Can I use j instead of i?

Yes. The calculator treats j as the imaginary unit, which is common in electrical engineering.

What is the difference between rectangular and polar form?

Rectangular form uses real and imaginary parts. Polar form uses modulus and argument, which are often better for multiplication, division, powers, and roots.

Why can the argument have more than one value?

Adding a full turn does not change the direction of a complex number, so arguments differ by 2π in radians or 360 degrees.

Does this solve arbitrary complex expressions?

No. It is built around structured operations on one or two complex numbers. Use a CAS for arbitrary symbolic expressions.

Complex Number Calculator

Calculate complex number arithmetic in rectangular or polar form. Supports addition, subtraction, multiplication, division, conjugates, inverse, powers, roots, modulus, argument, and form conversion.

Angle unit

Result form

Argument range

Argument display

First number

Second number

Examples

First number

3 + 4·i

Modulus: 5

Argument: 53.1301°

Real part: 3

Imaginary part: 4

Second number

1 − 2·i

Modulus: 2.236068

Argument: -63.4349°

Real part: 1

Imaginary part: -2

Operation result

4 + 2·i

Modulus: 4.472136

Argument: 26.5651°

Real part: 4

Imaginary part: 2

Step-by-step solution

Given: two entered complex numbers.

Addition: real parts and imaginary parts are added separately.

Real part of the result: 4

Imaginary part of the result: 2

What the calculator shows

Algebraic, trigonometric, exponential, and polar result forms.
Modulus, argument, real part, and imaginary part.
Step-by-step arithmetic logic without formula-heavy helper cards in the interface.
Conjugate, reciprocal, powers, and all complex roots.

Operations on the first number and a constantMultiplication by a real number

Real number

Selected operation result

6 + 8·i

Modulus: 10

Argument: 53.1301°

Real part: 6

Imaginary part: 8

Multiplication by a real number: both parts are scaled by the same factor.

Real part of the result: 6

Imaginary part of the result: 8

Powers and complex rootsPower calculation through modulus and argument

Integer power n (any integer)

Power result

-117 + 44·i

Modulus: 125

Argument: 159.3903°

Real part: -117

Imaginary part: 44

Given: modulus 5, argument 53.1301°

Method: modulus is raised to the selected power, and the argument is multiplied by that power.

Result modulus: 125

Result argument: 159.3903°

The result is shown in the selected form above.

What this complex number calculator does

English search intent expects a practical calculator for complex arithmetic, not only a formula sheet. Enter values as a+bi, a+bj, or polar r∠theta, then calculate operations and inspect the result in multiple forms.

Add, subtract, multiply, and divide two complex numbers.
Find conjugate, opposite, reciprocal, square, powers, and roots.
Show modulus, argument, rectangular form, polar form, trigonometric form, and exponential form.
Switch angle output between degrees and radians.

This is a structured calculator, not a full CAS. For a single conversion between rectangular, polar, trigonometric, and exponential notation, use the linked form converter.

Supported input forms

z = a + bi

a is the real part, b is the imaginary part, and i is the imaginary unit.

z = r ∠ θ

r is the modulus and theta is the argument in the selected angle unit.

Rectangular examples: 3+4i, 1-i, -2i, sqrt(2)+i.
Engineering notation with j is accepted, such as 3+4j.
Polar examples: 5∠53, sqrt(2)∠45, or equivalent radius-angle input.
Numeric parts can use fractions, pi, e, powers, and square roots.

Modulus, argument, and forms

∣ z ∣ = a^{2} + b^{2}

The modulus is the distance from the origin to the point on the Argand plane.

ar g (z) = atan2 (b, a)

The argument is the direction angle, with the quadrant handled by atan2.

z = r (cos θ + i sin θ) = r e^{i θ}

Polar, trigonometric, and exponential forms use the same modulus and argument.

Frequently Asked Questions

Sources and References

Calculations are based on the listed reference sources. Links open in a new tab.

Updated: June 2, 2026

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