Eighth grade >F.2 Identify rational and irrational numbers NV0 \[ \begin{array}{l}\text { Is } 4.111 \ldots \text { a rational number? } \\ \text { Submit nos no }\end{array} \]

1. Let \(x = 4.111\ldots\) where the digit \(1\) repeats indefinitely. 2. Write \(x\) as the sum of its integer and fractional parts: \[ x = 4 + 0.111\ldots \] 3. Denote the repeating part as \(y\): \[ y = 0.111\ldots \] 4. Multiply \(y\) by 10 to shift the decimal point: \[ 10y = 1.111\ldots \] 5. Subtract \(y\) from \(10y\): \[ 10y - y = 1.111\ldots - 0.111\ldots \] \[ 9y = 1 \] 6. Solve for \(y\): \[ y = \frac{1}{9} \] 7. Substitute back into \(x\): \[ x = 4 + \frac{1}{9} = \frac{36}{9} + \frac{1}{9} = \frac{37}{9} \] 8. Since \(x\) can be expressed as the fraction \(\frac{37}{9}\), it is a rational number. \[ \boxed{\text{Yes, } 4.111\ldots \text{ is a rational number.}} \]