Complex Number Calculator: Add, Subtract, Multiply, Divide, Powers, Roots and Polar Form

How this calculator works

Enter the real and imaginary parts of the first number, and of the second if the operation needs it.

Choose the operation, and the power or number of roots if needed.

The calculator gives the answer, its modulus and argument, and the roots listed where relevant.

Formulas used

• (a + bi)(c + di) = (ac - bd) + (ad + bc)i
• z^n = r^n(\cos n\theta + i\sin n\theta)
• |z| = \sqrt{a^2 + b^2}

Worked example

With the default numbers in the calculator above:

1. Modulus and argument of the first number: 5 at 53.1301 degrees
2. Multiply, using i squared = -1: 11 - 2i
3. Modulus of the result: 11.1803
4. Argument of the result: -10.3048 degrees (-0.179853 radians)
5. Real part and imaginary part of the result: 11 and -2

Change the numbers in the calculator to see the full working, every step, for your own case.

A second worked example

A different case, with these numbers: Operation z1 / z2.

1. Modulus and argument of the first number: 5 at 53.1301 degrees
2. Multiply top and bottom by the conjugate of the denominator: -1 + 2i (the denominator is 5)
3. Modulus of the result: 2.23607
4. Argument of the result: 116.565 degrees (2.03444 radians)
5. Real part and imaginary part of the result: -1 and 2

Common questions

Related calculators

• Complex Roots Calculator: All Roots of a Complex Number
• Trigonometric Identities Calculator: Check Any Identity for an Angle
• Quadratic and Cubic Equation Solver: Roots, Discriminant, Vertex and Factorised Form

Got a question about this? Ask an engineering question and get a plain-language answer with references.

Further reading: Any first-year mathematics or engineering mathematics text (complex numbers).

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