# Complex Number Multiplication Calculator

> Multiply or divide two complex numbers, with the result in rectangular form, its modulus and argument, and a check in polar form.

Category: Algebra, geometry & advanced math · Applies to: Worldwide · Last updated: 2026-10-29
Interactive calculator: https://calculators.geekmidst.com/calculators/complex-number-multiplication-calculator

Multiply complex numbers by expanding and using i² = −1; divide by multiplying top and bottom by the conjugate. In polar form, moduli multiply and angles add.

Also called: Complex number division calculator.

## Inputs
- Operation (options: z₁ × z₂, z₁ ÷ z₂; default mul)
- z₁ real part (a) (default 3)
- z₁ imaginary part (b) (default 2)
- z₂ real part (c) (default 1)
- z₂ imaginary part (d) (default 4)

## Results
- Result
- Real part
- Imaginary part
- Modulus |z|
- Argument
- Polar-form check

## Formula

```
(a + bi)(c + di) = (ac − bd) + (ad + bc)i;  (a + bi) ÷ (c + di) = [(ac + bd) + (bc − ad)i] ÷ (c² + d²)
```

Where:
- a, b: real and imaginary parts of z₁
- c, d: real and imaginary parts of z₂

## Worked example
1. Expand: 3·1 + 3·4i + 2i·1 + 2i·4i = 3 + 12i + 2i + 8i².
2. Use i² = −1: 3 − 8 + 14i = −5 + 14i.
3. Check: |z₁|·|z₂| = √13·√17 = √221 ≈ 14.866 = |−5 + 14i|.

**Result:** (3 + 2i)(1 + 4i) = −5 + 14i.

## What the result means

Multiplying by a complex number scales by its modulus and rotates by its argument; dividing does the reverse. The polar-form line shows this check.

## Assumptions and limits
- Inputs are in rectangular form a + bi.
- Division by 0 + 0i is undefined.

## Questions

### Why multiply by the conjugate when dividing?

Because (c + di)(c − di) = c² + d² is real, which clears i from the denominator.

### What is i × i?

By definition i² = −1, so i × i = −1.

## Related calculators
- [Complex Number Addition Calculator](https://calculators.geekmidst.com/calculators/complex-number-addition-calculator)
- [Complex Number Polar Form Converter](https://calculators.geekmidst.com/calculators/complex-number-polar-conversion-calculator)
- [Complex Number Modulus Calculator](https://calculators.geekmidst.com/calculators/complex-number-modulus-calculator)

Source: GM¹X CalcVerse — https://calculators.geekmidst.com/calculators/complex-number-multiplication-calculator