What is Z bar in complex numbers ?

In complex analysis, the bar symbol (a horizontal line over a variable) is used to represent the complex conjugate of a complex number.

For a complex number

=

+

Z=a+bi, where
a and
b are real numbers and
i is the imaginary unit, the complex conjugate of
Z, denoted as
‾
Z
or
∗
Z
∗
, is obtained by changing the sign of the imaginary part. In other words,
‾
=
−

Z
=a−bi.

The complex conjugate has several important properties. Let
Z and
W be complex numbers. Then:

‾
‾
=
Z

=Z (Taking the conjugate twice returns the original complex number).
+
‾
=
‾
+
‾
Z+W
​
=
Z
+
W
(The conjugate of a sum is the sum of the conjugates).

‾
=
‾
⋅
‾
ZW
=
Z
⋅
W
(The conjugate of a product is the product of the conjugates).
(

)
‾
=

‾
‾
(
W
Z
​
)
​
=
W

Z

​
(The conjugate of a quotient is the quotient of the conjugates).
The complex conjugate is often used in operations involving complex numbers, such as finding the modulus (absolute value) of a complex number or dividing complex numbers. It helps simplify expressions and work with complex numbers in various mathematical contexts.