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
- \(|z_1\,z_2|=|z_1||z_2|\)
- \(|\frac{z_1}{z_2}|=\frac{|z_1|}{|z_2|}\), 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
- \(\overline{z_1 + z_2} = \overline{z_1} + \overline{z_2}\)
- \(\overline{z_1 - z_2} = \overline{z_1} - \overline{z_2}\)
- \(\overline{z_1 \, z_2} = \overline{z_1} \, \overline{z_2}\)
- \(\overline{\left( \frac{z_1}{z_2} \right)} = \frac{\overline{z_1}}{\overline{z_2}}\)
Relation between modulus and conjugate
Let \(z=a+ib\) be a complex number. Then
\[z\,\overline{z}=|z|^2=a^2+b^2\]