Question

To convert 0.8 - to a rational number, we can write it as a fraction with a numerator and denominator.

Answer 1

To convert the repeating decimal \(0.8\overline{(8)}\) (which represents \(0.888...\)) to a rational number, we can follow these steps:

1. Let \(x = 0.888...\).

2. To eliminate the repeating part, multiply both sides of the equation by 10 (since there is one digit repeating):

\[
10x = 8.888...
\]

3. Now, we have two equations:

\[
x = 0.888...
\]
\[
10x = 8.888...
\]

4. Next, subtract the first equation from the second:

\[
10x - x = 8.888... - 0.888...
\]

This simplifies to:

\[
9x = 8
\]

5. Now, solve for \(x\):

\[
x = \frac{8}{9}
\]

So, the repeating decimal \(0.888...\) can be expressed as the rational number \(\frac{8}{9}\).