Given that z=2-i, confirm directly that 1/z = 1/z*

Question

It’s given that z=2-i. So I’ve worked out that 1/z=2/5 + i/5 in the form a+bi. I’m now asked to show that 1/z is equal to its conjugate, however I’ve worked out (1/z)* to be 2/5 - i/5, hence they are not equal. Unsure where I have gone wrong.

Answer 1

You are not wrong. The only complex numbers which are equal to their conjugates are the real numbers. And $\frac1z\notin\mathbb R$.