Product of Quotients of Real Numbers
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Theorem
- $\forall x, w \in \R, y, z \in \R_{\ne 0}: \dfrac x y \times \dfrac w z = \dfrac {x \times w} {y \times z}$
Proof
\(\ds \frac x y \times \frac w z\) | \(=\) | \(\ds x \times \frac 1 y \times w \times \frac 1 z\) | Definition of Real Division | |||||||||||
\(\ds \) | \(=\) | \(\ds x \times w \times \frac 1 y \times \frac 1 z\) | Real Number Axiom $\R \text M2$: Commutativity of Multiplication | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {x \times w} \times \paren {\frac 1 y \times \frac 1 z}\) | Real Number Axiom $\R \text M1$: Associativity of Multiplication | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {x \times w} \times \frac 1 {y \times z}\) | Product of Reciprocals of Real Numbers | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {x \times w} {y \times z}\) | Definition of Real Division |
$\blacksquare$
Sources
- 2000: James R. Munkres: Topology (2nd ed.) ... (previous) ... (next): $1$: Set Theory and Logic: $\S 4$: The Integers and the Real Numbers: Exercise $1 \ \text{(n)}$