Reciprocal of Quotient of Real Numbers

Theorem

$\forall x, y \in \R_{\ne 0}: \dfrac 1 {x / y} = \dfrac y x$

Proof

\(\ds \dfrac 1 {x / y}\)

\(=\)

\(\ds \frac 1 {x \times \dfrac 1 y}\)

Definition of Real Division

\(\ds \)

\(=\)

\(\ds 1 \times \frac 1 {x \times \dfrac 1 y}\)

Real Number Axiom $\R \text M3$: Identity Element for Multiplication

\(\ds \)

\(=\)

\(\ds \paren {y \times \frac 1 y} \times \frac 1 {x \times \dfrac 1 y}\)

Real Number Axiom $\R \text M4$: Inverses for Multiplication

\(\ds \)

\(=\)

\(\ds y \times \paren {\frac 1 y \times \frac 1 {x \times \dfrac 1 y} }\)

Real Number Axiom $\R \text M1$: Associativity of Multiplication

\(\ds \)

\(=\)

\(\ds y \times \frac 1 {y \times \paren {x \times \dfrac 1 y} }\)

Product of Reciprocals of Real Numbers

\(\ds \)

\(=\)

\(\ds y \times \frac 1 {x \times \paren {y \times \dfrac 1 y} }\)

Real Number Axiom $\R \text M2$: Commutativity of Multiplication

\(\ds \)

\(=\)

\(\ds y \times \frac 1 {x \times 1}\)

Real Number Axiom $\R \text M4$: Inverses for Multiplication

\(\ds \)

\(=\)

\(\ds y \times \frac 1 x\)

Real Number Axiom $\R \text M3$: Identity Element for Multiplication

\(\ds \)

\(=\)

\(\ds \frac y x\)

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 $1 \ \text{(q)}$