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Write the following infinite decimal expansion as a fraction. You do not need to simplify. Click for Help Video

0.111111...0.111111...0.111111...

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Let \( x = 0.111111\ldots \).

This repeating decimal can be expressed as a fraction. We will use the following method:

  1. Set \( x = 0.111111\ldots \).
  2. Multiply \( x \) by 10: \[ 10x = 1.111111\ldots \]
  3. Now we can subtract the first equation from the second: \[ 10x - x = 1.111111\ldots - 0.111111\ldots \] This simplifies to: \[ 9x = 1 \]
  4. Solving for \( x \) gives: \[ x = \frac{1}{9} \]

Now, if we consider the infinite decimal given in the question, it may represent \( 0.111111... \) as repeated indefinitely. Therefore, the expression simplifies to the same fraction.

Thus, the fraction representation of \( 0.111111... \) is: \[ \frac{1}{9} \]