Which of the following is correct ?

Options

A. `2 + 3 i gt 1 + 4 i `
B. ` 6 + 2 i gt 3 + 3i `
C. ` 5 + 8 i gt 5 + 7 i `
D. none of these

Answer

To determine which of the given options is correct regarding the comparison of complex numbers, we need to analyze each option carefully.

### Step-by-Step Solution:

1. **Understanding Complex Numbers**:
A complex number is of the form \( a + bi \), where \( a \) is the real part and \( b \) is the imaginary part. When comparing complex numbers, we cannot directly say one is greater than the other without additional context, such as comparing their magnitudes or real parts.

2. **Analyzing Each Option**:
- **Option A**: \( 2 + 3i \) vs \( 1 + 4i \)
- Real parts: \( 2 \) and \( 1 \) (compare these)
- Imaginary parts: \( 3i \) and \( 4i \) (compare these)
- Since \( 2 > 1 \) but \( 3i < 4i \), we cannot definitively say which complex number is greater.

- **Option B**: \( 6 + 2i \) vs \( 3 + 3i \)
- Real parts: \( 6 \) and \( 3 \) (compare these)
- Imaginary parts: \( 2i \) and \( 3i \) (compare these)
- Since \( 6 > 3 \) but \( 2i < 3i \), we cannot definitively say which complex number is greater.

- **Option C**: \( 5 + 8i \) vs \( 5 + 7i \)
- Real parts: \( 5 \) and \( 5 \) (equal)
- Imaginary parts: \( 8i \) and \( 7i \) (compare these)
- Here, \( 8i > 7i \), so \( 5 + 8i > 5 + 7i \) is true.

- **Option D**: None of these
- This option suggests that none of the previous comparisons are correct.

3. **Conclusion**:
- From the analysis, we found that option C is correct because \( 5 + 8i > 5 + 7i \) based on the comparison of imaginary parts when the real parts are equal.

### Final Answer:
**Option C is correct: \( 5 + 8i > 5 + 7i \)**.