Key Concepts and Formulas

• Conjugate of a Complex Number: If $z = a + bi$, where $a$ and $b$ are real numbers, then its conjugate is $\bar{z} = a - bi$.
• Conjugate of a Conjugate: $\overline{(\bar{z})} = z$
• Conjugate of a Quotient: $\overline{\left(\frac{z_1}{z_2}\right)} = \frac{\overline{z_1}}{\overline{z_2}}$

Step-by-Step Solution

Step 1: Express the Unknown Complex Number $z$ in terms of its Conjugate

Why this step? We are given $\bar{z}$ and need to find $z$. Using the property that the conjugate of a conjugate is the original number, we can directly relate $z$ to the given $\bar{z}$.

Using the property $z = \overline{(\bar{z})}$, we substitute the given expression for $\bar{z}$: $z = \overline{\left(\frac{1}{i - 1}\right)}$

Step 2: Calculate the Conjugate of the Given Complex Expression

Why this step? We need to evaluate the conjugate of $\frac{1}{i-1}$. We use the property that the conjugate of a quotient is the quotient of the conjugates.

Applying the property $\overline{\left(\frac{z_1}{z_2}\right)} = \frac{\overline{z_1}}{\overline{z_2}}$: $z = \frac{\overline{1}}{\overline{i - 1}}$

Now, let's find the conjugate of the numerator and the denominator separately:

• Conjugate of the Numerator: The numerator is $1$, which is a real number. The conjugate of a real number is itself. So, $\overline{1} = 1$.
• Conjugate of the Denominator: The denominator is $i - 1$, which can be written as $-1 + i$. The conjugate is formed by changing the sign of the imaginary part, resulting in $-1 - i$. So, $\overline{i - 1} = -1 - i$.

Substitute these conjugates back into the expression for $z$: $z = \frac{1}{-1 - i}$

Step 3: Simplify the Complex Number by Rationalizing the Denominator

Why this step? To express the complex number in standard form $a + bi$, we need to rationalize the denominator by multiplying the numerator and denominator by the conjugate of the denominator.

The denominator is $-1 - i$. Its conjugate is $-1 + i$. Multiply the numerator and denominator of $z = \frac{1}{-1 - i}$ by $\frac{-1 + i}{-1 + i}$: $z = \frac{1}{-1 - i} \times \frac{-1 + i}{-1 + i}$

Now, perform the multiplication:

• Numerator: $1 \times (-1 + i) = -1 + i$
• Denominator: Use the difference of squares identity $(a - b)(a + b) = a^2 - b^2$. Here, $a = -1$ and $b = i$. $(-1 - i)(-1 + i) = (-1)^2 - (i)^2 = 1 - (-1) = 1 + 1 = 2$

So, the expression for $z$ becomes: $z = \frac{-1 + i}{2}$

Finally, separate the real and imaginary parts to express $z$ in the standard $a + bi$ form: $z = -\frac{1}{2} + \frac{1}{2}i$

Step 4: Compare the Result with the Given Options

Why this step? We need to identify which of the given options matches our simplified expression for $z$, which is $z = -\frac{1}{2} + \frac{1}{2}i$.

Let's simplify option (A):

• (A) ${{ - 1} \over {i - 1}}$: Multiply the numerator and denominator by the conjugate of $i-1$, which is $-1-i$: $\frac{-1}{i-1} = \frac{-1}{-1+i} \times \frac{-1-i}{-1-i} = \frac{1+i}{(-1)^2 - (i)^2} = \frac{1+i}{1-(-1)} = \frac{1+i}{2} = \frac{1}{2} + \frac{1}{2}i$

This does NOT match our answer. Let's try option (C), which was incorrectly identified as the answer previously:

• (C) ${{ - 1} \over {i + 1}}$: Multiply the numerator and denominator by the conjugate of $i+1$, which is $1-i$: $\frac{-1}{i+1} = \frac{-1}{1+i} \times \frac{1-i}{1-i} = \frac{-1+i}{(1)^2 - (i)^2} = \frac{-1+i}{1-(-1)} = \frac{-1+i}{2} = -\frac{1}{2} + \frac{1}{2}i$

This matches our calculated value for $z$.

Common Mistakes & Tips

• Be careful with signs when taking conjugates and rationalizing denominators. A small sign error can lead to an incorrect answer.
• Remember to multiply both the numerator and the denominator by the conjugate of the denominator when rationalizing.
• Don't forget that $i^2 = -1$.

Summary

We are given the conjugate of a complex number and asked to find the complex number itself. We used the property that taking the conjugate twice returns the original number. We then rationalized the denominator to express the complex number in the standard $a+bi$ form and compared it with the given options. The correct answer is $-\frac{1}{2} + \frac{1}{2}i$, which corresponds to option (C).

The final answer is \boxed{-\frac{1}{2} + \frac{1}{2}i}, which corresponds to option (C).

I apologize for the error in the previous response. The correct answer is indeed option (C).