Complex Conjugate Calculator

How Complex Conjugate Calculator Works?

A complex number is written as $z=a+\mathrm{bi}$, where a is the real part, b is the imaginary part, and $\sqrt{-1}$ is represented by i. Enter the real and imaginary parts separately, including the sign of the imaginary part. The calculator then determines the conjugate, modulus, and principal argument.

The complex conjugate keeps the real part unchanged and reverses the sign of the imaginary part:

For example, if $z=3+4i$, its conjugate is $\overline{z}=3-4i$. The calculator reports the conjugate real part as 3 and the conjugate imaginary part as −4.

The modulus is the distance from the origin to the point $(a,b)$ on the complex plane. It is calculated using the Pythagorean theorem:

The principal argument is the counterclockwise angle from the positive real axis to the complex number. The calculator uses the two-argument arctangent function, which correctly identifies the quadrant:

The result is provided in radians and degrees. For $z=3+4i$, the modulus is 5 and the principal argument is approximately 0.9273 radians, or 53.130 degrees.

Practical Use Cases

Students can use the calculator to check homework involving conjugates, complex-plane geometry, polar form, and operations such as division of complex numbers. The separate real and imaginary outputs make it easier to verify each algebraic step.

Electrical and electronics engineers can use complex conjugates and modulus values when working with phasors, impedance, frequency-response calculations, and power relationships. The argument output gives the phase angle associated with a complex quantity.

Mathematicians and researchers can use the calculator to inspect complex-number properties, compare a number with its conjugate, and confirm polar representations before applying more advanced algebra or analysis.

Signal-processing and physics learners can use the modulus to assess the magnitude of a complex amplitude and the argument to identify its orientation or phase relative to the positive real axis.

Accuracy and Limitations

The calculator evaluates the conjugate exactly at the displayed sign and component level, while the modulus and arguments are subject to normal floating-point rounding. It accepts real and imaginary inputs from −1,000,000,000,000 to 1,000,000,000,000 and reports the principal argument selected by atan2, typically in the range −π to π radians. The argument of the zero complex number, where both inputs are 0, is mathematically undefined, so no meaningful direction exists for that case. Results with very large or highly precise decimal inputs may show minor rounding in the displayed modulus or angle.

Quick Examples