Let 𝑍, 𝜔 ∈ ℂ. And also 𝑍 ≠ 𝜔 and 𝑍, 𝜔 ≠ 0. Show that |𝑍|^2 = 𝑍. 𝑍^-. 

To show that \(|Z|^2 = Z \cdot \overline{Z}\) for \(Z \in \mathbb{C}\), where \(\overline{Z}\) denotes the complex conjugate of \(Z\), we will use the definition of the modulus (absolute value) of a complex number and the properties of complex conjugation.

### Step-by-Step Proof:

1. **Definition of a Complex Number**:
   A complex number \(Z\) can be written in the form:
   \[
   Z = x + yi
   \]
   where \(x, y \in \mathbb{R}\) (real numbers), and \(i\) is the imaginary unit with the property that \(i^2 = -1\).

2. **Complex Conjugate**:
   The complex conjugate of \(Z = x + yi\) is defined as:
   \[
   \overline{Z} = x - yi.
   \]

3. **Product of a Complex Number and its Conjugate**:
   The product of a complex number \(Z\) and its conjugate \(\overline{Z}\) is given by:
   \[
   Z \cdot \overline{Z} = (x + yi)(x - yi).
   \]

4. **Simplifying the Product**:
   Using the distributive property (also known as FOIL method), we calculate:
   \[
   Z \cdot \overline{Z} = x^2 - xyi + xyi - y^2i^2.
   \]
   Notice that the middle terms \(-xyi\) and \(+xyi\) cancel each other out. Also, since \(i^2 = -1\), we have:
   \[
   Z \cdot \overline{Z} = x^2 - (-y^2) = x^2 + y^2.
   \]

5. **Definition of the Modulus of a Complex Number**:
   The modulus (or absolute value) of a complex number \(Z = x + yi\) is defined as:
   \[
   |Z| = \sqrt{x^2 + y^2}.
   \]
   Therefore, the square of the modulus is:
   \[
   |Z|^2 = (|Z|)^2 = (\sqrt{x^2 + y^2})^2 = x^2 + y^2.
   \]

6. **Conclusion**:
   From steps 4 and 5, we see that:
   \[
   Z \cdot \overline{Z} = x^2 + y^2 = |Z|^2.
   \]
   Therefore, we have shown that:
   \[
   |Z|^2 = Z \cdot \overline{Z}.
   \]

This completes the proof that the square of the modulus of a complex number \(Z\) is equal to the product of the number and its complex conjugate.

### Would you like more details or have any questions?

Here are some related questions you might find interesting:

1. How do you compute the modulus of a complex number in polar form?
2. What is the geometric interpretation of the complex conjugate on the complex plane?
3. How can the product \(Z \cdot \overline{Z}\) be used to find the distance from a complex number to the origin?
4. What happens to the modulus of a complex number when it is multiplied by another complex number?
5. How can we extend the concept of modulus to complex functions?

**Tip:** When dealing with complex numbers, remember that the modulus represents the distance of the number from the origin on the complex plane, while the argument represents the angle it makes with the positive real axis.

Understand and prove the relation |Z|^2 = Z \cdot \overline{Z} for complex numbers Z. Explore the definition of modulus and complex conjugate with detailed step-by-step explanation. Suitable for undergraduate students and those interested in complex number theory.

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