1. **Problem statement:** Provide two alternative definitions for the set of rational numbers using the idea of ordered pairs. 2. **Definition 1:** Rational numbers can be defined as the set of ordered pairs $(a,b)$ where $a$ and $b$ are integers, and $b \neq 0$. Each pair $(a,b)$ represents the rational number $\frac{a}{b}$. 3. **Important rule:** Two pairs $(a,b)$ and $(c,d)$ represent the same rational number if and only if $ad = bc$. 4. **Definition 2:** Alternatively, rational numbers can be defined as equivalence classes of ordered pairs of integers $(a,b)$ with $b \neq 0$, under the equivalence relation $(a,b) \sim (c,d)$ if $ad = bc$. 5. **Explanation:** This means that instead of considering each pair separately, we group all pairs that represent the same fraction into one class, which corresponds to a unique rational number. 6. **Summary:** Thus, rational numbers are either represented as pairs $(a,b)$ with $b \neq 0$ or as equivalence classes of such pairs under the relation $ad = bc$.