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Modulus & conjugate of complex number

Modulus of a complex number:

Let \(z=a+ib\) be a complex number. Then its modulus \(|z|=\sqrt{a^2+b^2}\)

Properties:

Let \(z_1\) and \(z_2\) be two complex numbers, then

  1. \(|z_1\,z_2|=|z_1||z_2|\)
  2. \(|\frac{z_1}{z_2}|=\frac{|z_1|}{|z_2|}\), where \(|z_2|\neq 0\)

This isn't true for addition and subtraction.

Conjugate of a complex number:

Let \(z=a+ib\) be a complex number. Then its conjugate \(\overline{z}=a-ib\)

Properties:

Let \(z_1\) and \(z_2\) be two complex numbers, then

  1. \(\overline{z_1 + z_2} = \overline{z_1} + \overline{z_2}\)
  2. \(\overline{z_1 - z_2} = \overline{z_1} - \overline{z_2}\)
  3. \(\overline{z_1 \, z_2} = \overline{z_1} \, \overline{z_2}\)
  4. \(\overline{\left( \frac{z_1}{z_2} \right)} = \frac{\overline{z_1}}{\overline{z_2}}\)

Relation between modulus and conjugate

Let \(z=a+ib\) be a complex number. Then

\[z\,\overline{z}=|z|^2=a^2+b^2\]