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 $\dfrac a b$ and $\dfrac c d$ are equal.
$(2): \quad$ The integers $a d$ and $b c$ are equal.


Proof

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 $\Z$.

$\Box$


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 $\Q$.

$\blacksquare$


Sources