Solve the equation $z + \overline{z} = 4$ for the complex number $z$.

1. Problem Description

Solve the equation $z + \overline{z} = 4$ for the complex number $z$.

2. Solution Steps

Let $z = a + bi$, where $a$ and $b$ are real numbers.

Then the complex conjugate of $z$ is $\overline{z} = a - bi$.

Substituting these expressions into the given equation, we have

$a + bi + a - bi = 4$.

Combining like terms, we get

$2a = 4$.

Dividing both sides by 2, we find

$a = 2$.

Therefore, $z = 2 + bi$, where $b$ can be any real number. The complex numbers satisfying the equation are of the form $2 + bi$.

3. Final Answer

$z = 2 + bi$, where $b$ is any real number.