https://file.aiquickdraw.com/mathbot/file/17273716129055a76als8.jpg

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.

Which of the following statements are true for all z ≠ 0?
1. z + z* is real
2. z - z* is real
3. zz* is real
4. z / z* is real

Analyze which of the statements about the real and imaginary parts of complex numbers is true for non-zero z. Explore the behavior of complex conjugates and modulus in various operations. Learn how z + z*, z - z*, zz*, and z / z* behave.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation