Product of Reciprocals of Real Numbers
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Theorem
- $\forall x, y \in \R_{\ne 0}: \dfrac 1 x \times \dfrac 1 y = \dfrac 1 {x \times y}$
Proof
\(\ds \frac 1 {x \times y} \times \paren {x \times y}\) | \(=\) | \(\ds 1\) | Real Number Axiom $\R \text M4$: Inverses for Multiplication | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \frac 1 {x \times y} \times \paren {x \times y} \times \frac 1 y\) | \(=\) | \(\ds 1 \times \frac 1 y\) | as $y \ne 0$ | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \paren {\frac 1 {x \times y} \times x} \times \paren {y \times \frac 1 y}\) | \(=\) | \(\ds 1 \times \frac 1 y\) | Real Number Axiom $\R \text M1$: Associativity of Multiplication | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \paren {\frac 1 {x \times y} \times x} \times 1\) | \(=\) | \(\ds 1 \times \frac 1 y\) | Real Number Axiom $\R \text M4$: Inverses for Multiplication | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \frac 1 {x \times y} \times x\) | \(=\) | \(\ds \frac 1 y\) | Real Number Axiom $\R \text M3$: Identity Element for Multiplication | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \paren {\frac 1 {x \times y} \times x} \times \frac 1 x\) | \(=\) | \(\ds \frac 1 y \times \frac 1 x\) | as $x \ne 0$ | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \frac 1 {x \times y} \times \paren {x \times \frac 1 x}\) | \(=\) | \(\ds \frac 1 y \times \frac 1 x\) | Real Number Axiom $\R \text M1$: Associativity of Multiplication | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \frac 1 {x \times y} \times 1\) | \(=\) | \(\ds \frac 1 y \times \frac 1 x\) | Real Number Axiom $\R \text M4$: Inverses for Multiplication | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \frac 1 {x \times y}\) | \(=\) | \(\ds \frac 1 y \times \frac 1 x\) | Real Number Axiom $\R \text M3$: Identity Element for Multiplication | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \frac 1 {x \times y}\) | \(=\) | \(\ds \frac 1 x \times \frac 1 y\) | Real Number Axiom $\R \text M2$: Commutativity of 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 $1 \ \text{(m)}$