Complex numbers, extending beyond the realm of real numbers, offer a fascinating exploration of mathematical concepts. In this section, focuses on the geometric interpretations of complex number operations, unveiling their effects and representations on the Argand plane.

## Geometric Effects of Conjugation

- Conjugation flips the complex number across the real axis.
- Changes imaginary part's sign, keeps real part the same.

### Example:

Conjugate of $z = a + bi$ is $z^* = a - bi$.

## Addition and Subtraction

- Similar to vector addition/subtraction.

### Example:

Add $z_1 = a + bi$ and $z_2 = c + di$ to form a parallelogram with vertices at $0, z_1, z_2, z_1 + z_2$.

## Multiplication and Division

- Seen as rotations and scalings.

### Example:

Multiplying $z_1$ by $z_2$ rotates $z_1$ by the argument of $z_2$ and scales by the modulus of $z_2$.

## Visual Understanding on Argand Plane

### Argand Plane Basics:

- Represents complex numbers as points.
- Real part on x-axis, imaginary part on y-axis.

### Example:

Multiply $z_1 = 3 + 4i$ and $z_2 = 1 - 2i$ and plot on Argand plane.

**Solution:**

**1. Complex Number Multiplication:**

- Multiply parts of complex numbers like algebraic expressions.
- Use $i^2 = -1$.

**2. Calculate **$z_1 \times z_2$**:**

- Expand: $(3 + 4i)(1 - 2i)$.
- Simplify: $3 - 6i + 4i + 8 = 11 - 2i$.
- Result: $z_1 \times z_2 = 11 - 2i$.

**3. Plot on Argand Plane:**

- Plot $11 - 2i$ at point (11, -2).

**4. Physical Interpretation:**

- Shows rotation and scaling of $z_1$ by $z_2$.