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.

Explore how to solve the equation |Z| = 1, which represents a unit circle in the complex plane. This problem covers key concepts in complex numbers, including the modulus and the geometry of the complex plane, suitable for high school students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation