Which of the following is the largest number ?
$0.9, 0.\overline{9}, 0.0\overline{9}, 0.\overline{09}$

Q: Which of the following is the largest number ? $0.9, 0.\overline{9}, 0.0\overline{9}, 0.\overline{09}$

The correct answer is Option 2: $0.\overline{9}$. 1. **Explanation for Option 2: $0.\overline{9}$** - The notation $0.\overline{9}$ represents a repeating decimal where the digit 9 repeats indefinitely. Mathematically, $0.\overline{9}$ is equivalent to 1. This can be shown through various methods, such as algebraic manipulation or understanding limits in calculus. For instance, if we let $x = 0.\overline{9}$, then $10x = 9.\overline{9}$. Subtracting the first equation from the second gives $9x = 9$, leading to $x = 1$. Therefore, $0.\overline{9} = 1$, which is the largest possible value among the given options. 2. **Explanation for why the other options are incorrect:** - **Option 1: 0.9** - The number 0.9 is a finite decimal and is exactly equal to 0.9. It is less than 1, which makes it smaller than $0.\overline{9}$ (which equals 1). - **Option 3: $0.0\overline{9}$** - The notation $0.0\overline{9}$ represents a repeating decimal where the digit 9 repeats indefinitely starting after the first zero. This is equivalent to 0.099999..., which is less than 0.1. Therefore, it is significantly smaller than $0.\overline{9}$. - **Option 4: $0.\overline{09}$** - The notation $0.\overline{09}$ represents a repeating decimal where the sequence "09" repeats indefinitely. This is equivalent to 0.090909..., which is less than 0.1. Hence, it is also smaller than $0.\overline{9}$. In conclusion, Option 2, $0.\overline{9}$, is the largest number among the given options because it is equivalent to 1, which is greater than any of the other values listed.

Question

Accepted Answer

The correct answer is Option 2: $0.\overline{9}$.

1. **Explanation for Option 2: $0.\overline{9}$**
   - The notation $0.\overline{9}$ represents a repeating decimal where the digit 9 repeats indefinitely. Mathematically, $0.\overline{9}$ is equivalent to 1. This can be shown through various methods, such as algebraic manipulation or understanding limits in calculus. For instance, if we let $x = 0.\overline{9}$, then $10x = 9.\overline{9}$. Subtracting the first equation from the second gives $9x = 9$, leading to $x = 1$. Therefore, $0.\overline{9} = 1$, which is the largest possible value among the given options.

2. **Explanation for why the other options are incorrect:**

- **Option 1: 0.9**
     - The number 0.9 is a finite decimal and is exactly equal to 0.9. It is less than 1, which makes it smaller than $0.\overline{9}$ (which equals 1).

- **Option 3: $0.0\overline{9}$**
     - The notation $0.0\overline{9}$ represents a repeating decimal where the digit 9 repeats indefinitely starting after the first zero. This is equivalent to 0.099999..., which is less than 0.1. Therefore, it is significantly smaller than $0.\overline{9}$.

- **Option 4: $0.\overline{09}$**
     - The notation $0.\overline{09}$ represents a repeating decimal where the sequence "09" repeats indefinitely. This is equivalent to 0.090909..., which is less than 0.1. Hence, it is also smaller than $0.\overline{9}$.

In conclusion, Option 2, $0.\overline{9}$, is the largest number among the given options because it is equivalent to 1, which is greater than any of the other values listed.