The conjugate of the complex number `(a+ib)` is :

Options

A. `-a-ib`
B. `-a+ib`
C. `a-ib`
D. None of these

Answer

To find the conjugate of the complex number \( z = a + ib \), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Definition of Conjugate**:
The conjugate of a complex number is defined as the reflection of that number across the real axis in the complex plane.

2. **Identify the Components of the Complex Number**:
In the complex number \( z = a + ib \), \( a \) is the real part and \( b \) is the imaginary part.

3. **Visualize the Complex Plane**:
In the complex plane, the real part \( a \) is plotted along the horizontal axis (real axis), and the imaginary part \( b \) is plotted along the vertical axis (imaginary axis).

4. **Reflect Across the Real Axis**:
When reflecting \( z = a + ib \) across the real axis, the real part \( a \) remains unchanged, while the imaginary part \( b \) changes its sign. Thus, the imaginary part becomes \( -b \).

5. **Write the Conjugate**:
Therefore, the conjugate of \( z \) is given by:
\[ z^* = a - ib \]

6. **Conclusion**:
The conjugate of the complex number \( a + ib \) is \( a - ib \).

### Final Answer:
The conjugate of the complex number \( a + ib \) is \( a - ib \).

---