The conjugate of a complex number is 1/(i-1) then that complex number is (A) -1/(i-1) (B) 1/(i+1) (C

Q: Wait, why can't I just multiply by the conjugate of the denominator first to simplify $\bar{z}$?

You absolutely can! You would get $\bar{z} = \frac{1}{i-1} \cdot \frac{-i-1}{-i-1} = \frac{-i-1}{1+1} = \frac{-i-1}{2}$. Then, to find $z$, you would take the conjugate of that result. Both methods are mathematically sound, but using the property $\overline{(\frac{z_1}{z_2})} = \frac{\bar{z_1}}{\bar{z_2}}$ is often faster and less prone to arithmetic errors.

Q: Does the conjugate of $i-1$ become $i+1$ or $-i-1$?

Great question! The conjugate is defined by changing the sign of the imaginary part only. In $i-1$, the imaginary part is $i$ (or $1i$). Changing its sign gives $-i$. The real part, $-1$, remains unchanged. Thus, the conjugate is $-i-1$.

Q: Why is the conjugate of 1 just 1?

A real number $a$ can be written as $a + 0i$. Its conjugate is $a - 0i$, which is still just $a$. Since 1 is a purely real number, its conjugate is indeed 1.

Q: Is there a difference between $-\frac{1}{i+1}$ and $\frac{-1}{i+1}$?

Not at all! They are identical. In algebra, the negative sign can be placed in front of the fraction, in the numerator, or in the denominator. We usually place it in front or in the numerator for clarity.

Q: I keep getting confused with the signs. How do I stay organized?

The best way is to write out the real and imaginary parts explicitly before applying the conjugate. If you see $i-1$, rewrite it as $-1 + i$. Now, the conjugate is clearly $-1 - i$. This small step prevents the sign errors that often plague students in the heat of an exam.

Question

The conjugate of a complex number is $\frac{1}{i-1}$ then that complex number is

Options:

(A) $-1/(i-1)$

(B) $1/(i+1)$

(C) $-1/(i+1)$

(D) $1/(i-1)$

Sigma Percentile · Accepted Answer

### Analyzing the Setup

To find the original complex number $z$ given its conjugate $\bar{z} = \frac{1}{i-1}$, we must utilize the fundamental properties of complex conjugation. The most critical property here is the involution property of conjugates, which states that the conjugate of a conjugate returns the original value:

$$\overline{\bar{z}} = z$$

Therefore, to recover $z$, we simply apply the conjugate operation to the given expression:

$$z = \overline{\left( \frac{1}{i-1} \right)}$$

### The Power of Distribution

We utilize the quotient property of complex conjugates, which allows the bar to distribute over division:

$$\overline{\left( \frac{z_1}{z_2} \right)} = \frac{\bar{z_1}}{\bar{z_2}}$$

Applying this to our expression, we obtain:

$$z = \frac{\bar{1}}{\overline{i-1}}$$

Since $1$ is a purely real number, its conjugate remains unchanged, so $\bar{1} = 1$. For the denominator, we identify the imaginary part of $i-1$ as $i$ and negate it to find the conjugate:

$$\overline{i-1} = -i-1$$

### The Final Simplification

Substituting these values back into our equation, we arrive at:

$$z = \frac{1}{-i-1}$$

To express this in a more standard form, we factor out the negative sign from the denominator:

$$z = \frac{1}{-(i+1)}$$

By moving the negative sign to the front, we reach our **final result**:

$$z = -\frac{1}{i+1}$$

### Why This Matters

By leveraging the algebraic properties of conjugates, we bypassed the need for immediate rationalization or complex arithmetic. In the context of the JEE, recognizing these structural symmetries is essential for maintaining speed and accuracy.

**Mastering these properties** allows you to decompose complex expressions into manageable components, ensuring that you spend your time on logic rather than tedious calculation. Keep this intuition sharp, as it is the key to unlocking more advanced problems in the complex plane.

Animated Solution for Mathematics - Complex Numbers: The conjugate of a complex number is $\frac{1}{i - 1}$ then that complex number is

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Visualized Solution

Given $\overset{z}{ˉ} = \frac{1}{i - 1}$

Property: $\overline{\overset{z}{ˉ}} = z$

Applying the Conjugate

Conjugate of a Quotient

Distributing the Bar

Conjugate of $i - 1$

Simplifying the Signs

Final Conclusion

The Sigma Insight: Algebraic Operations on Complex Numbers

Analyzing the Setup

The Power of Distribution

The Final Simplification

Why This Matters

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Animated Solution for Mathematics - Complex Numbers: .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The conjugate of a complex number is i−11​ then that complex number is

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Visualized Solution

Given .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } zˉ=i−11​

Property: .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } zˉ=z