The conjugate of a complex number is -1/(i - 1). Then the complex number is

• -1/(i - 1)
• -1/(i + 1)
• -1/(i - 1)
• 1/(i + 1)

To find the complex number given its conjugate, let's start with the conjugate itself, which is represented as:

Understanding the Conjugate

The conjugate of a complex number is formed by changing the sign of the imaginary part. In this case, the conjugate is:

Conjugate = -1/(i - 1)

Finding the Complex Number

The complex number can be derived from the conjugate. We need to simplify the expression:

Complex Number = -1/(i - 1) - 1/(i + 1) - 1/(i - 1) + 1/(i + 1)

Simplifying the Expression

• First, combine like terms:
• Notice that -1/(i - 1) appears twice, and 1/(i + 1) appears once.

Now, let's simplify:

Complex Number = -2/(i - 1) + 2/(i + 1)

Finding a Common Denominator

The common denominator for the fractions is:

(i - 1)(i + 1)

Combining the Fractions

Now, rewrite the fractions with the common denominator:

Complex Number = [-2(i + 1) + 2(i - 1)] / [(i - 1)(i + 1)]

Final Simplification

Expanding the numerator:

Complex Number = [-2i - 2 + 2i - 2] / [(i - 1)(i + 1)]

This simplifies to:

Complex Number = [0 - 4] / [(i - 1)(i + 1)]

Thus, the complex number is:

Complex Number = -4 / [(i - 1)(i + 1)]

Conclusion

In summary, the complex number derived from the given conjugate is:

-4 / [(i - 1)(i + 1)]