Multiply Complex Numbers Calculator

Complex numbers can be represented in two primary formats: rectangular form (z = a + bi) and polar form (z = r∠θ). When working in rectangular form, multiplication follows standard algebraic expansion combined with the fundamental rule that the imaginary unit squared equals negative one (i² = -1). For two numbers, (a₁ + b₁i) and (a₂ + b₂i), the product is calculated as:

Real Part: (a₁a₂ - b₁b₂)

Imaginary Part: (a₁b₂ + b₁a₂)

When working in polar form (magnitude r and phase angle θ), multiplication becomes notably streamlined: you multiply the individual magnitudes together (r₁ × r₂) and add their respective phase angles together (θ₁ + θ₂).

Let's multiply two complex numbers in rectangular form: Z₁ = 3 + 4i and Z₂ = 1 - 2i.

Step 1: Apply the FOIL method (First, Outer, Inner, Last).
Product = (3 × 1) + (3 × -2i) + (4i × 1) + (4i × -2i)

Step 2: Simplify the terms.
= 3 - 6i + 4i - 8i²

Step 3: Substitute i² = -1.
= 3 - 2i - 8(-1)
= 3 - 2i + 8

Step 4: Combine real and imaginary components.
= (3 + 8) - 2i = 11 - 2i

Convert When Convenient: If you are dealing with high powers or heavy rotation, converting rectangular coordinates into polar form before multiplying will drastically simplify your arithmetic.

Watch Your Signs: The most frequent mistake when expanding binomials with imaginary numbers is dropping or miscalculating negative signs, especially when substituting i² = -1.

Double-Check Units: Ensure that your angle inputs are set consistently to either degrees or radians before computing polar products to avoid skewed results.