Let's break down each of the given statements to determine which are true for any non-zero complex number $z$.

Let $z$ be a complex number. A complex number $z$ can be expressed in the form:

$z = x + iy$

where $x$ and $y$ are real numbers and $i$ is the imaginary unit ($i^2 = -1$). The complex conjugate $z^*$ of $z$ is defined as:

$z^* = x - iy$

Now, let's analyze each statement:

1. $z + z^*$ is real:

$z + z^* = (x + iy) + (x - iy) = 2x$ Since $2x$ is a real number (as $x$ is real), this statement is true.

2. $z - z^*$ is real:

$z - z^* = (x + iy) - (x - iy) = 2iy$ Since $2iy$ is purely imaginary, this statement is false.

3. $zz^*$ is real:

$zz^* = (x + iy)(x - iy) = x^2 - (iy)^2 = x^2 + y^2$ The result, $x^2 + y^2$, is always real (it’s the modulus squared of $z$), so this statement is true.

4. $\frac{z}{z^*}$ is real:

$\frac{z}{z^*} = \frac{x + iy}{x - iy}$ Multiplying numerator and denominator by the conjugate of the denominator:

$\frac{z}{z^*} = \frac{(x + iy)(x + iy)}{(x - iy)(x + iy)} = \frac{x^2 + 2ixy - y^2}{x^2 + y^2}$ This simplifies to a complex number unless $y = 0$ (i.e., unless $z$ is real). Since it’s not always real, this statement is false.

Final Conclusion:

• $z + z^*$ is real: True
• $z - z^*$ is real: False
• $zz^*$ is real: True
• $\frac{z}{z^*}$ is real: False

Would you like a further breakdown or have any questions? Here are some related questions for further practice:

1. What is the geometric interpretation of $z$ and $z^*$ in the complex plane?
2. How is the modulus of a complex number $z$ related to $zz^*$?
3. Can $z - z^*$ ever be zero, and if so, under what conditions?
4. What does the result $\frac{z}{z^*}$ represent geometrically?
5. How can you find the real and imaginary parts of $z$ using $z + z^*$ and $z - z^*$?

Tip: For any complex number $z$, the product $zz^*$ is always the square of the modulus of $z$, a useful fact in many calculations.