The number \(0.989889888988889888889...\) is rational because it can be represented as a decimal that exhibits a repeating pattern. The sequence shows that the digits after the decimal point contain a repeating structure of "98" followed by some variation. Specifically, the number is periodic if it eventually repeats indefinitely.
To determine if a decimal is rational, we consider that a number is rational if it can be expressed as a fraction of two integers. Repeating decimals can be converted into fractions, which demonstrates their rational nature.
While the decimal representation may appear complex due to the repeating digits, it can ultimately be expressed as \(\frac{a}{b}\), where \(a\) and \(b\) are integers, confirming that it is rational.
In summary, the number \(0.989889888988889888889...\) is rational because it can be expressed as a repeating decimal, and all repeating decimals are rational numbers.
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