Show that the decimal \( 3,21 \dot{8} \) is a rational number

To demonstrate that the decimal \( 3.21 \dot{8} \) (which means \( 3.218888...\) with the 8 repeating indefinitely) is a rational number, we can express it as a fraction. Let's denote \( x = 3.21 \dot{8} \). First, we isolate the repeating part by setting up the equations: 1. \( x = 3.218888...\) 2. Multiply both sides by 10 to move the decimal point: \( 10x = 32.18888...\) 3. Now, multiply both sides of the original \( x \) by 1000 (to move beyond the repeating decimal): \( 1000x = 3218.888...\) Next, we can subtract the first equation from the second: \[ 1000x - 10x = 3218.888... - 32.18888... \] This simplifies to: \[ 990x = 3186.7 \] To isolate \( x \): \[ x = \frac{3186.7}{990} \] Convert \( 3186.7 \) to a fraction: \[ 3186.7 = \frac{31867}{10} \] Thus: \[ x = \frac{\frac{31867}{10}}{990} = \frac{31867}{9900} \] Since \( 31867 \) and \( 9900 \) are both integers, \( 3.21 \dot{8} \) can be expressed as a fraction of two integers, which confirms that it is indeed a rational number.