Equality of Rational Numbers
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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