Question

What is \[Z\] bar in complex numbers ?

Answer 1

Hint: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.