Add, subtract, multiply or divide two complex numbers written in the form a + bi.
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From the article Complex Numbers — Modulus, Argument and Operations.
a = 3, b = 4, both positive → first quadrant.
|z| = √(3² + 4²) = √(9 + 16) = √25 = 5
α = tan⁻¹(4/3) = tan⁻¹(1.3333) = 53.13°, and since it is the first
quadrant, θ = 53.13°.
Polar form: z = 5 (cos 53.13° + i sin 53.13°)
Check: 5 cos 53.13° = 5 × 0.6 = 3 ✓ and 5 sin 53.13° = 5 × 0.8 = 4 ✓
Example 2 — a second-quadrant number: z = −1 + i√3
|z| = √(1 + 3) = √4 = 2
Reference angle α = tan⁻¹(√3 / 1) = 60°. The point is in the second quadrant (a negative,
b positive), so θ = 180° − 60° = 120°.
Polar form: z = 2 (cos 120° + i sin 120°).
Check: 2 cos 120° = 2 × (−0.5) = −1 ✓ and 2 sin 120° = 2 × 0.8660 = 1.732 = √3 ✓. A calculator that returned −60° would have given the point 1 − i√3 — the wrong number entirely.
Worked in full in Complex Numbers — Modulus, Argument and Operations.
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