Equality of Rational Numbers

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Theorem

Let $a, b, c, d$ be integers, with $b$ and $d$ nonzero.


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.


Proof

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 $\Z$.

$\Box$


2 implies 1

Let $bd = ac$ in $\Z$.

By definition of ring homomorphism, $bc = ad$ in $\Q$.

Thus Let $\dfrac ab = \dfrac cd$ in $\Q$.

$\blacksquare$