Complex Number Multiplication Calculator

Effortlessly calculate the product of two complex numbers.

Enter the real and imaginary parts of two complex numbers in the fields below.

Complex Number 1 (a + bi)

Re(z1)
+
Im(z1)
i

Complex Number 2 (c + di)

Re(z2)
+
Im(z2)
i

Product of Complex Numbers (z1 * z2):

+ i

Result is in the standard form (Real + Imaginary i).

Understanding Complex Number Multiplication

Complex numbers are numbers that extend the real numbers with an imaginary unit 'i', where i² = -1. A complex number is in the form a + bi, where 'a' is the real part and 'b' is the imaginary part.

To multiply two complex numbers, (a + bi) and (c + di), you use the distributive property, similar to multiplying binomials:

  • (a + bi) * (c + di) = a*c + a*di + bi*c + bi*di
  • = ac + adi + bci + bd*i²
  • Since i² = -1, we substitute it: ac + adi + bci - bd
  • Group real and imaginary parts: (ac - bd) + (ad + bc)i

This calculator simplifies this process, providing you with the product of two complex numbers instantly. Use it to check your homework, explore complex number arithmetic, or for any calculations involving complex numbers.