Calculadora de Número Complexo
- Created by
- Renato Passos, Eng. de Software
- Reviewed by
- Renato Passos, Eng. de Software
Last updated: Apr 18, 2026
Formula
z = r·(cos θ + i·sen θ) com r = √(a² + b²), θ = atan2(b, a)
About this calculator
The Complex Number Calculator is an online tool that performs basic operations with complex numbers in the algebraic form z = a + bi. It automatically calculates the modulus (r), argument (θ), and converts the number to polar form r·(cos θ + i·sen θ). The modulus represents the distance from the point (a, b) to the origin, while the argument is the angle with the positive real axis. The tool uses the formulas r = √(a² + b²) and θ = atan2(b, a), ensuring correct quadrant placement.
To use the calculator, simply enter the real part (a) and imaginary part (b) in the corresponding fields and click 'Calculate'. The result will display the modulus, argument in degrees and radians, and the complete polar representation. This tool is useful for students in engineering, physics, and mathematics who need to solve problems involving electrical circuits, waves, or signal analysis, where complex numbers are common.
Note that the argument is calculated in the range from -π to π (or -180° to 180°), following the principal value convention. If the complex number has both real and imaginary parts zero, the calculator returns zero modulus and an undefined argument. It is recommended to verify the entered values, especially for numbers with negative imaginary parts, as the sign directly affects the argument.
Frequently asked questions
What should I do if the argument shows as 'undefined'?
This happens when the complex number is zero (a = 0 and b = 0). In this case, the modulus is zero and the argument is not defined.
Can I use decimal numbers in the fields?
Yes, the calculator accepts decimal numbers. Use a dot (.) as the decimal separator.
Is the result of the argument in degrees or radians?
The calculator displays the argument in both degrees and radians for easier interpretation.
What is the difference between algebraic and polar forms?
The algebraic form (a + bi) is based on Cartesian coordinates, while the polar form (r·(cos θ + i·sen θ)) uses distance and angle. Both represent the same number.
Does the calculator handle complex conjugates?
Not directly. It only calculates the modulus, argument, and polar form of the entered number. For the conjugate, change the sign of the imaginary part.