Plotting complex numbers, converting between rectangular and polar form, multiplying and dividing in polar form, De Moivre's Theorem, and finding nth roots — with worked examples.
Every complex number z = a + bi corresponds to a unique point in the complex plane (also called the Argand plane). The plane looks like a standard coordinate grid, but the two axes represent different things:
Horizontal axis
Real axis
Represents the real part a of the complex number a + bi.
Vertical axis
Imaginary axis
Represents the imaginary part b of the complex number a + bi.
Plotting Examples
3 + 4i
point (3, 4)
−2 + i
point (−2, 1)
5
point (5, 0) — real axis
Why this matters
Treating complex numbers as points (or vectors) in 2D space makes geometric operations — rotation, scaling, finding distance — concrete and visual. The distance from the origin to (a, b) is the modulus r = √(a² + b²), and the angle is the argument θ.
| Rectangular Form | Polar Form | |
|---|---|---|
| Notation | a + bi | r(cos θ + i sin θ) or r·cis θ |
| Components | real part a, imaginary part b | modulus r, argument θ |
| Best for | Addition, subtraction, plotting | Multiplication, division, powers, roots |
| Example | 1 + √3 i | 2(cos 60° + i sin 60°) = 2·cis 60° |
Note: "cis θ" is shorthand for cos θ + i sin θ. Some textbooks use re^(iθ) (Euler form); all three are equivalent.
Modulus
r = √(a² + b²)
Distance from origin to (a, b)
Argument (base formula)
θ = arctan(b/a)
Then adjust for quadrant (see table)
Quadrant Adjustment Table
| Quadrant | Signs (a, b) | Adjustment |
|---|---|---|
| I | a > 0, b > 0 | None |
| II | a < 0, b > 0 | Add 180° (π) |
| III | a < 0, b < 0 | Add 180° (π) |
| IV | a > 0, b < 0 | Add 360° (2π) |
Always verify: r·cos θ should equal a, and r·sin θ should equal b.
Real part
a = r · cos θ
Imaginary part
b = r · sin θ
Example: Convert 4·cis 150° to rectangular
a = 4 · cos 150° = 4 · (−√3/2) = −2√3
b = 4 · sin 150° = 4 · (1/2) = 2
4·cis 150° = −2√3 + 2i
a = −1, b = √3 (Quadrant II: a < 0, b > 0)
Step 1 — Modulus: r = √((−1)² + (√3)²) = √(1 + 3) = √4 = 2
Step 2 — Base angle: arctan(√3 / −1) = arctan(−√3)
Calculator gives −60°. But a < 0, so add 180°: θ = −60° + 180° = 120°
Step 3 — Verify: 2·cos 120° = 2·(−1/2) = −1 ✓ | 2·sin 120° = 2·(√3/2) = √3 ✓
−1 + √3 i = 2·cis 120° = 2(cos 120° + i sin 120°)
a = −√2, b = −√2 (Quadrant III: a < 0, b < 0)
Step 1 — Modulus: r = √((−√2)² + (−√2)²) = √(2 + 2) = √4 = 2
Step 2 — Base angle: arctan(−√2 / −√2) = arctan(1) = 45°
But a < 0, so add 180°: θ = 45° + 180° = 225°
Step 3 — Verify: 2·cos 225° = 2·(−√2/2) = −√2 ✓ | 2·sin 225° = 2·(−√2/2) = −√2 ✓
−√2 − √2 i = 2·cis 225° = 2·cis(5π/4)
a = 2, b = −2 (Quadrant IV: a > 0, b < 0)
Step 1 — Modulus: r = √(2² + (−2)²) = √(4 + 4) = √8 = 2√2
Step 2 — Base angle: arctan(−2 / 2) = arctan(−1) = −45°
a > 0 and b < 0, so add 360°: θ = −45° + 360° = 315°
Step 3 — Verify: 2√2·cos 315° = 2√2·(√2/2) = 2 ✓ | 2√2·sin 315° = 2√2·(−√2/2) = −2 ✓
2 − 2i = 2√2·cis 315° = 2√2·cis(7π/4)
Multiplication
r₁·cis θ₁ × r₂·cis θ₂
= r₁r₂ · cis(θ₁ + θ₂)
Multiply moduli · add arguments
Division
r₁·cis θ₁ ÷ r₂·cis θ₂
= (r₁/r₂) · cis(θ₁ − θ₂)
Divide moduli · subtract arguments
Worked Example — Multiplication
Compute: 3·cis 40° × 2·cis 110°
= (3 × 2) · cis(40° + 110°)
= 6 · cis 150°
Rectangular check: 6·cos 150° + 6i·sin 150° = −3√3 + 3i
Worked Example — Division
Compute: 10·cis 200° ÷ 5·cis 80°
= (10/5) · cis(200° − 80°)
= 2 · cis 120°
Rectangular: 2(cos 120° + i sin 120°) = −1 + √3 i
De Moivre's Theorem
[r(cos θ + i sin θ)]ⁿ = rⁿ(cos nθ + i sin nθ)
(r · cis θ)ⁿ = rⁿ · cis(nθ)
Raise the modulus to the nth power · multiply the argument by n
De Moivre's Theorem is the engine behind computing powers of complex numbers efficiently. Without it, computing (1+i)⁸ would require 7 multiplications with FOIL. With it: convert once, raise r to 8, multiply θ by 8, convert back.
Worked Example: Compute (1 + i)⁸
Step 1 — Convert 1 + i to polar form
r = √(1² + 1²) = √2
θ = arctan(1/1) = 45° (Quadrant I, no adjustment)
1 + i = √2 · cis 45°
Step 2 — Apply De Moivre's with n = 8
(√2 · cis 45°)⁸ = (√2)⁸ · cis(8 × 45°)
= (2^(1/2))⁸ · cis 360°
= 2⁴ · cis 360°
= 16 · cis 360°
Step 3 — Convert back to rectangular
cis 360° = cos 360° + i sin 360° = 1 + 0i = 1
(1 + i)⁸ = 16
Formula — All n distinct nth roots
zₖ = r^(1/n) · [cos((θ + 2πk)/n) + i sin((θ + 2πk)/n)]
for k = 0, 1, 2, ..., n−1
• Every non-zero complex number has exactly n distinct nth roots.
• All n roots lie on a circle of radius r^(1/n).
• The roots are equally spaced by angles of 2π/n (or 360°/n).
Worked Example: Find all 3 cube roots of 8
Step 1 — Write 8 in polar form
8 = 8 + 0i → r = 8, θ = 0°
8 = 8 · cis 0°
Step 2 — Apply the nth root formula with n = 3
zₖ = 8^(1/3) · cis((0° + 360°k)/3) = 2 · cis(120°k)
Step 3 — Compute for k = 0, 1, 2
k = 0: z₀ = 2 · cis 0° = 2
k = 1: z₁ = 2 · cis 120°
= 2(cos 120° + i sin 120°) = 2(−1/2 + i·√3/2)
= −1 + √3 i
k = 2: z₂ = 2 · cis 240°
= 2(cos 240° + i sin 240°) = 2(−1/2 − i·√3/2)
= −1 − √3 i
Verify: All 3 roots cube to 8. The roots form an equilateral triangle on a circle of radius 2, spaced 120° apart.
Tip 1 — Quadrant Check
After computing θ = arctan(b/a), always verify by plugging back in: r·cos θ must equal a and r·sin θ must equal b. This catches quadrant errors before they cost you points.
Tip 2 — nth Roots Count
The nth root formula always produces exactly n roots (k = 0 through n−1). If you're asked for "all square roots", expect 2 answers. "All cube roots": 3 answers. "All 4th roots": 4 answers. Missing roots means you stopped k too early.
Tip 3 — De Moivre's Shortcut
For large powers (n ≥ 3), always use De Moivre's — never FOIL repeatedly. The conversion: find r and θ, raise r to n, multiply θ by n. A 5-step problem becomes 3 lines.
For a complex number z = a + bi, the modulus (also called absolute value) is r = √(a² + b²) — the distance from the origin to the point (a, b) on the complex plane. The argument θ is the angle that the line segment from the origin to (a, b) makes with the positive real axis, measured counterclockwise. To find it, use θ = arctan(b/a) and then adjust for the correct quadrant: if a < 0 add 180° (or π); if a > 0 and b < 0 add 360° (or 2π). Always verify by checking that r·cos θ = a and r·sin θ = b.
Polar form makes multiplication and division straightforward. To multiply z₁ = r₁(cos θ₁ + i sin θ₁) and z₂ = r₂(cos θ₂ + i sin θ₂): multiply the moduli and add the arguments — z₁z₂ = r₁r₂(cos(θ₁+θ₂) + i sin(θ₁+θ₂)). To divide: divide the moduli and subtract the arguments — z₁/z₂ = (r₁/r₂)(cos(θ₁−θ₂) + i sin(θ₁−θ₂)). This is far more efficient than expanding in rectangular form for powers and roots.
Every non-zero complex number z = r(cos θ + i sin θ) has exactly n distinct nth roots. They are given by the formula: zₖ = r^(1/n) · (cos((θ + 2πk)/n) + i sin((θ + 2πk)/n)) for k = 0, 1, 2, ..., n−1. The n roots are equally spaced around a circle of radius r^(1/n) in the complex plane, separated by angles of 2π/n. For example, the 3 cube roots of 8 are 2, 2(cos 120° + i sin 120°) = −1 + i√3, and 2(cos 240° + i sin 240°) = −1 − i√3.
Imaginary unit i, operations, conjugates, and the modulus
Sine, cosine, identities, and the unit circle — foundations for polar form
The polar plane, graphing polar curves, roses, limaçons, and cardioids
Exact values for sin and cos at all standard angles
Interactive problems on polar conversion, De Moivre's Theorem, and nth roots — with step-by-step solutions and private tutoring. Free to try.
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