-2
$\begingroup$

We have $|z|=1$ does this mean $\overline{z} = 1/z$ is true?

I've tried this problem with the properties of $1/z = \overline{z}/|z|^2$ however I didn't get anywhere. Anyone can help prove this?

  • 3
    What are $z$ and $x$ w.r.t. each other? – 2017-01-06
  • 1
    No, it's not true. If $z=1$ and $x=2$, then $\overline x=\frac1x$ is not true. – 2017-01-06

2 Answers 2

0

I think you are asking if the modulus of a complex number $z = 1$, then is the conjugate of $z$ equal to its inverse ($\frac{1}{z}$)?

The answer is yes.

Let $z=a+bi$ where $a^2+b^2=1$ (in other words, $|z|=1$). Then:

$\frac{1}{a+bi}=\frac{1}{a+bi}\cdot\frac{a-bi}{a-bi}=\frac{a-bi}{a^2+b^2}$.

Since $a^2+b^2=1$, $\frac{1}{a+bi}=a-bi$, which is $\bar{z}$.

Note this result only holds if $z$ is on the complex unit circle.

-2

We have: $z\cdot \bar{z} = |z|^2 = 1 \implies \bar{z} = \dfrac{1}{z}$. $x = \bar{x}$, but $x^2$ may or may not be $1$. So its not true in general.