Enter two complex numbers, pick an operation, and get the result instantly — plus the polar form of the answer.
Complex Number Calculator
How It’s Calculated
Addition and subtraction combine real and imaginary parts separately. Multiplication uses (a+bi)(c+di) = (ac−bd) + (ad+bc)i, applying i² = −1. Division multiplies both numerator and denominator by the conjugate of the denominator to clear the imaginary part from below the line. The modulus √(re²+im²) and argument (the angle from the positive real axis) together give the polar form of any complex number.
Worked Example
For z₁ = 3+4i and z₂ = 1+2i, multiplying: (3×1 − 4×2) + (3×2 + 4×1)i = (3−8) + (6+4)i = −5 + 10i.
Frequently Asked Questions
Q: Why does dividing complex numbers involve the conjugate?
Multiplying denominator and numerator by the conjugate turns the denominator into a real number (c²+d²), since (c+di)(c−di) = c²+d² — that’s what makes the division possible.
Q: What do modulus and argument actually represent?
Plot a complex number as a point (re, im). The modulus is its distance from the origin; the argument is the angle that line makes with the positive real axis — together they’re an alternative way of describing the same point.
This is exactly the arithmetic covered in 1st PUC Complex Numbers & Quadratic Equations. See how Daniel Classes teaches 1st and 2nd PUC Mathematics →