# If z is a complex number, show that the sum of z and its conjugate is a real number.

Question
Complex numbers
If z is a complex number, show that the sum of z and its conjugate is a real number.

2021-02-01
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.

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