If z is a complex number, then

Options

A. `|z^(2)|gt|z|`
B. `|z^(2)| = |z^(2)|`
C. `|z^(2)|lt|z|^(2)`
D. `|z^(2)|ge|z|^(2)`

Explanation

To solve the problem, we need to analyze the relationship between \( z^2 \) and \( |z|^2 \) where \( z \) is a complex number. Let's denote \( z = x + iy \), where \( x \) and \( y \) are real numbers.

### Step-by-Step Solution:

1. **Define the complex number**:
Let \( z = x + iy \), where \( x \) and \( y \) are real numbers.

2. **Calculate the modulus of \( z \)**:
The modulus of \( z \) is given by:
\[ |z| = \sqrt{x^2 + y^2} \]
Therefore, the square of the modulus is:
\[ |z|^2 = (|z|)^2 = x^2 + y^2 \]

3. **Calculate \( z^2 \)**:
Now, we calculate \( z^2 \):
\[ z^2 = (x + iy)^2 = x^2 + 2xyi - y^2 = (x^2 - y^2) + 2xyi \]

4. **Find the modulus of \( z^2 \)**:
The modulus of \( z^2 \) is:
\[ |z^2| = |(x^2 - y^2) + 2xyi| = \sqrt{(x^2 - y^2)^2 + (2xy)^2} \]

5. **Expand the expression**:
Expanding the expression inside the square root:
\[ |z^2| = \sqrt{(x^2 - y^2)^2 + 4x^2y^2} \]
This can be simplified to:
\[ |z^2| = \sqrt{x^4 - 2x^2y^2 + y^4 + 4x^2y^2} = \sqrt{x^4 + 2x^2y^2 + y^4} = \sqrt{(x^2 + y^2)^2} \]

6. **Final simplification**:
Therefore, we have:
\[ |z^2| = x^2 + y^2 = |z|^2 \]

7. **Conclusion**:
From the above steps, we conclude that:
\[ |z^2| = |z|^2 \]
This implies that the modulus of \( z^2 \) is equal to the square of the modulus of \( z \).

### Final Answer:
Thus, we have shown that:
\[ |z^2| = |z|^2 \]