How to use this calculator
- Enter or choose operation.
- Enter or choose z₁ real part (a).
- Enter or choose z₁ imaginary part (b).
- Enter or choose z₂ real part (c).
- Enter or choose z₂ imaginary part (d).
- The result updates as you type. Use Copy result or Share to keep it.
Method
(a + bi)(c + di) = (ac − bd) + (ad + bc)i; (a + bi) ÷ (c + di) = [(ac + bd) + (bc − ad)i] ÷ (c² + d²)
- a, b
- real and imaginary parts of z₁
- c, d
- real and imaginary parts of z₂
Worked example
- Expand: 3·1 + 3·4i + 2i·1 + 2i·4i = 3 + 12i + 2i + 8i².
- Use i² = −1: 3 − 8 + 14i = −5 + 14i.
- Check: |z₁|·|z₂| = √13·√17 = √221 ≈ 14.866 = |−5 + 14i|.
(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.