Which of the following numbers is the greatest of all?
`0.9 ,0.bar(9), 0.0bar(9), 0.bar(09)`

Options

A. 0.9
B. `0.bar(9)`
C. `0.bar(09)`
D. `0.bar(09)`

Answer

To determine which of the given numbers is the greatest among `0.9`, `0.9̅`, `0.0̅9`, and `0.0̅9̅`, we need to analyze each number carefully.

### Step 1: Understand the notation
- `0.9` is simply 0.9.
- `0.9̅` means 0.999999..., which is a repeating decimal where 9 repeats infinitely.
- `0.0̅9` means 0.099999..., which is a repeating decimal where 9 repeats infinitely after the initial 0.0.
- `0.0̅9̅` means 0.099999..., which is the same as `0.0̅9`.

### Step 2: Convert repeating decimals to fractions
- `0.9̅` can be converted to a fraction:
- Let \( x = 0.9̅ \)
- Then, \( 10x = 9.9̅ \)
- Subtracting these gives \( 10x - x = 9.9̅ - 0.9̅ \)
- This simplifies to \( 9x = 9 \), so \( x = 1 \).
- Therefore, \( 0.9̅ = 1 \).

### Step 3: Compare the values
- Now we have:
- `0.9` = 0.9
- `0.9̅` = 1
- `0.0̅9` = 0.099999... = 0.1 (since it approaches 0.1 as the 9s repeat)
- `0.0̅9̅` = 0.099999... = 0.1 (same as above)

### Step 4: List the values for comparison
- `0.9` = 0.9
- `0.9̅` = 1
- `0.0̅9` = 0.1
- `0.0̅9̅` = 0.1

### Step 5: Identify the greatest number
- Among the values:
- `0.9` (0.9)
- `0.9̅` (1)
- `0.0̅9` (0.1)
- `0.0̅9̅` (0.1)

The greatest number is `0.9̅`, which equals 1.

### Conclusion
The greatest number among `0.9`, `0.9̅`, `0.0̅9`, and `0.0̅9̅` is **`0.9̅` (or 1)**.

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