Is \( 0.040040004 \ldots \) a rational number? Explain your reasoning.

To determine whether the number \( 0.040040004 \ldots \) is a rational number, we first need to analyze its decimal representation. The decimal \( 0.040040004 \ldots \) can be expressed as \( 0.0\overline{0400} \), where the sequence "0400" repeats indefinitely. This indicates that the decimal is not terminating but is repeating. A rational number is defined as a number that can be expressed as the quotient of two integers, \( \frac{p}{q} \), where \( p \) and \( q \) are integers and \( q \neq 0 \). One of the key characteristics of rational numbers is that their decimal representations either terminate or eventually repeat. Since \( 0.040040004 \ldots \) has a repeating decimal part, it fits the criteria for being a rational number. To further confirm this, we can convert the repeating decimal into a fraction. Let \( x = 0.040040004 \ldots \). To eliminate the repeating part, we can multiply \( x \) by \( 10000 \) (since the repeating block "0400" has 4 digits): \[ 10000x = 400.400040004 \ldots \] Now, we can set up the equation: \[ 10000x = 400 + x \] Subtract \( x \) from both sides: \[ 10000x - x = 400 \] This simplifies to: \[ 9999x = 400 \] Now, divide both sides by \( 9999 \): \[ x = \frac{400}{9999} \] Since both 400 and 9999 are integers, \( x \) can be expressed as a fraction of two integers, confirming that \( 0.040040004 \ldots \) is indeed a rational number. Thus, the conclusion is: \[ \text{Yes, } 0.040040004 \ldots \text{ is a rational number.} \]