what is as 0.106 with the 06 repeating indefintely as a fraction

To convert \( 0.\overline{106} \) (where "106" repeats indefinitely) into a fraction, follow these steps:

### Step 1: Let \( x = 0.\overline{106} \).
This means:
\[
x = 0.106106106\ldots
\]

### Step 2: Multiply both sides by 1000 to shift the decimal point three places to the right:
\[
1000x = 106.106106106\ldots
\]

### Step 3: Subtract the original equation from this new one:
\[
1000x - x = 106.106106106\ldots - 0.106106106\ldots
\]
\[
999x = 106
\]

### Step 4: Solve for \( x \):
\[
x = \frac{106}{999}
\]

So, \( 0.\overline{106} \) as a fraction is \( \frac{106}{999} \).

This fraction is already in its simplest form because 106 and 999 have no common divisors (106 is prime, and 999 is not divisible by 106).

### Final answer:
\[
0.\overline{106} = \frac{106}{999}
\]

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Would you like further explanation on repeating decimals or fractions? Here are a few related questions:

1. How do you convert other repeating decimals into fractions?
2. Why does multiplying by powers of 10 work to eliminate the decimal?
3. What happens if the repeating part has a different length, e.g., \( 0.\overline{1} \)?
4. Can you simplify fractions like \( \frac{106}{999} \) further? If so, how?
5. How do you handle non-repeating decimals when converting them to fractions?

**Tip:** Always check if the repeating part can be expressed with fewer digits, as it makes the calculation easier.

what is as 0.106 with the 06 repeating indefinitely as a fraction

Learn how to convert the repeating decimal 0.106 (with 06 repeating indefinitely) into a fraction. Follow a step-by-step algebraic method to express it as 106/999. Suitable for students in Grades 7-10 studying fractions and repeating decimals.

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