Rational Numbers - kipawaa/Proof-Tree GitHub Wiki
Statement
Common Definition
The rational numbers are the set $\mathbb{Q} = \{ \frac{a}{b} \mid a \in \mathbb{Z}, b \in \mathbb{Z} \setminus \{0\}\}$.
Explanation
The rational numbers are the set of numbers that can be written as a quotient (division) of two Integers.
Proof(s)
Set-Theoretic Construction
The rational numbers are given by the Equivalence Relation $(a, b) \sim (c, d)$ iff $ad = bc$ for $(a, b), (c, d) \in \mathbb{Z} \times (\mathbb{Z} \setminus \{0\})$.