If `z_1,z_2` and `z_3,z_4` are two pairs of conjugate complex numbers then `arg(z_1/z_4)+arg(z_2/z_3)=`

Explanation

To solve the problem, we need to find the value of \( \arg\left(\frac{z_1}{z_4}\right) + \arg\left(\frac{z_2}{z_3}\right) \) given that \( z_1, z_2 \) and \( z_3, z_4 \) are pairs of conjugate complex numbers.

### Step-by-step Solution:

1. **Understanding the Conjugate Pairs**:
- Since \( z_1 \) and \( z_2 \) are conjugates, we can write \( z_2 = \overline{z_1} \).
- Similarly, since \( z_3 \) and \( z_4 \) are conjugates, we can write \( z_4 = \overline{z_3} \).

2. **Expressing the Arguments**:
- We need to evaluate \( \arg\left(\frac{z_1}{z_4}\right) + \arg\left(\frac{z_2}{z_3}\right) \).
- Using the property of arguments, we have:
\[ \arg\left(\frac{z_1}{z_4}\right) = \arg(z_1) - \arg(z_4) \]
\[ \arg\left(\frac{z_2}{z_3}\right) = \arg(z_2) - \arg(z_3) \]

3. **Substituting the Conjugates**:
- Substitute \( z_2 = \overline{z_1} \) and \( z_4 = \overline{z_3} \):
\[ \arg(z_2) = \arg(\overline{z_1}) = -\arg(z_1) \]
\[ \arg(z_4) = \arg(\overline{z_3}) = -\arg(z_3) \]

4. **Combining the Arguments**:
- Now substituting back into our expression:
\[ \arg\left(\frac{z_1}{z_4}\right) + \arg\left(\frac{z_2}{z_3}\right) = \left(\arg(z_1) - \arg(z_4)\right) + \left(-\arg(z_1) - \arg(z_3)\right) \]
- This simplifies to:
\[ \arg(z_1) - (-\arg(z_3)) - \arg(z_1) - \arg(z_3) = 0 \]

5. **Final Result**:
- Therefore, the final result is:
\[ \arg\left(\frac{z_1}{z_4}\right) + \arg\left(\frac{z_2}{z_3}\right) = 0 \]

### Conclusion:
Thus, the answer is:
\[ \arg\left(\frac{z_1}{z_4}\right) + \arg\left(\frac{z_2}{z_3}\right) = 0 \]