Complex Number Calculator – Arithmetic & Modulus

Complex numbers pair a real part with an imaginary part, and arithmetic follows i² = −1. Adding is component-wise: (3+2i) + (1−4i) = 4−2i. Multiplying distributes and swaps signs: (1+2i)(3+i) = 3 + i + 6i + 2i² = 3 + 7i − 2 = 1+7i. The conjugate flips the imaginary sign, and z × z̄ = |z|²: (3+4i)(3−4i) = 25, so |3+4i| = 5 — the distance from the origin in the complex plane. Division multiplies by the conjugate of the denominator, converting (1+2i)/(3−4i) into (−5+10i)/25 = −0.2+0.4i. The calculator prints each intermediate product and the modulus/argument pair, linking rectangular work to the geometry: the argument of 3+4i is arctan(4/3) ≈ 53.13°. Results stay exact in fraction form where possible.

1. 1

   Enter two complex numbers

   Form a+bi for each, e.g. 1+2i and 3+i

2. 2

   Pick the operation

   Add, subtract, multiply or divide

3. 3

   Follow the distribution

   Every cross-product is shown, applying i² = −1

4. 4

   Read modulus & argument

   |z| and θ = atan(b/a) accompany each result

• 1

  i² = −1 is the only new rule; everything else is algebra

• 2

  Multiply by the conjugate to divide complex numbers

• 3

  z·z̄ = |z|², so |3+4i| = 5 without touching square roots

• Writing i² as +1 instead of −1

• Adding real to imaginary parts directly: (3+2i)+(1+4i) is 4+6i, components stay separate

• Forgetting the conjugate when dividing, leaving i in the denominator

• Computing modulus as a+b instead of √(a²+b²)