हिन्दी

Complex Number Calculator: Operations, Modulus, Argument and Polar Form

Enter z₁ = a + bi and z₂ = c + di and pick an operation. The result comes with its modulus, argument, polar form and the working.

Result

11 − 2i

Modulus |z|
11.18034
Argument θ
−10.3048° = −0.179853 rad
Polar form
11.1803(cos −10.3° + i sin −10.3°)
Conjugate
11 + 2i
Show working
  1. z₁ = 3 + 4i, z₂ = 1 − 2i
  2. (ac − bd) + (ad + bc)i
  3. (3·1 − 4·(−2)) + (3·(−2) + 4·1)i
  4. = 11 − 2i
  5. |z| = √(11² + (−2)²) = 5√5 ≈ 11.18034
  6. arg z = atan2(−2, 11) = −10.3048°

How to use

  1. Choose +, −, × or ÷.
  2. Enter the real and imaginary parts of z₁ and z₂.
  3. Read the result and its polar form; open “Show working” for the steps.

Good to know

Rules: (a + bi)(c + di) = (ac − bd) + (ad + bc)i, because i² = −1. To divide, multiply top and bottom by the conjugate c − di, so the denominator becomes the real number c² + d².

Modulus |z| = √(x² + y²) is the distance from 0 on the Argand plane. The argument θ = atan2(y, x) is the angle from the positive real axis, given as the principal value between −180° and 180°.

Polar form z = r(cos θ + i sin θ) makes multiplication easy: multiply the moduli and add the arguments. Exact values are shown where possible, e.g. |11 − 2i| = 5√5.

Questions

What is i?

The imaginary unit, defined by i² = −1. A complex number has a real part and an imaginary part: z = a + bi.

How do I divide complex numbers?

Multiply numerator and denominator by the conjugate of the denominator. Example: (3 + 4i)/(1 − 2i) = (−5 + 10i)/5 = −1 + 2i.

What is the argument of a complex number?

The angle the point makes with the positive real axis. For 1 + i it is 45° or π/4.

Which range does the argument use?

The principal value, from −180° to 180° (−π to π), as in NCERT Class 11.

Is my data stored or sent anywhere?

No. The calculation runs entirely in your browser; nothing you type leaves your device.

Last updated: