The equation $|Z| = 1$ represents the condition that the complex number $Z$ has a magnitude (or modulus) of 1. Let's break this down step by step.

1. A complex number $Z$ can be expressed as $Z = x + iy$, where:

   • $x$ is the real part,
   • $y$ is the imaginary part,
   • $i$ is the imaginary unit ($i^2 = -1$).

2. The modulus of a complex number $Z = x + iy$ is defined as: $|Z| = \sqrt{x^2 + y^2}$

3. Given $|Z| = 1$, we have: $\sqrt{x^2 + y^2} = 1$

4. Squaring both sides gives: $x^2 + y^2 = 1$ This equation describes a circle in the complex plane (also known as the Argand plane) with a radius of 1, centered at the origin.

In summary, the complex numbers that satisfy $|Z| = 1$ lie on a unit circle centered at the origin in the complex plane.

Would you like more details or have any questions?

Here are 5 questions to deepen understanding:

1. How do we calculate the argument (angle) of a complex number on the unit circle?
2. What is the polar form of a complex number on the unit circle?
3. How would you represent $Z$ in terms of Euler's formula for this case?
4. Can you find specific examples of complex numbers that satisfy $|Z| = 1$?
5. How does this relate to rotations in the complex plane?

Tip: Any complex number with modulus 1 can be expressed as $Z = e^{i\theta}$, where $\theta$ is the angle in radians.