Argand Plane and Polar Form: Explained with Examples

The Argand plane plots z = x + iy as the point (x, y): real part on the horizontal axis, imaginary part on the vertical. Addition becomes vector addition, conjugation is reflection across the real axis, and multiplication gains a geometric meaning through the polar form z = r(cos θ + i sin θ) = re^{iθ}, where r = |z| and θ = arg z.

Polar form matters because multiplication becomes "multiply radii, add angles," turning messy algebra into geometry. Rotations, oscillations, and waves all become transparent, which is why engineers reach for re^{iθ} whenever phases enter a problem.

Key Concepts

• Modulus r: r = √(x² + y²), distance from the origin, always non-negative.
• Argument θ: angle counterclockwise from the positive real axis; the principal argument Arg z lies in (−π, π].
• Polar form: z = r(cos θ + i sin θ) = re^{iθ} via Euler's formula e^{iθ} = cos θ + i sin θ.
• Multiplication: r_1e^{iθ_1}·r_2e^{iθ_2} = r_1r_2e^{i(θ_1+θ_2)}: multiply lengths, add angles.
• Division: r_1e^{iθ_1}/r_2e^{iθ_2} = (r_1/r_2)e^{i(θ_1−θ_2)}: divide lengths, subtract angles.

Formulas

• r = √(x² + y²); θ = atan2(y, x) picks the correct quadrant.
• De Moivre: (re^{iθ})^n = r^ne^{inθ} for integer n; n-th roots divide the angle by n.
• Conjugate of re^{iθ} is re^{−iθ} (negate the angle).
• Back to rectangular: x = r cos θ, y = r sin θ.

Worked Example

Convert z = 1 + i√3 to polar form: r = √(1 + 3) = 2, and tan θ = √3 with x, y > 0 gives θ = π/3, so z = 2e^{iπ/3}. Check: 2cos(π/3) = 1, 2sin(π/3) = √3. Multiply z_1 = √2e^{iπ/4} (= 1 + i) by z_2 = 2e^{iπ/3}: modulus 2√2, angle π/4 + π/3 = 7π/12, giving 2√2e^{i7π/12}. Rectangular check: (1 + i)(1 + i√3) = (1 − √3) + i(1 + √3), with modulus √((1−√3)² + (1+√3)²) = √8 = 2√2, confirming the polar multiplication.

Common Misconceptions

• "The argument is unique." Adding 2πk gives the same point; only the principal value Arg z is pinned to (−π, π].
• "Arguments always add exactly." The sum of principal arguments may leave (−π, π] and must be reduced modulo 2π.
• "Polar form changes the number." It is the same point: 1 + i, √2(cos π/4 + i sin π/4), and √2e^{iπ/4} are one number in three notations.

Real-World Applications

• AC analysis: phasor multiplication and division become angle addition and subtraction.
• Computer graphics: rotating 2D points is a single multiplication by e^{iθ}.
• Wave physics: oscillations written Re(Ae^{i(ωt+φ)}) expose amplitude and phase directly.

The Argand plane makes complex numbers into vectors, and polar form makes multiplication into stretch-and-rotate. Together they replace pages of algebra with a picture.