Complex Number Calculator

What is the Complex Number Calculator?

A complex number a + bi has a real part a and an imaginary part b, where i is the square root of −1. Complex numbers give every polynomial equation a solution and are essential in electrical engineering (impedance of AC circuits), signal processing, control systems and quantum physics.

Enter two complex numbers and an operation. Along with the answer in a + bi form, the calculator gives the modulus |z| (distance from the origin), the argument (angle from the positive real axis, in degrees) and the complex conjugate.

How to use the Complex Number Calculator

1. First number: real part (a) (sample: 3)
2. First number: imaginary part (b) — the coefficient of i; enter a negative number for a − bi. (sample: 4)
3. Operation (sample: Multiply (×))
4. Second number: real part (c) (sample: 1)
5. Second number: imaginary part (d) — the coefficient of i in the second number. (sample: -2)
6. Press Calculate (results also refresh as you type). The result card shows the Result, along with Real part, Imaginary part, Modulus |z| and Argument (angle).

Formula used

(a + bi)(c + di) = (ac − bd) + (ad + bc)i

• i = √−1, so i² = −1

Division multiplies top and bottom by the conjugate of the divisor: (a + bi) ÷ (c + di) = [(ac + bd) + (bc − ad)i] ÷ (c² + d²). For a result x + yi, the modulus is √(x² + y²) and the argument is atan2(y, x).

Worked example

These are the sample values already filled in the calculator above, and the result it returns for them:

Tips for using the Complex Number Calculator

• A number times its conjugate is always real: (3 + 4i)(3 − 4i) = 9 + 16 = 25 = |z|².
• In polar form, multiplying complex numbers multiplies their moduli and adds their angles.
• Electrical engineers often write j instead of i to avoid confusion with current.