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.

$\blacksquare$