Question

prove that modulus of a complex number and its conjugate are equal​

Answer 1

Answer:

Approach 1

$\displaystyle |z|^2=z\bar z = \bar zz=\bar z\bar{\bar z} = |\bar z|^2\\\\\Rightarrow |z| = \pm|\bar z|.\\\\\text{But the modulus is non-negative, so it follows that}\ |z|=|\bar z|.$

Approach 2

Suppose z = a + bi.

Then the conjugate is z' = a - bi.

So |z| = √(a²+b²) = √(a²+(-b)²) = |z'|.

Approach 3

Suppose z = r cis θ.

Then the conjugate is z' = r cis (-θ).

So |z| = r = |z'|.