In this section, focus into the Argand diagram, a crucial tool for visualizing and understanding complex numbers. This diagram not only represents complex numbers graphically but also provides a clear insight into their operations, making it an essential concept for students.

Argand Diagram Basics

1. Argand Diagram: A 2D plane to represent complex numbers, where the horizontal axis (Real Axis) shows the real part and the vertical axis (Imaginary Axis) shows the imaginary part of a complex number.

2. Real Axis: The 'Re' axis. In a complex number $z = a + bi$, 'a' is plotted here.

3. Imaginary Axis: The 'Im' axis. In $z = a + bi$, 'bi' goes here.

4. Point Representation: Each complex number $z = a + bi$ is a unique point $a, b$ on the diagram, useful for visualizing complex numbers as vectors.

5. Modulus and Argument:

• Modulus $|z|$: Distance from the point to the origin.
• Argument $\arg(z)$: Angle from the positive real axis to the line connecting the point and the origin.

Image courtesy of Online Math Learning

Operations on the Argand Diagram

1. Addition: Add complex numbers by adding their real and imaginary parts separately. Geometrically, it's like vector addition: the sum $z_1 + z_2$ is the diagonal of the parallelogram formed by $z_1$ and $z_2$.

2. Subtraction: Subtract by taking the difference of real and imaginary parts. Geometrically, it's the vector from the head of the subtracted number to the other number's head.

3. Multiplication: Multiply by rotating and scaling. Multiply the angle of $z_1$ by the argument of $z_2$ and the modulus of $z_1$ by the modulus of $z_2$. Results in rotating and stretching $z_1's$ vector.

4. Division: Inverse of multiplication. Divide the angle of $z_1$ by the argument of $z_2$ and the modulus of $z_1$ by the modulus of $z_2$. Leads to rotating and shrinking $z_1's$ vector.

5. Conjugation: Conjugate of $z = a + bi$ is $z* = a - bi$. Reflect the point $a, b$ across the real axis to get $a, -b$.

Example Questions

Question 1: Representation of $3 + 4i$ on an Argand Diagram

Solution:

• Understand: $3 + 4i$ has real part 3, imaginary part 4i.
• Plot: Point $3, 4$ on diagram, 3 on real axis, 4 on imaginary axis.
• Modulus: Distance from origin. Calculate as $√3² + 4² = 5$.
• Argument: Angle with positive real axis. Calculate as $\tan^{-1}\left(\frac{4}{3}\right) ≈ 53.13^\circ$.

Question 2: Adding $1 + 2i$ and $3 + 4i$ on the Argand Diagram

Solution:

• Understand: Add real parts (1 and 3) and imaginary parts (2i and 4i) separately.
• Plot: Points (1, 2) and (3, 4) on diagram.
• Parallelogram: Draw using (1, 2) and (3, 4) as vertices.
• Sum: Diagonal from origin to opposite vertex of parallelogram. Algebraically, $4 + 6i$.
• Interpretation: Sum is like vector addition of (1, 2) and (3, 4).