Quotient of Quotients of Real Numbers
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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 $1 \ \text{(r)}$