Sum of Quotients of Real Numbers

Theorem

$\forall x, w \in \R, y, z \in \R_{\ne 0}: \dfrac x y + \dfrac w z = \dfrac {\paren {x \times z} + \paren {y \times w} } {y \times z}$

Proof

\(\ds \frac x y + \frac w z\)

\(=\)

\(\ds \paren {x \times \frac 1 y} + \paren {w \times \frac 1 z}\)

Definition of Real Division

\(\ds \)

\(=\)

\(\ds \paren {x \times \frac 1 y \times 1} + \paren {1 \times w \times \frac 1 z}\)

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

\(\ds \)

\(=\)

\(\ds \paren {x \times \frac 1 y \times z \times \frac 1 z} + \paren {y \times \frac 1 y \times w \times \frac 1 z}\)

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

\(\ds \)

\(=\)

\(\ds \paren {x \times z \times \frac 1 y \times \frac 1 z} + \paren {y \times w \times \frac 1 y \times \frac 1 z}\)

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

\(\ds \)

\(=\)

\(\ds \paren {\paren {x \times z} \times \paren {\frac 1 y \times \frac 1 z} } + \paren {\paren {y \times w} \times \paren {\frac 1 y \times \frac 1 z} }\)

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

\(\ds \)

\(=\)

\(\ds \paren {\paren {x \times z} + \paren {y \times w} } \times \paren {\frac 1 y \times \frac 1 z}\)

Real Number Axiom $\R \text D$: Distributivity of Multiplication over Addition

\(\ds \)

\(=\)

\(\ds \paren {\paren {x \times z} + \paren {y \times w} } \times \frac 1 {y z}\)

Product of Reciprocals of Real Numbers

\(\ds \)

\(=\)

\(\ds \frac {\paren {x \times z} + \paren {y \times w} } {y \times z}\)

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 $1 \ \text{(o)}$