Question

What is $Z$ bar in complex numbers ?

Answer 1

Here in this question, we have to explain what $Z$ bar ($\overline Z$) represents in the complex number. Take any general representation of a complex number i.e., $Z = x + iy$ and write its conjugate by changing the sign of the imaginary part that the resultant complex number is represented as $Z$ bar ($\overline Z$).

Complete answer:
A complex number generally denoted as Capital Z $\left( Z \right)$ is any number that can be written in the form $x + iy$ it’s always represented in binomial form. Where, $x$ and $y$ are real numbers. ‘$x$’ is called the real part of the complex number, ‘$y$’ is called the imaginary part of the complex number, and ‘$i$’ (iota) is called the imaginary unit.

We define another complex number $\overline Z$ such that $\overline Z = x - iy$. We call $\overline Z$ or the complex number obtained by changing the sign of the imaginary part (positive to negative or vice versa). Thus, the complex number $Z = x + iy$, the conjugate of $Z$ or complex number can be written as $x - iy$ it is denoted by Z bar i.e., $\overline Z$.

Some properties of conjugate complex number $\left( {\overline Z } \right)$ are: If $Z$, ${Z_1}$ and ${Z_2}$ are complex numbers, then
-If $\overline Z$ be the conjugate of $\overline Z$, then $\overline {\left( {\overline Z } \right)} = Z$.
$\Rightarrow \overline {{Z_1} \pm {Z_2}} = \overline {{Z_1}} \pm \overline {{Z_2}}$
$\Rightarrow\overline {{Z_1} \cdot {Z_2}} = \overline {{Z_1}} \cdot \overline {{Z_2}}$
$\Rightarrow\overline {\left( {\frac{{{Z_1}}}{{{Z_2}}}} \right)} = \frac{{\overline {{Z_1}} }}{{\overline -{{Z_2}} }}$, but ${Z_2} \ne 0$
$\Rightarrow \left| {\overline Z } \right| = Z$
$\Rightarrow Z\overline Z = {\left| Z \right|^2}$
$\Rightarrow {Z^{ - 1}} = \frac{{\overline Z }}{{{{\left| Z \right|}^2}}}$, but $Z \ne 0$.

Hence, Z bar is a conjugate of complex number Z.

Note: The conjugate of complex numbers is found by reflecting $Z$ across the real axis. the conjugate of a complex number Z bar ($\overline Z$) is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign so we can notice easily that the complex conjugate of a complex number is obtained by just changing the sign of the imaginary part.