Complex Conjugate and Modulus Calculator

Find the conjugate, modulus, and argument of a complex number z = a + bi, shown on the complex plane.

Input

Enter the real part a and imaginary part b to compute the conjugate, modulus, and argument of z = a + bi.

Real part a

The real component of the complex number

Imaginary part b

The imaginary component (coefficient of i)

Input complex number z = 3 + 4i

Conjugate z bar

3 - 4i

Modulus |z|

Argument (degrees)

53.13010235 deg

Argument (radians)

0.92729522 rad

Real part a

Imaginary part b

Modulus calculation

sqrt(3² + 4²) = 5

The argument is measured counterclockwise from the real axis, in the range minus 180 to 180 degrees. At the origin the argument is undefined.

The conjugate of z = a + bi is a − bi, obtained by flipping the sign of the imaginary part.
The modulus is |z| = sqrt(a squared plus b squared), the distance from the origin.
The argument arg(z) is the angle from the real axis, found with atan2(b, a) to handle quadrants correctly, in the range minus 180 to 180 degrees.
At the origin (a = 0, b = 0) the argument is undefined, so it is shown as undefined.

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