If z is a complex number then

Options

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

Answer

To solve the problem, we need to analyze the properties of complex numbers and their moduli. Let's denote the complex number \( z \) as \( z = x + iy \), where \( x \) and \( y \) are real numbers.

### Step 1: Calculate \( z^2 \)
First, we compute \( z^2 \):
\[ z^2 = (x + iy)^2 = x^2 + 2xyi - y^2 = (x^2 - y^2) + 2xyi \]

### Step 2: Calculate the modulus of \( z^2 \)
Next, we find the modulus of \( z^2 \):
\[ |z^2| = |(x^2 - y^2) + 2xyi| = \sqrt{(x^2 - y^2)^2 + (2xy)^2} \]
Calculating the expression inside the square root:
\[ |z^2| = \sqrt{(x^2 - y^2)^2 + 4x^2y^2} \]

### Step 3: Calculate \( |z|^2 \)
Now, we calculate \( |z|^2 \):
\[ |z| = |x + iy| = \sqrt{x^2 + y^2} \]
Thus,
\[ |z|^2 = x^2 + y^2 \]

### Step 4: Calculate \( |z|^2 \) squared
Next, we compute \( |z|^2 \) squared:
\[ |z|^4 = (x^2 + y^2)^2 = x^4 + 2x^2y^2 + y^4 \]

### Step 5: Compare \( |z^2| \) and \( |z|^2 \)
Now we need to compare \( |z^2|^2 \) and \( |z|^4 \):
\[ |z^2|^2 = (x^2 - y^2)^2 + 4x^2y^2 \]
Expanding \( |z^2|^2 \):
\[ |z^2|^2 = (x^4 - 2x^2y^2 + y^4) + 4x^2y^2 = x^4 + 2x^2y^2 + y^4 \]
Thus, we find:
\[ |z^2|^2 = |z|^4 \]

### Conclusion
From the calculations, we have shown that:
\[ |z^2|^2 = |z|^4 \]
This leads us to conclude that:
\[ |z^2| = |z|^2 \]
Thus, the statement \( |z^2| \) is equal to \( |z|^2 \) is verified.