It's rational because of the repeating digits.
We have
<em>x</em> = 2.732<u>343434</u>…
Then
1,000<em>x</em> = 2,732.343434…
and
100,000<em>x</em> = 273,234.343434…
Subtracting these makes the fractional part go away, leaving you with
100,000<em>x</em> - 1,000<em>x</em> = 273,234.343434… - 2,732.343434…
100,000<em>x</em> - 1,000<em>x</em> = 273,234 - 2,732
99,000<em>x</em> = 270,502
<em>x</em> = 270,502/99,000 = 135,251/49,500
which is clearly a rational number.
