Overview
A complex number has a real part and an imaginary part and is written as a + bi. Complex numbers add and subtract componentwise, multiply like binomials with i² = −1, and divide by multiplying numerator and denominator by the conjugate of the denominator. Polar form is useful for multiplication, division, oscillations and phases.
Formula
Addition: (a + bi) + (c + di) = (a + c) + (b + d)i. Multiplication: (a + bi)(c + di) = (ac − bd) + (ad + bc)i.
How it works
Enter the first number
Set the real and imaginary parts of z₁.
Enter the second number
Set the real and imaginary parts of z₂.
Choose the operation
Addition, subtraction, multiplication or division.
Read the result
The algebraic form, modulus, argument and polar form appear.
Worked examples
Example 1: (1 + 2i)(3 + 4i)
The result is −5 + 10i, with modulus ≈ 11.1803 and argument ≈ 116.5651°.
Example 2: (1 + 2i) + (3 + 4i)
The result is 4 + 6i.
Frequently asked questions
What does i mean?
i is the imaginary unit, defined by i² = −1.
Why can't I divide by a zero complex number?
Division requires a nonzero denominator, and the zero complex number is 0 + 0i.
How do I convert to polar form?
Compute the modulus √(a² + b²) and argument atan2(b, a), then write z = r(cos θ + i sin θ).
This calculator is for educational and informational purposes only and is not a substitute for professional advice.