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Complex Number Calculator

Calculate with complex numbers (a + bi). Perform basic operations and convert between rectangular and polar forms.

Calculate with Complex Numbers

Real part
+
Imaginary part
i
Real part
+
Imaginary part
i

Complex Number Operations

Addition & Subtraction

Add or subtract the real and imaginary parts separately:

  • (a + bi) + (c + di) = (a+c) + (b+d)i
  • (a + bi) − (c + di) = (a−c) + (b−d)i

Example: (3 + 2i) + (1 + 4i) = 4 + 6i

Multiplication & Division

Multiplication: Use FOIL method

  • (a + bi)(c + di) = (ac−bd) + (ad+bc)i

Division: Multiply by conjugate

  • (a + bi) ÷ (c + di) = [(a + bi)(c − di)] / (c² + d²)

Forms & Properties

Rectangular Form

Standard form: a + bi

  • a = real part
  • b = imaginary part
  • i = √(−1), where i² = −1

Polar Form

Polar form: r(cos θ + i sin θ)

  • r = magnitude = √(a² + b²)
  • θ = angle = arctan(b/a)
  • Also written as: r∠θ or r·e^(iθ)

Examples

  • Addition: (2 + 3i) + (4 + 5i) = 6 + 8i
  • Subtraction: (5 + 7i) − (2 + 3i) = 3 + 4i
  • Multiplication: (1 + 2i) × (3 + 4i) = −5 + 10i
  • Division: (4 + 2i) ÷ (1 + 1i) = 3 − 1i
  • Conjugate: conjugate of (3 + 4i) = 3 − 4i
  • Magnitude: |3 + 4i| = 5

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