x y \times \frac z w\) Reciprocal of Quotient of Real Numbers
\(\ds \) \(=\) \(\ds \dfrac {x \times z} {y \times w}\) Product of Quotients of Real Numbers

$\blacksquare$


Sources

  • 2000: James R. Munkres: Topology (2nd ed.) ... (previous) ... (next): $1$: Set Theory and Logic: $\S 4$: The Integers and the Real Numbers: Exercise $1 \ \text{(r)}$
This article is issued from ProofWiki. The text is available under Creative Commons Attribution-ShareAlike License unless otherwise noted. Additional terms may apply for the media files.