The decimal \( 0.\overline{8} \) (where the digit 8 repeats infinitely) can be converted into a fraction using the following steps:

### Step 1: Assign a variable to the repeating decimal
Let \( x = 0.\overline{8} \).

### Step 2: Multiply by 10 to move the decimal point
Since the repeating part is a single digit (8), multiply both sides of the equation by 10:
\[
10x = 8.\overline{8}
\]
Now we have:
\[
x = 0.\overline{8}
\quad \text{and} \quad 10x = 8.\overline{8}.
\]

### Step 3: Subtract the equations
Subtract the first equation from the second:
\[
10x - x = 8.\overline{8} - 0.\overline{8}
\]
\[
9x = 8.
\]

### Step 4: Solve for \( x \)
Now solve for \( x \) by dividing both sides by 9:
\[
x = \frac{8}{9}.
\]

Thus, the fractional form of \( 0.\overline{8} \) is \( \frac{8}{9} \).

Would you like any further details or have any questions?

### Related questions:
1. How can I convert other repeating decimals to fractions?
2. How can I express \( 0.\overline{123} \) as a fraction?
3. What is the fractional form of \( 0.\overline{6} \)?
4. How can I convert non-repeating decimals to fractions?
5. How do I simplify fractions?
6. Can all repeating decimals be converted to fractions?
7. How can I prove that a repeating decimal is a rational number?
8. How does the method differ for repeating and non-repeating decimals?

### Tip:
When converting repeating decimals, the number of digits in the repeating block determines the power of 10 used in the multiplication step!

Learn how to convert the repeating decimal 0.⁢8¯ (where the digit 8 repeats infinitely) into a fraction, step-by-step. Detailed explanation with mathematical proofs and related questions answered. Suitable for Grades 7-9.

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