If `Z= - bar(Z)` , then which one of the following is correct?

Options

A. A. The real part of z is zero
B. B. The imaginary part of Z is zero
C. C. The real part of Z is equal to imaginary of part
D. D. The sum of real and imaginary parts of Z is Z

Answer

To solve the problem where \( Z = -\overline{Z} \), we will follow these steps:

**Step 1: Define the complex number.**
Let \( Z = x + iy \), where \( x \) is the real part and \( y \) is the imaginary part of the complex number \( Z \).

**Hint:** Remember that a complex number can be expressed in the form \( a + bi \), where \( a \) is the real part and \( b \) is the imaginary part.

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**Step 2: Write the conjugate of \( Z \).**
The conjugate of \( Z \) is given by \( \overline{Z} = x - iy \).

**Hint:** The conjugate of a complex number changes the sign of the imaginary part.

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**Step 3: Set up the equation.**
According to the problem, we have:
\[ Z = -\overline{Z} \]
Substituting the expressions for \( Z \) and \( \overline{Z} \), we get:
\[ x + iy = - (x - iy) \]

**Hint:** Make sure to distribute the negative sign correctly when substituting.

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**Step 4: Simplify the equation.**
This simplifies to:
\[ x + iy = -x + iy \]

**Hint:** Combine like terms carefully to isolate real and imaginary parts.

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**Step 5: Equate the real and imaginary parts.**
From the equation \( x + iy = -x + iy \), we can equate the real parts and the imaginary parts:
1. Real part: \( x = -x \)
2. Imaginary part: \( iy = iy \) (which is always true)

**Hint:** When equating parts, remember that the imaginary parts must be equal, but they do not provide new information in this case.

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**Step 6: Solve for \( x \).**
From the equation \( x = -x \), we can add \( x \) to both sides:
\[ 2x = 0 \]
Thus, we find:
\[ x = 0 \]

**Hint:** If you have an equation like \( a = -a \), adding \( a \) to both sides is a good way to solve it.

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**Step 7: Conclusion about \( Z \).**
Since \( x \) (the real part of \( Z \)) is 0, we conclude that the real part of \( Z \) is 0.

**Final Answer:** The correct option is that the real part of \( Z \) is 0.

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