Solution 1.1:7d

There is, admittedly, a repeating pattern in the decimal expansion

\displaystyle 0\textrm{.}\underline{10}\ \underline{100}\ \underline{1000}\ \underline{10000}\ \underline{100000}\,\ldots

but for it to be a rational number, the decimal expansion must, after a certain decimal place, consist of a fixed combination of digits that repeat themselves indefinitely. There is no such repetition in the decimal expansion given above (the digit groups 10, 100, 1000, 10000, ... increase in size all the time). The number is therefore not rational.