Equality of Rational Numbers

Theorem

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

The following statements 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

• 1971: Wilfred Kaplan and Donald J. Lewis: Calculus and Linear Algebra ... (previous) ... (next): Introduction: Review of Algebra, Geometry, and Trigonometry: $\text{0-1}$: The Real Numbers