Math

Multiply Complex Numbers Calculator

Multiply two complex numbers (a + bi)(c + di).

Calculated Output

Real part 14

Imaginary part (i) -5

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

Step-by-Step Breakdown

1. Values used:

2. a = 2

3. b = 3 i

4. c = 1

5. d = -4 i

7. Real part = 14

8. Imaginary part = -5i

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The math behind it

(a+bi)(c+di) = ac + adi + bci + bdi² = (ac − bd) + (ad + bc)i.

(2 + 3i)(1 − 4i) = 14 − 5i.

Why does the sign change?

Because i² = −1, the product of the imaginary parts becomes a negative real number.