Plot the conjegate of the complex number -7-4i on the Argand plane. ]

Answer

To plot the conjugate of the complex number \(-7 - 4i\) on the Argand plane, we will follow these steps:

### Step 1: Identify the complex number and its conjugate
The given complex number is:
\[ z = -7 - 4i \]
The conjugate of a complex number \(z = a + bi\) is given by:
\[ \overline{z} = a - bi \]
For our complex number:
\[ \overline{z} = -7 + 4i \]

### Step 2: Plot the conjugate on the Argand plane
The Argand plane is a two-dimensional plane where the x-axis represents the real part and the y-axis represents the imaginary part of a complex number.

1. **Real part**: The real part of \(\overline{z}\) is \(-7\).
2. **Imaginary part**: The imaginary part of \(\overline{z}\) is \(4\).

### Step 3: Locate the point on the Argand plane
- Start from the origin \((0, 0)\).
- Move 7 units to the left along the x-axis (since the real part is negative).
- Move 4 units up along the y-axis (since the imaginary part is positive).

The point \((-7, 4)\) represents the conjugate of the complex number \(-7 - 4i\) on the Argand plane.

### Step 4: Label the point
Label the point \((-7, 4)\) as \(\overline{z}\).

### Step 5: Calculate the distance from the origin
The distance from the origin to the point \((-7, 4)\) can be calculated using the formula for the modulus of a complex number:
\[ |\overline{z}| = \sqrt{(-7)^2 + (4)^2} = \sqrt{49 + 16} = \sqrt{65} \]

### Final Result
The conjugate of the complex number \(-7 - 4i\) is \(-7 + 4i\), and it is plotted at the point \((-7, 4)\) on the Argand plane.

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