Given z is a complex number.

Let z=x+iy

We know that conjugate of a complex number is given by changing sign of imaginary part.

Conjugate of given complex number z will be:

z=x−iy

Now, sum of complex number z and its conjugate will be:

\(\displaystyle{z}+\overline{{z}}={\left({x}+{i}{y}\right)}+{\left({x}-{i}{y}\right)}={x}+{i}{y}+{x}-{i}{y}={2}{x}\)

Therefore, sum of z and it conjugate is real number.

Let z=x+iy

We know that conjugate of a complex number is given by changing sign of imaginary part.

Conjugate of given complex number z will be:

z=x−iy

Now, sum of complex number z and its conjugate will be:

\(\displaystyle{z}+\overline{{z}}={\left({x}+{i}{y}\right)}+{\left({x}-{i}{y}\right)}={x}+{i}{y}+{x}-{i}{y}={2}{x}\)

Therefore, sum of z and it conjugate is real number.