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