Last updated: 2026-07-01

Complex number arithmetic

A complex number combines a real part and an imaginary part into a single quantity, written a + bi, where i is the imaginary unit defined by i² = −1. The number a is the real part and b is the coefficient of the imaginary part. This calculator takes two complex numbers, z₁ = a + bi and z₂ = c + di, applies the operation you select — addition, subtraction, multiplication, or division — and reports the result together with its modulus and argument.

Complex numbers extend the real number line into a two-dimensional plane, which makes them indispensable in electrical engineering, signal processing, control theory, and quantum mechanics.

The four operations

Addition and subtraction work component by component, exactly like adding vectors:

• z₁ + z₂ = (a + c) + (b + d)i
• z₁ − z₂ = (a − c) + (b − d)i

Multiplication expands as two binomials, using i² = −1 to fold the i² term back into the real part:

(a + bi)(c + di) = ac + adi + bci + bdi² = (ac − bd) + (ad + bc)i

Division multiplies numerator and denominator by the conjugate of the denominator, c − di. This clears the imaginary part from the denominator, leaving the real value c² + d²:

z₁ / z₂ = ((ac + bd) + (bc − ad)i) / (c² + d²)

Division is defined for every z₂ except 0 + 0i, where c² + d² = 0.

Worked example

Problem: Multiply z₁ = 3 + 4i by z₂ = 1 + 2i.

Using the multiplication rule with a = 3, b = 4, c = 1, d = 2:

• Real part: ac − bd = (3)(1) − (4)(2) = 3 − 8 = −5
• Imaginary part: ad + bc = (3)(2) + (4)(1) = 6 + 4 = 10

So z₁ z₂ = −5 + 10i.

The modulus of the result is √((−5)² + 10²) = √125 ≈ 11.18, and its argument is atan2(10, −5) ≈ 116.57°, which sits in the second quadrant because the real part is negative and the imaginary part is positive.

Modulus and argument

Every complex number corresponds to a point in the plane, with the real part on the horizontal axis and the imaginary part on the vertical axis. Two polar quantities describe that point:

• The modulus r = √(x² + y²) is the distance from the origin. It measures the size of the number.
• The argument φ is the angle measured from the positive real axis. Using the two-argument arctangent, φ = atan2(y, x), places the angle in the correct quadrant and returns a value between −180° and 180° (or −π and π in radians).

Rectangular versus polar form

Rectangular form, x + yi, lists the horizontal and vertical coordinates directly. Polar form, r(cos φ + i sin φ), uses the distance and angle instead. The two forms describe the same number; choosing between them is a matter of convenience.

Addition and subtraction are simplest in rectangular form, since the parts combine independently. Multiplication, division, and powers are simplest in polar form: moduli multiply or divide while arguments add or subtract. De Moivre's theorem builds directly on this, turning the cumbersome task of raising a complex number to a power into a single multiplication of the argument.