Yes, \( 89.777... \) is a rational number. A rational number is defined as any number that can be expressed as the quotient of two integers, where the denominator is not zero.
In this case, \( 89.777... \) can be represented as \( 89.777\overline{7} \), where the \( 7 \) repeats indefinitely. This can be converted into a fraction.
Let \( x = 89.77777...\).
To eliminate the repeating part, we can multiply by 10 (to shift the decimal place one position to the right):
\[ 10x = 897.77777... \]
Now, we can subtract the original \( x \) from this equation:
\[ 10x - x = 897.77777... - 89.77777... \] \[ 9x = 808 \]
Now, solving for \( x \):
\[ x = \frac{808}{9} \]
Since both 808 and 9 are integers and the denominator is not zero, \( 89.777...\) is indeed a rational number.
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