Question
\begin{tabular}{l} Identify the number as rational or irrational. \\ \( 0.234223422234 \ldots \) \\ \hline\end{tabular}
Ask by Mitchell Gough. in the United States
Mar 10,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The number \(0.234223422234\ldots\) is irrational because its decimal expansion does not repeat in a fixed pattern.
Solution
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, \(\textbf{irrational}\).
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Simplify this solution Mind Expander
The number \( 0.234223422234 \ldots \) is a rational number. While it may initially seem complex due to its decimal expansion, it actually has a repeating pattern after the initial "0.234." The sequence “234” repeats infinitely, which is a hallmark of rational numbers; they can always be expressed as a fraction of two integers. In this case, the repeating part allows us to represent \( 0.234223422234 \ldots \) in fractional form, confirming its rational nature. To further illustrate, you can convert it by recognizing the repeating decimals and using algebra to find its fractional equivalent, turning that infinite dilemma into a neat mathematical equation!
