Equality of Rational Numbers
The following are equivalent:
- $(1): \quad$ The rational numbers $\dfrac a b$ and $\dfrac c d$ are equal.
- $(2): \quad$ The integers $a d$ and $b c$ are equal.
Note that by definition, $\Q$ is the field of quotients of $\Z$.
1 implies 2
Let $\dfrac a b = \dfrac c d$ in $\Q$.
Then $b c = a d$ in $\Q$.
By Canonical Mapping to Field of Quotients is Injective, $b c = a d$ in $\Z$.
2 implies 1
Let $bd = ac$ in $\Z$.
By definition of ring homomorphism, $b c = a d$ in $\Q$.
Thus Let $\dfrac a b = \dfrac c d$ in $\Q$.