Complex Number Calculator

A complex number has a real part and an imaginary part, written a + bi, where i² = −1. This complex number calculator adds, subtracts, multiplies and divides complex numbers with exact fractions, and gives the modulus, argument and polar form. An Argand diagram shows each number as an arrow.

Try:

How to use the complex number calculator

  1. Type z₁, such as 3 + 4i.
  2. Choose an operation and type z₂ if needed.
  3. Press Calculate to see the result, its modulus and argument, and the diagram.

Formula

i2=−1i^2 = -1
∣a+bi∣=a2+b2|a + bi| = \sqrt{a^2 + b^2}

Worked example: (3+4i)(1−2i)

  • z₁: 3+4i
  • Operation: z₁ × z₂
  • z₂: 1-2i

✓ Answer checked

11−2i11 - 2i
Modulus
55≈11.180345\sqrt{5} \approx 11.18034
Argument
−0.1798535 rad=−10.304846∘-0.1798535\text{ rad} = -10.304846^\circ
Polar form
55(cos⁡−0.17985+isin⁡−0.17985)5\sqrt{5}\left(\cos -0.17985 + i\sin -0.17985\right)
−4−22468101214−4−2246z₁z₂result
Argand diagram: real part across, imaginary part up. Each number is an arrow from 0.
Step-by-step working (3 steps)

  1. Expand the brackets

    Multiply every term by every term (FOIL).

    (3+4i)(1−2i)=3−6i+4i−8i2(3 + 4i)(1 - 2i) = 3 - 6i + 4i - 8i^2

  2. Use i² = −1 i² = −1

    Replace i² with −1 and collect terms.

    =11−2i= 11 - 2i

  3. Modulus and argument |a+bi| = √(a²+b²)

    The modulus is the distance from 0; the argument is the angle from the positive real axis.

    ∣z∣=(11)2+(−2)2=55,arg⁡z≈−0.179853 rad|z| = \sqrt{(11)^2 + (-2)^2} = 5\sqrt{5},\quad \arg z \approx -0.179853\text{ rad}
Formulas used
i2=−1i^2 = -1
a+bic+di=(a+bi)(c−di)c2+d2\frac{a+bi}{c+di} = \frac{(a+bi)(c-di)}{c^2+d^2}
How this was checked
  • ✓ Floating-point arithmetic gives the same real and imaginary parts.

Frequently asked questions

How do you multiply complex numbers?

Expand the brackets like algebra, then replace i² with −1. For example, (3 + 4i)(1 − 2i) = 3 − 6i + 4i + 8 = 11 − 2i.

How do you divide complex numbers?

Multiply the top and bottom by the conjugate of the bottom, which makes the denominator a real number.

What is the modulus of a complex number?

Its distance from 0 on the Argand diagram: √(a² + b²). The modulus of 3 + 4i is 5.

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