Sum of Quotients of Real Numbers

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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

\(\displaystyle \frac x y + \frac w z\) \(=\) \(\displaystyle \paren {x \times \frac 1 y} + \paren {w \times \frac 1 z}\) Definition of Real Division
\(\displaystyle \) \(=\) \(\displaystyle \paren {x \times \frac 1 y \times 1} + \paren {1 \times w \times \frac 1 z}\) Real Number Axioms: $\R M 3$: Identity
\(\displaystyle \) \(=\) \(\displaystyle \paren {x \times \frac 1 y \times z \times \frac 1 z} + \paren {y \times \frac 1 y \times w \times \frac 1 z}\) Real Number Axioms: $\R M 4$: Inverses
\(\displaystyle \) \(=\) \(\displaystyle \paren {x \times z \times \frac 1 y \times \frac 1 z} + \paren {y \times w \times \frac 1 y \times \frac 1 z}\) Real Number Axioms: $\R M 2$: Commutativity
\(\displaystyle \) \(=\) \(\displaystyle \paren {\paren {x \times z} \times \paren {\frac 1 y \times \frac 1 z} } + \paren {\paren {y \times w} \times \paren {\frac 1 y \times \frac 1 z} }\) Real Number Axioms: $\R M 1$: Associativity
\(\displaystyle \) \(=\) \(\displaystyle \paren {\paren {x \times z} + \paren {y \times w} } \times \paren {\frac 1 y \times \frac 1 z}\) Real Number Axioms: $\R D$: Distributivity
\(\displaystyle \) \(=\) \(\displaystyle \paren {\paren {x \times z} + \paren {y \times w} } \times \frac 1 {y z}\) Product of Reciprocals of Real Numbers
\(\displaystyle \) \(=\) \(\displaystyle \frac {\paren {x \times z} + \paren {y \times w} } {y \times z}\) Definition of Real Division

$\blacksquare$


Sources