Product of Reciprocals of Real Numbers

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 $1 \ \text{(m)}$