`0.bar(001)` is equal to

Options

A. `(1)/(1000)`
B. `(1)/(999)`
C. `(1)/(99)`
D. `(1)/(9)`

Answer

To find the value of \( 0.\overline{001} \), we will follow these steps:

### Step 1: Understand the notation
The notation \( 0.\overline{001} \) means that the digits "001" repeat indefinitely. This can be expressed as:
\[ 0.001001001001\ldots \]

### Step 2: Set up an equation
Let \( x = 0.\overline{001} \). Therefore, we can write:
\[ x = 0.001001001001\ldots \]

### Step 3: Multiply by a power of 10
Since "001" has 3 digits, we multiply both sides of the equation by \( 1000 \) (which is \( 10^3 \)):
\[ 1000x = 1.001001001001\ldots \]

### Step 4: Subtract the original equation from this new equation
Now we subtract the original equation from this new equation:
\[ 1000x - x = 1.001001001001\ldots - 0.001001001001\ldots \]
This simplifies to:
\[ 999x = 1 \]

### Step 5: Solve for \( x \)
Now, we solve for \( x \):
\[ x = \frac{1}{999} \]

### Conclusion
Thus, the value of \( 0.\overline{001} \) is:
\[ 0.\overline{001} = \frac{1}{999} \]