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 $z_2$.