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 find the relationship between the modulus of a complex number \( z \) and its square. Let's denote the complex number \( z \) as:

\[ z = x + iy \]

where \( x \) and \( y \) are real numbers.

### Step 1: Find the modulus of \( z \)

The modulus of \( z \) is defined as:

\[ |z| = \sqrt{x^2 + y^2} \]

### Step 2: Find the modulus squared

The modulus squared of \( z \) is:

\[ |z|^2 = (|z|)^2 = (\sqrt{x^2 + y^2})^2 = x^2 + y^2 \]

### Step 3: Find \( z^2 \)

Next, we calculate \( z^2 \):

\[ z^2 = (x + iy)^2 = x^2 + 2xyi + (iy)^2 = x^2 - y^2 + 2xyi \]

### Step 4: Find the modulus of \( z^2 \)

Now, we find the modulus of \( z^2 \):

\[ |z^2| = \sqrt{(\text{Real part})^2 + (\text{Imaginary part})^2} \]
\[ |z^2| = \sqrt{(x^2 - y^2)^2 + (2xy)^2} \]

### Step 5: Expand the expression

Expanding the expression gives:

\[ |z^2| = \sqrt{(x^2 - y^2)^2 + 4x^2y^2} \]

Now, we can simplify \( (x^2 - y^2)^2 \):

\[ (x^2 - y^2)^2 = x^4 - 2x^2y^2 + y^4 \]

Thus,

\[ |z^2| = \sqrt{x^4 - 2x^2y^2 + y^4 + 4x^2y^2} = \sqrt{x^4 + 2x^2y^2 + y^4} \]

### Step 6: Recognize the perfect square

Notice that:

\[ x^4 + 2x^2y^2 + y^4 = (x^2 + y^2)^2 \]

So we have:

\[ |z^2| = \sqrt{(x^2 + y^2)^2} = |x^2 + y^2| \]

Since \( x^2 + y^2 \) is non-negative, we can write:

\[ |z^2| = x^2 + y^2 \]

### Step 7: Relate \( |z^2| \) and \( |z|^2 \)

From our earlier steps, we found:

\[ |z|^2 = x^2 + y^2 \]

Thus, we have:

\[ |z^2| = |z|^2 \]

### Conclusion

From the above steps, we conclude that:

\[ |z^2| = |z|^2 \]

### Final Answer

The correct option is that the modulus of \( z^2 \) is equal to the modulus of \( z \) squared.