Show that the decimal \( 3,2 \ddot{8} \) is a rational number

To show that the decimal \( 3,2 \ddot{8} \) is a rational number, we start by recognizing that \( 3,2 \ddot{8} \) can be expressed as \( 3.288888... \). Let's break it down: 1. Let \( x = 3.288888... \). 2. We can separate the decimal part into two parts: \( x = 3.28 + 0.000000...288888... \). 3. The number \( 0.000000...288888... \) can be represented as \( 0.28 + 0.000000...088888... \). 4. Let \( y = 0.288888... \). To convert \( y \) into a fraction, we can manipulate \( y \): - Multiply \( y \) by 10: \( 10y = 2.88888... \). - Subtract \( y \) from \( 10y \): \[ 10y - y = 2.88888... - 0.28888... = 2.6. \] - This simplifies to \( 9y = 2.6 \) or \( y = \frac{2.6}{9} = \frac{26}{90} = \frac{13}{45} \). 5. Therefore, returning to \( x \): \[ x = 3.28 + \frac{13}{45} = \frac{328}{100} + \frac{13}{45}. \] To combine these fractions, find a common denominator (which would be 900): - Convert \( 3.28 \): \( \frac{328}{100} = \frac{328 \times 9}{900} = \frac{2952}{900} \). - Convert \( \frac{13}{45} = \frac{13 \times 20}{900} = \frac{260}{900} \). So, \[ x = \frac{2952 + 260}{900} = \frac{3212}{900}. \] Since \( x \) can be expressed as a fraction with integers in the numerator and denominator, \( 3,2 \ddot{8} \) is a rational number!