\begin{tabular}{l} Identify the number as rational or irrational. \ $$ 0.234223422234 \ldots $$ \ \hline\end{tabular}

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1. A number is rational if its decimal representation either terminates or eventually becomes periodic (repeating a fixed block over and over).

2. The given number is  
   $$
   0.234223422234\ldots
   $$
   Examining the digits shows that after the initial block “234”, the next block is “2234”, followed by “22234”, and so on. In other words, the number appears to be
   $$
   0.234,\;2234,\;22234,\;222234,\ldots
   $$
   where the number of $$2$$’s preceding “34” increases by one in each successive segment.

3. Since the repeating portion is not fixed (the block of digits that repeats varies in length), the decimal expansion does not eventually settle into a periodic pattern.

4. Therefore, the decimal expansion does not correspond to that of a rational number.

5. The number is, hence, $$\textbf{irrational}$$.
The number $$0.234223422234\ldots$$ is irrational because its decimal expansion does not repeat in a fixed pattern.

\begin{tabular}{l} Identify the number as rational or irrational. \\ \( 0.234223422234 \ldots \) \\ \hline\end{tabular}

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