In binary system the decimal number 0.3 can be expressed as

Options

A. `(0.01001…..)_(2)`
B. `(0.10110……)_(2)`
C. `(0.11001…..)_(2)`
D. `(0.10101…..)_(2)`

Explanation

To convert the decimal number 0.3 into its binary equivalent, we will follow a systematic approach by repeatedly multiplying the decimal fraction by 2 and noting down the integer part of the result. Here’s a step-by-step solution:

### Step-by-Step Solution:

1. **Start with the decimal number**: We have the decimal number 0.3.

2. **Multiply by 2**:
- \( 0.3 \times 2 = 0.6 \)
- The integer part is 0. So, we write down 0.

3. **Repeat the process with the fractional part**:
- Take the fractional part (0.6) and multiply by 2:
- \( 0.6 \times 2 = 1.2 \)
- The integer part is 1. Write down 1.

4. **Continue with the new fractional part**:
- Take the fractional part (0.2) and multiply by 2:
- \( 0.2 \times 2 = 0.4 \)
- The integer part is 0. Write down 0.

5. **Repeat again**:
- Take the fractional part (0.4) and multiply by 2:
- \( 0.4 \times 2 = 0.8 \)
- The integer part is 0. Write down 0.

6. **Continue the multiplication**:
- Take the fractional part (0.8) and multiply by 2:
- \( 0.8 \times 2 = 1.6 \)
- The integer part is 1. Write down 1.

7. **Repeat the process**:
- Take the fractional part (0.6) and multiply by 2:
- \( 0.6 \times 2 = 1.2 \)
- The integer part is 1. Write down 1.

8. **Notice the pattern**:
- We see that we are back to the fractional part 0.2, which we have already calculated. This indicates that the sequence will repeat.

9. **Compile the results**:
- From the steps above, we have collected the digits: 0 (from 0.3), 1 (from 0.6), 0 (from 0.2), 0 (from 0.4), 1 (from 0.8), 1 (from 0.6).
- Thus, the binary representation of 0.3 is approximately \( 0.0100110011... \) (with the sequence "0011" repeating).

### Final Answer:
The decimal number 0.3 can be expressed in binary as \( 0.0100110011... \) (where "0011" repeats).