Question

Convert 0.888... to a rational number (fraction).

Answer 1

To convert the repeating decimal \( 0.888... \) into a fraction, we can use the following method:

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

Next, multiply both sides of the equation by 10 to shift the decimal point one place to the right:

\[
10x = 8.888...
\]

Now we have two equations:

1. \( x = 0.888... \)
2. \( 10x = 8.888... \)

Next, we can subtract the first equation from the second:

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

This simplifies to:

\[
9x = 8
\]

Now, to solve for \( x \), we divide both sides by 9:

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

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