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.