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Complex Number Calculator

Add, subtract, multiply and divide complex numbers, with modulus, argument and polar form.

By Nitin Raturi · Last reviewed

Your worksheet

How to use the Complex Number Calculator

Add, subtract, multiply or divide two complex numbers and see the polar form.

  1. Enter the real and imaginary parts of z₁.
  2. Enter the real and imaginary parts of z₂.
  3. Choose the operation.
  4. Read the result and its modulus and argument.

Good to know

  • Use a minus sign for a negative imaginary part, as in 1 − 2i.

z₁ = a + bi and z₂ = c + di. Remember i² = −1.

z₁ real part (a)

The real part of the first number, a + bi.

How to fill it in

  • Enter any number, negative allowed.

z₁ imaginary part (b)

The number multiplying i in the first number.

How to fill it in

  • Enter 4 for 3 + 4i, or −4 for 3 − 4i.

Example

3 + 4i has modulus and argument 5 ∠ 53.1301°.

z₂ real part (c)

The real part of the second number, c + di.

How to fill it in

  • Enter any number.

z₂ imaginary part (d)

The imaginary part of the second number.

How to fill it in

  • Enter any number.

Operation

Add, subtract, multiply or divide z₁ by z₂.

How to fill it in

  • Division multiplies top and bottom by the conjugate of z₂.

Example

(3 + 4i) and (1 − 2i): sum 4 + 2i, product 11 − 2i, quotient −1 + 2i. And i × i = −1.

(3 + 4i) × (1 − 2i)

11 − 2i

(3 + 4i) × (1 − 2i) = 11 − 2i, which in polar form is 11.1803 ∠ −10.3048°.

|z₁| and arg z₁
5 ∠ 53.1301°
|z₂| and arg z₂
2.2361 ∠ −63.4349°
Conjugate of z₁
3 − 4i
Result in polar form
11.1803 ∠ −10.3048°
Result, full digits
11, -2i

Exact rules of complex arithmetic; figures are shown to 4 decimal places. Angles are in degrees, from −180° to 180°.

ChangelogHow we test this calculatorEmbed this calculator

How this calculator works

  1. Step 1Add(a + c) + (b + d)iReal with real, imaginary with imaginary.
  2. Step 2Multiply(ac − bd) + (ad + bc)iExpand and use i² = −1.
  3. Step 3Divide× conjugateClear i from the denominator.

A complex number a + bi has a real part a and an imaginary part b, where i² = −1. Adding and subtracting work part by part.

Multiplying expands the brackets, and every i² becomes −1. To divide, multiply the top and bottom by the conjugate of the bottom, which leaves a real denominator.

Worked example: (3 + 4i)(1 − 2i)

Multiplying two complex numbers
Product
(ac − bd) + (ad + bc)i
  1. Real part3 × 1 − 4 × (−2)11
  2. Imaginary part3 × (−2) + 4 × 1−2
  3. Product11 − 2i

In polar form the result is 11.1803 ∠ −10.3048°: the moduli multiply and the angles add.

Read the result

  • Sum4 + 2iAdd the parts separately.
  • Quotient−1 + 2i(3 + 4i) ÷ (1 − 2i), via the conjugate 1 + 2i.

The defining rule: i × i = −1.

Assumptions and limits

  • DegreesArguments are in degrees, −180° to 180°.
  • Principal angleMeasured from the positive real axis.
  • No division by 0Dividing by 0 + 0i is undefined.
  • Rounded displayFull digits are shown in the details.

Complex numbers appear as roots of quadratics and cubics with no real solution.

To find those roots, use the quadratic formula calculator or the cubic equation calculator. The vector calculator works with directions in three dimensions.

Common mistakes

  • Mistake: Treating i² as +1.

    Instead: i² = −1, so bd is subtracted in the real part.

  • Mistake: Dividing parts separately.

    Instead: Multiply by the conjugate first.

Checking the method

A separate script redoes every case with Python’s built-in complex numbers and its polar-form library, and matches each golden value.

Read how we test our calculators, or browse more maths calculators.

Frequently asked questions

What is (3 + 4i)(1 − 2i)?

11 − 2i: real part 3 + 8 = 11, imaginary part −6 + 4 = −2.

What is (3 + 4i) ÷ (1 − 2i)?

−1 + 2i, after multiplying top and bottom by the conjugate 1 + 2i.

How do you add complex numbers?

Add real parts and imaginary parts separately: (3 + 4i) + (1 − 2i) = 4 + 2i.

What is i squared?

i × i = −1, by definition.

What is the modulus of 3 + 4i?

√(3² + 4²) = 5, at an argument of about 53.13°: 5 ∠ 53.1301°.

Sources and methodology

The formulas are standard time-value-of-money results. We check every figure against golden values worked out independently, by closed form and by month-by-month simulation; see the methodology.

Written and maintained by Nitin Raturi, a software engineer (about the author). Last reviewed 4 Nov 2026; next review due by . Found an error? Tell us.

Changelog

  • : Initial release.