Complex Number Calculator

Compute complex number operations — arithmetic, modulus, argument, and conjugate.

Enter Complex Numbers

A =+i

B =+i

Results

A + B4 + 6i

A - B2 + 2i

A × B-5 + 10i

A ÷ B2.2 - 0.4i

|A| (modulus)5

arg(A)53.1301°

conj(A)3 - 4i

Polar Form of A

5∠53.1301°

r∠θ° where r = |A| and θ = arg(A)

About Complex Number Calculator

Perform arithmetic and analysis on complex numbers in the form a + bi. Supports addition, subtraction, multiplication, division, modulus (absolute value), argument (phase angle in degrees), conjugate, and polar form conversion. All results are rounded to 6 significant figures. Inputs are preserved in the URL so you can share or bookmark any calculation.

Built and maintained by Meet Shah · Last updated 30 July 2026

What this tool is used for

Multiplying or dividing complex numbers without expanding by hand.
Finding a modulus and argument for a polar conversion.
Checking a homework result with the working visible.
Working through an electrical impedance calculation.
Confirming a conjugate or a division result.

Frequently Asked Questions

How is complex division carried out?: By multiplying top and bottom by the conjugate of the divisor, which makes the denominator real: (a+bi)/(c+di) = ((ac+bd) + (bc−ad)i) / (c²+d²). When c and d are both zero that denominator vanishes, and the tool says undefined rather than printing NaN.
Why is the argument found with atan2 rather than atan?: Because atan(b/a) cannot tell 1+i from −1−i — both give a ratio of 1. atan2 takes the two components separately and returns the right quadrant, over the range −180° to +180°. The result is converted from radians to degrees for display.
What is the argument of zero?: Mathematically undefined — the origin points nowhere — but atan2(0, 0) returns 0 by convention, so 0° is displayed. Whenever the modulus is 0, treat the angle as meaningless rather than as a direction.
What does the polar form tell me?: r∠θ is the same number written as a distance from the origin and an angle: 3+4i has modulus 5 and argument 53.13°. Multiplying multiplies the moduli and adds the angles, which is why polar form is the practical one for powers and roots.
What is the conjugate for?: Flipping the sign of the imaginary part gives a partner whose product with the original is real and equals the modulus squared: (3+4i)(3−4i) = 25. That is what clears the denominator in division, and what keeps power calculations real-valued in signal processing.
When is polar form easier than rectangular?: For multiplication, division and powers — magnitudes multiply and angles add, which turns a messy expansion into two simple operations. Addition is the reverse case, where rectangular form is trivial and polar is not.

Common errors and gotchas

Forgetting to multiply by the conjugate when dividing, which is what removes i from the denominator.
Getting the argument's quadrant wrong, which a naive arctangent does not resolve.
Confusing the modulus with the real part.
Mixing degrees and radians in the argument.
Assuming complex numbers are ordered, so comparing two by size is meaningless.