Complex Numbers#0

We can rephrase the problem in numerical form to be:

$z +$ $\frac{1}{z*}$ $= |z|$

By letting $z = x + iy$ and using Pythagoras to evaluate the absolute value:

$x + iy +$ $\frac{1}{x - iy}$ $= \sqrt{x^2 + y^2}$

Removing the fractions and simplifying:

$x^2 + y^2 + 1 = \sqrt{x^2 + y^2}$

Let: $a = x^2$ and $b = y^2$ ,

$(a + b + 1)^2 = a + b$

$a^2 + 2ab + b^2 + 2a + 2b + 1 = a + b$

$a^2 + 2ab + b^2 = - a - b - 1$

$(a + b)^2 = - a - b - 1$

As we have defined $z = x + iy$ , $x$ and $y$ are real numbers and since $a$ and $b$ are the squares of real numbers, $a$ and $b$ are positive real numbers. Therefore, on this last line, the left hand side must be positive and the right hand side must be negative. Hence, there are $\boxed{0}$ values for $z$ .