To convert the infinite decimal expansion \(0.555...\) into a fraction, we can use the following method:
Let \(x\) be equal to the decimal:
\[ x = 0.555... \]
Next, we multiply both sides of the equation by 10:
\[ 10x = 5.555... \]
Now, observe that we can rewrite \(5.555...\) in terms of \(x\):
\[ 10x = 5 + 0.555... = 5 + x \]
Now, we can set up the equation:
\[ 10x = 5 + x \]
To isolate \(x\), subtract \(x\) from both sides:
\[ 10x - x = 5 \]
This simplifies to:
\[ 9x = 5 \]
Now, divide both sides by 9:
\[ x = \frac{5}{9} \]
Thus, the infinite decimal expansion \(0.555...\) can be expressed as the fraction:
\[ \frac{5}{9} \]
1 answer