Complex Conjugate Calculator

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The real part of the complex number z = a + bi.
The coefficient of i in z = a + bi. Enter a negative number for z = a - |b|i.
Unit for the argument (angle) of the complex number.
Complex conjugate (z-bar)
3 - 4i

The conjugate of z = a + bi is z-bar = a - bi

Original number (z)3 + 4i
Modulus |z|5
Modulus squared |z|^225
Argument (angle)53.1301°
Polar form5 * (cos(53.1301°) + i*sin(53.1301°))
z * z-bar25
Modulus |z|5
z * z-bar = |z|^225

The conjugate of z = 3 + 4i is z-bar = 3 - 4i.

  • The conjugate z-bar = 3 - 4i is a reflection of z = 3 + 4i across the real axis on the complex plane.
  • The modulus |z| = 5.0000 is the distance from the origin. Both z and z-bar are exactly this far from 0.
  • Multiplying z by z-bar eliminates the imaginary part completely, giving the real number 25.0000 (which equals |z|^2).
  • This product property is why the conjugate is essential for dividing complex numbers: multiply numerator and denominator by the denominator's conjugate to get a real denominator.

Next stepTo divide (p + qi) by z = a + bi, multiply both by z-bar = a - bi. The denominator becomes |z|^2 = 25.00, a real number, making the division straightforward.

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