The decimal number 0.77 in binary system is:

Options

A. `(-0.1100101)_(2)`
B. `(0.11000101)_(2)`
C. `(0.10100101)_(2)`
D. None of these

Answer

To convert the decimal number 0.77 into binary, we will follow a systematic approach of multiplying by 2 and extracting the integer part. Here’s a step-by-step breakdown of the process:

### Step-by-Step Solution:

1. **Start with the decimal number**:
- We begin with 0.77.

2. **Multiply by 2**:
- \( 0.77 \times 2 = 1.54 \)
- **Integer part**: 1

3. **Take the fractional part and multiply by 2 again**:
- \( 0.54 \times 2 = 1.08 \)
- **Integer part**: 1

4. **Continue with the fractional part**:
- \( 0.08 \times 2 = 0.16 \)
- **Integer part**: 0

5. **Repeat the process**:
- \( 0.16 \times 2 = 0.32 \)
- **Integer part**: 0

6. **Continue multiplying**:
- \( 0.32 \times 2 = 0.64 \)
- **Integer part**: 0

7. **Keep going**:
- \( 0.64 \times 2 = 1.28 \)
- **Integer part**: 1

8. **Next multiplication**:
- \( 0.28 \times 2 = 0.56 \)
- **Integer part**: 0

9. **Continue multiplying**:
- \( 0.56 \times 2 = 1.12 \)
- **Integer part**: 1

10. **Final multiplication**:
- \( 0.12 \times 2 = 0.24 \)
- **Integer part**: 0

11. **Final multiplication**:
- \( 0.24 \times 2 = 0.48 \)
- **Integer part**: 0

12. **Final multiplication**:
- \( 0.48 \times 2 = 0.96 \)
- **Integer part**: 0

13. **Final multiplication**:
- \( 0.96 \times 2 = 1.92 \)
- **Integer part**: 1

14. **Final multiplication**:
- \( 0.92 \times 2 = 1.84 \)
- **Integer part**: 1

15. **Final multiplication**:
- \( 0.84 \times 2 = 1.68 \)
- **Integer part**: 1

16. **Final multiplication**:
- \( 0.68 \times 2 = 1.36 \)
- **Integer part**: 1

17. **Final multiplication**:
- \( 0.36 \times 2 = 0.72 \)
- **Integer part**: 0

18. **Final multiplication**:
- \( 0.72 \times 2 = 1.44 \)
- **Integer part**: 1

19. **Final multiplication**:
- \( 0.44 \times 2 = 0.88 \)
- **Integer part**: 0

20. **Final multiplication**:
- \( 0.88 \times 2 = 1.76 \)
- **Integer part**: 1

21. **Final multiplication**:
- \( 0.76 \times 2 = 1.52 \)
- **Integer part**: 1

22. **Final multiplication**:
- \( 0.52 \times 2 = 1.04 \)
- **Integer part**: 1

23. **Final multiplication**:
- \( 0.04 \times 2 = 0.08 \)
- **Integer part**: 0

### Collecting the Integer Parts:
Now, we collect all the integer parts obtained from each multiplication:

- The integer parts in order are: 1, 1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 0.

### Final Binary Representation:
Thus, the binary representation of the decimal number 0.77 is approximately:
**0.1100011101011110...** (and it continues)