Math

Complex Number Calculator

Add, subtract, multiply or divide two complex numbers.

Real part of z₁

Imag part of z₁ (i)

Real part of z₂

Imag part of z₂ (i)

Operation

Calculated Output

Real part 11

Imaginary part (i) -10

Formula: (a+bi) ⊕ (c+di) per chosen operation

Step-by-Step Breakdown

1. Values used:

2. Real part of z₁ = 3

3. Imag part of z₁ = 2 i

4. Real part of z₂ = 1

5. Imag part of z₂ = -4 i

7. Real part = 11

8. Imaginary part = -10i

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The math behind it

Multiplication: (a+bi)(c+di) = (ac − bd) + (ad + bc)i. Division multiplies by the conjugate of the denominator: result = (a+bi)(c−di) ÷ (c² + d²).

(3 + 2i)(1 − 4i) = (3 + 8) + (−12 + 2)i = 11 − 10i.

What is i?

The imaginary unit, defined by i² = −1.