Complex Number Multiplication Calculator

Multiply two complex numbers using the FOIL method with detailed step-by-step solutions.

Enter Complex Numbers

First Complex Number (z1 = a + bi)
Second Complex Number (z2 = c + di)

Result

z1 × z2
-5 + 14i
(3 + 2i)(1 + 4i)
z1 3 + 2i
z2 1 + 4i
Real Part -5
Imaginary Part 14i
Magnitude |z| 14.866

Step-by-Step FOIL Solution

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

How to Multiply Complex Numbers

Complex number multiplication follows the same distributive property as multiplying binomials, with the key identity that i2 = -1. Given two complex numbers z1 = a + bi and z2 = c + di, their product is computed using the FOIL method (First, Outer, Inner, Last).

The FOIL Method for Complex Numbers

First Terms

Multiply the real parts of both numbers together.

First = a × c

Outer Terms

Multiply the real part of the first by the imaginary part of the second.

Outer = a × di = adi

Inner Terms

Multiply the imaginary part of the first by the real part of the second.

Inner = bi × c = bci

Last Terms

Multiply the imaginary parts. Since i² = -1, this becomes a real number.

Last = bi × di = bdi² = -bd

The General Formula

Combining all FOIL terms and using i2 = -1:

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

The real part of the product is (ac - bd) and the imaginary part is (ad + bc).

Example

Multiply (3 + 2i) by (1 + 4i):

  1. First: 3 × 1 = 3
  2. Outer: 3 × 4i = 12i
  3. Inner: 2i × 1 = 2i
  4. Last: 2i × 4i = 8i2 = -8
  5. Combine real: 3 + (-8) = -5
  6. Combine imaginary: 12i + 2i = 14i
  7. Result: -5 + 14i

Geometric Interpretation

Multiplying complex numbers has a beautiful geometric meaning. When you multiply two complex numbers, you multiply their magnitudes (distances from the origin) and add their angles (arguments). If z1 has magnitude r1 and angle θ1, and z2 has magnitude r2 and angle θ2, then their product has magnitude r1r2 and angle θ1 + θ2.

Applications

  • Electrical Engineering: AC circuit analysis uses complex impedance where multiplication represents circuit interactions.
  • Signal Processing: Fourier transforms and signal convolution involve complex multiplication.
  • Quantum Mechanics: Quantum states are described by complex amplitudes that are multiplied during operations.
  • Computer Graphics: Rotations in 2D can be performed by multiplying by a unit complex number.
  • Control Theory: Transfer functions use complex number arithmetic for system analysis.