Author: Rishabh Kumar (IIT + ISI | Global Math Mentor)**
Published: October 2025
Category: Complex Numbers | Algebra Meets Geometry
🔹 Introduction
Complex numbers are more than just algebraic expressions — they represent points in a plane.
This geometric representation is called the Argand Diagram.
The Argand plane lets us see complex numbers — turning abstract algebra into geometry.
It’s named after Jean-Robert Argand, who first visualized complex numbers this way in the early 19th century.
🧭 1️⃣ Representing a Complex Number
A complex number z=x+iyz = x + iyz=x+iy can be represented as a point (x, y) on a coordinate plane:
- The x-axis represents the real part (Re(z))
- The y-axis represents the imaginary part (Im(z))
So, z=x+iy↔P(x,y)z = x + iy \quad \leftrightarrow \quad P(x, y)z=x+iy↔P(x,y)
🔹 Example
z=3+4iz = 3 + 4iz=3+4i
On the Argand diagram:
- Move 3 units along the real axis,
- Move 4 units up the imaginary axis,
- Mark point P(3,4)P(3, 4)P(3,4).
The point PPP represents the complex number 3+4i3 + 4i3+4i.
🔹 Geometric Representation
(Illustration: Real axis (horizontal), imaginary axis (vertical), point P(x,y)P(x, y)P(x,y), vector OPOPOP)
Each complex number corresponds to a position vector from the origin OOO to the point P(x,y)P(x, y)P(x,y). OP→=xi+yj\overrightarrow{OP} = x\mathbf{i} + y\mathbf{j}OP=xi+yj
⚡️ 2️⃣ Modulus and Argument
The modulus and argument describe a complex number in terms of length and direction. z=x+iy⇒∣z∣=x2+y2,arg(z)=tan−1 (yx)z = x + iy \Rightarrow |z| = \sqrt{x^2 + y^2}, \quad \arg(z) = \tan^{-1}\!\left(\frac{y}{x}\right)z=x+iy⇒∣z∣=x2+y2,arg(z)=tan−1(xy)
- ∣z∣|z|∣z∣ = distance from origin to point PPP
- arg(z)\arg(z)arg(z) = angle made by line OPOPOP with positive real axis
✅ Together, they give the polar form: z=r(cosθ+isinθ)z = r(\cos\theta + i\sin\theta)z=r(cosθ+isinθ)
where r=∣z∣, θ=arg(z)r = |z|, \ \theta = \arg(z)r=∣z∣, θ=arg(z)
🔹 Example
z=1+i3z = 1 + i\sqrt{3}z=1+i3 r=12+(3)2=2,θ=tan−1 (31)=60°=π3r = \sqrt{1^2 + (\sqrt{3})^2} = 2, \quad \theta = \tan^{-1}\!\left(\frac{\sqrt{3}}{1}\right) = 60° = \frac{\pi}{3}r=12+(3)2=2,θ=tan−1(13)=60°=3π z=2(cosπ3+isinπ3)\boxed{z = 2(\cos\frac{\pi}{3} + i\sin\frac{\pi}{3})}z=2(cos3π+isin3π)
🌀 3️⃣ Operations on Argand Diagram
| Operation | Algebraic | Geometric Effect |
|---|---|---|
| Addition | z1+z2z_1 + z_2z1+z2 | Vector addition (parallelogram rule) |
| Subtraction | z1−z2z_1 – z_2z1−z2 | Vector from z2z_2z2 to z1z_1z1 |
| Multiplication | z1z2=r1r2[cos(θ1+θ2)+isin(θ1+θ2)]z_1z_2 = r_1r_2[\cos(\theta_1+\theta_2)+i\sin(\theta_1+\theta_2)]z1z2=r1r2[cos(θ1+θ2)+isin(θ1+θ2)] | Rotate by θ2\theta_2θ2, scale by r2r_2r2 |
| Division | z1z2=r1r2[cos(θ1−θ2)+isin(θ1−θ2)]\frac{z_1}{z_2} = \frac{r_1}{r_2}[\cos(\theta_1-\theta_2)+i\sin(\theta_1-\theta_2)]z2z1=r2r1[cos(θ1−θ2)+isin(θ1−θ2)] | Rotate by –θ2\theta_2θ2, shrink by 1r2\frac{1}{r_2}r21 |
| Conjugate | zˉ=x−iy\bar{z} = x – iyzˉ=x−iy | Reflection across real axis |
🔹 Example — Addition
Let z1=2+3i, z2=1+iz_1 = 2 + 3i, \ z_2 = 1 + iz1=2+3i, z2=1+i: z1+z2=(2+1)+(3+1)i=3+4iz_1 + z_2 = (2+1) + (3+1)i = 3 + 4iz1+z2=(2+1)+(3+1)i=3+4i
Graphically, z₁ + z₂ = diagonal of parallelogram with sides z1z_1z1 and z2z_2z2.
🔹 Example — Multiplication
Let z1=2(cos30°+isin30°)z_1 = 2(\cos30° + i\sin30°)z1=2(cos30°+isin30°), z2=3(cos45°+isin45°)z_2 = 3(\cos45° + i\sin45°)z2=3(cos45°+isin45°) z1z2=6(cos75°+isin75°)z_1z_2 = 6(\cos75° + i\sin75°)z1z2=6(cos75°+isin75°)
✅ Multiply moduli, add arguments → rotation by 45°, scaling by ×3.
🔺 4️⃣ Loci on Argand Diagram
Many exam problems involve loci of complex numbers satisfying given conditions.
| Condition | Locus Type |
|---|---|
| ( | z |
| ( | z – a |
| arg(z)=θ\arg(z) = \thetaarg(z)=θ | Half-line making angle θ with real axis |
| ℜ(z)=c\Re(z) = cℜ(z)=c | Vertical line x=cx = cx=c |
| ℑ(z)=c\Im(z) = cℑ(z)=c | Horizontal line y=cy = cy=c |
🔹 Example — Circle Locus
Find the locus of points zzz satisfying ∣z−(2+i)∣=3|z – (2 + i)| = 3∣z−(2+i)∣=3.
Center = (2, 1), Radius = 3
→ Circle with center at (2, 1) and radius 3.
🔹 Example — Argument Locus
Find locus of arg(z)=π4\arg(z) = \frac{\pi}{4}arg(z)=4π.
→ Straight line through origin making angle 45° with real axis.
🎯 5️⃣ Geometric Meaning of Complex Operations
- Multiplying by iii rotates a complex number 90° anticlockwise.
- Multiplying by –1 reflects across origin.
- Complex conjugation reflects across real axis.
- Multiplying by real k > 1 scales the vector (stretch).
🔹 Example — Rotation
If z=1+iz = 1 + iz=1+i, then iz=i(1+i)=i−1=−1+iiz = i(1 + i) = i – 1 = -1 + iiz=i(1+i)=i−1=−1+i.
✅ This is a rotation of zzz by 90° anticlockwise about the origin.
🧮 6️⃣ Summary Table
| Concept | Algebraic Form | Geometric Meaning |
|---|---|---|
| Complex number | z=x+iyz = x + iyz=x+iy | Point (x, y) |
| Modulus | ( | z |
| Argument | arg(z)=tan−1(y/x)\arg(z) = \tan^{-1}(y/x)arg(z)=tan−1(y/x) | Angle with real axis |
| Conjugate | zˉ=x−iy\bar{z} = x – iyzˉ=x−iy | Reflection across real axis |
| Multiplication | z1z2z_1z_2z1z2 | Rotation + scaling |
| Division | z1z2\frac{z_1}{z_2}z2z1 | Reverse rotation + scaling |
| ( | z – a | = r ) |
| arg(z)=θ\arg(z) = \thetaarg(z)=θ | Ray / line through origin |
🔹 Common Mistakes
- ❌ Mixing degrees and radians in arguments.
- ❌ Forgetting that argument can be negative.
- ❌ Confusing conjugate reflection (real vs imaginary axis).
- ❌ Ignoring quadrant when finding arg(z)\arg(z)arg(z).
🌟 Why It Matters
The Argand diagram transforms complex numbers into a geometric language —
connecting algebra, trigonometry, and geometry seamlessly.
It’s used to:
- Visualize roots of unity, rotations, transformations
- Solve loci and geometry of modulus/argument problems
- Understand De Moivre’s theorem and Euler’s formula visually
Once you see complex numbers geometrically, you never go back to just algebra.
📘 Learn Beyond Visualization
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In the Mathematics Elevate Mentorship Program, you’ll:
✅ Visualize algebra through geometry,
✅ Solve advanced loci & argument problems,
✅ Build intuition for STEP, IB, and A Level challenges.
🚀 Think complex — geometrically.
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