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Multiply complex numbers

Type two complex numbers and get their product, with the real and imaginary part, the modulus and the argument.

Write as a + bi, for example 3 − 2i, −i or 4i.

Fill in the fields; the answer appears here straight away.

How it works

  1. Enter the values

    Type the numbers or the formula. Decimals may use a point or a comma. No idea? Click Fill in an example.

  2. Instant answer

    The answer appears as you type, with the most important intermediate values.

  3. See the working

    Under "How it's worked out" you see the steps, handy for checking your own calculation.

How it works

Multiply out the brackets and use i² = −1: (a + bi)(c + di) = (ac − bd) + (ad + bc)i. For (3 + 2i)(1 − 4i) = 3 − 12i + 2i − 8i² = 3 − 10i + 8 = 11 − 10i.

Geometric meaning

When you multiply, the moduli multiply and the arguments add up. Multiplying by i turns a number a quarter turn (90°) around the origin.

See also divide complex numbers.

Worked examples

  • (1 + i)(1 − i) = 1 − i² = 2
  • i · i = −1
  • (2 + 3i)² = −5 + 12i

Rotation and scaling

Multiplying by a complex number rotates and scales. Multiplying by i is a quarter turn (90°) anticlockwise, since i · (a + bi) = −b + ai. Multiplying by 1 + i rotates by 45° and stretches by √2. Because moduli multiply and arguments add, polar form makes products easy: a number with modulus 2 and angle 30° times one with modulus 3 and angle 60° has modulus 6 and angle 90°.

Where it is used

Multiplication of complex numbers underlies AC circuit analysis, signal processing (the Fourier transform) and fractals such as the Mandelbrot set, which repeats z² + c. To go from polar to Cartesian form, see the polar to Cartesian converter, and for the reverse operation, divide complex numbers.

Frequently asked questions

Why is i² = −1?

It is the definition of i, the number whose square is −1. It is what makes the formula (a + bi)(c + di) = (ac − bd) + (ad + bc)i work.

How do I square a complex number?

Enter it as both numbers. (2 + 3i)² = 4 + 12i + 9i² = −5 + 12i.

What happens when I multiply by the conjugate?

You always get a real, non-negative number: |z|². For 3 + 4i, (3 + 4i)(3 − 4i) = 9 + 16 = 25, which is 5 squared. That is the trick used when dividing.

How can I check the result?

The modulus of the product equals the product of the moduli. For (3 + 2i)(1 − 4i), √13 × √17 = √221 ≈ 14.866, which is the modulus the calculator shows for 11 − 10i.