Commensurability is Transitive
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Theorem
Let $a$, $b$, $c$ be three real numbers.
Let $a$ and $b$ be commensurable, and $b$ and $c$ be commensurable.
Then $a$ and $c$ are commensurable.
Proof
From the definition of commensurablility:
- $\dfrac a b, \dfrac b c \in \Q$
where $\Q$ denotes the set of all rational numbers.
From Rational Multiplication is Closed:
- $\dfrac a b \times \dfrac b c \in \Q$
Cancelling $b$, we have:
- $\dfrac a c \in \Q$
Hence the result.
$\blacksquare$