Complex Number Calculator

The four operations

For z₁ = a + bi and z₂ = c + di, the arithmetic follows directly from treating i as a symbol with i² = −1:

• Sum: (a + c) + (b + d)i
• Difference: (a − c) + (b − d)i
• Product: (ac − bd) + (ad + bc)i
• Quotient: multiply top and bottom by the conjugate of z₂: [(ac + bd) + (bc − ad)i] / (c² + d²)

Division is only defined when the denominator c² + d² is non-zero, that is, whenever z₂ is not 0 + 0i.

Modulus and argument of z₁

The modulus is the distance from the origin to the point (a, b) in the complex plane: |z₁| = √(a² + b²). The argument is the angle that z₁ makes with the positive real axis, computed with the two-argument arctangent arg(z₁) = atan2(b, a) and shown here in degrees, in the range −180° to 180°.

Worked example

Let z₁ = 3 + 2i and z₂ = 1 + 4i.

Pitfalls to avoid

• Forgetting i² = −1. When multiplying, the bd·i² term flips sign and joins the real part, that is why the real part of the product is ac − bd, not ac + bd.
• Dividing without the conjugate. You cannot split (a + bi)/(c + di) term by term; multiply by (c − di)/(c − di) first.
• Mixing degrees and radians. The argument here is in degrees; many programming languages return radians from atan2. Multiply by 180/π to convert.
• Quadrant errors. Use atan2, not plain arctan(b/a), or the angle for points with a negative real part lands in the wrong quadrant.