Complex Conjugate Calculator

The complex conjugate of a number changes the sign of its imaginary part while leaving its real part unchanged. If z = a + bi, then its conjugate is a - bi. For example, the conjugate of 3 + 4i is 3 - 4i, and the conjugate of -2 - 2i is -2 + 2i. This reflection across the real axis preserves the number's distance from the origin while reversing the sign of its angle. Conjugates are valuable because multiplying a complex number by its conjugate removes the imaginary component. The product (a + bi)(a - bi) equals a² + b², a nonnegative real number. That identity is used to divide complex numbers: multiply the numerator and denominator by the denominator's conjugate, then simplify the denominator to a real value. It also connects directly to modulus because the square of the modulus is a² + b². The modulus, written as the absolute value of z, is the Euclidean distance from the origin to the point (a, b) on the complex plane. It is calculated with the Pythagorean formula square root of a² + b². Conjugate pairs always have the same modulus. This calculator uses the hypot function for a stable calculation and displays a rounded decimal when the result is not an integer. The argument is the directed angle from the positive real axis to the complex number. A simple inverse tangent cannot distinguish all quadrants, so the calculator uses atan2(b, a). This returns the principal argument from negative pi through positive pi radians and handles points on the axes correctly. The result is shown in both radians and degrees. For the zero complex number, atan2 conventionally returns zero, although mathematically the argument of zero is undefined. Complex conjugation distributes over addition and multiplication: the conjugate of a sum is the sum of the conjugates, and the conjugate of a product is the product of the conjugates. Applying conjugation twice returns the original number. A complex number equals its own conjugate exactly when its imaginary part is zero, which characterizes the real numbers inside the complex-number system. Conjugates appear throughout algebra, electrical engineering, signal processing, quantum mechanics, Fourier analysis, and polynomial theory. Real-coefficient polynomials have nonreal roots in conjugate pairs, and engineers use conjugation when calculating power and impedance. Enter the coefficients separately to avoid ambiguity about signs, then use the displayed modulus, principal argument, and real-valued product to verify the result.