Question

The conjugate of a complex number is \[\dfrac{1}{{i - 1}}\]. Then that complex number will be
A. \[\dfrac{1}{{(i - 1)}}\]
B. \[\dfrac{{ - 1}}{{(i - 1)}}\]
C. \[\dfrac{1}{{(i + 1)}}\]
D. \[\dfrac{{ - 1}}{{(i + 1)}}\]

Answer 1

Hint:A Complex Number means just by adding two numbers together (a Real and an Imaginary Number). The conjugate of the complex number means that only the sign of the imaginary part differs. For the given complex number is in the form \[z = x + iy\], then the conjugate of that number will be \[\overline z = x - iy\]. Here, if you take the conjugate number as a complex number and then solve it, you will still get the correct answer.

Complete step by step answer:
Given the conjugate of the complex number as below,
\[\overline z = \dfrac{1}{{i - 1}}\]
Multiply by \[i + 1\]in both numerator and denominator to remove the imaginary part from the denominator.
\[\overline z = \dfrac{1}{{i - 1}} \times \dfrac{{i + 1}}{{i + 1}}\]
\[\Rightarrow \overline z = \dfrac{{1(i + 1)}}{{(i - 1)(i + 1)}}\]
Removing the brackets, we will get,
\[\overline z = \dfrac{{i + 1}}{{{i^2} - {1^2}}}\]
We will use this known formula: \[(a - b)(a + b) = {a^2} - {b^2}\] and also we know that \[{i^2} = ( - 1)\].
\[\overline z = \dfrac{{i + 1}}{{ - 1 - 1}}\]
\[\Rightarrow \overline z = \dfrac{{i + 1}}{{ - 2}}\]
Thus, \[\overline z = \dfrac{{i + 1}}{{ - 2}}\]

Now, we will find the complex number of the given conjugate number.So, the complex number of z is as below:
\[\overline z = \dfrac{{ - i + 1}}{{ - 2}}\]
\[\Rightarrow \overline z = \dfrac{{ - i + 1}}{{ - 2}} \times \dfrac{{ - i - 1}}{{ - i - 1}}\]
\[\overline z = \dfrac{{( - i - 1)( - i - 1)}}{{2( - i - 1)}}\]
We will this known formula: \[(a - b)(a + b) = {a^2} - {b^2}\]
\[\overline z = \dfrac{{{{( - i)}^2} - {1^2}}}{{ - 2( - i - 1)}}\]
And also we know that\[{i^2} = ( - 1)\]
\[\overline z = \dfrac{{ - 1 - 1}}{{ - 2( - i - 1)}}\]
\[\Rightarrow \overline z = \dfrac{{ - 2}}{{ - 2( - i - 1)}}\]
\[\Rightarrow \overline z = \dfrac{1}{{ - i - 1}}\]
\[\Rightarrow \overline z = \dfrac{1}{{ - (i + 1)}}\]
\[\Rightarrow \overline z = \dfrac{{ - 1}}{{i + 1}} \\\]
Other method is very simple and easy too, which is shown below.
Given the conjugate of a complex number as below:
\[\overline z = \dfrac{1}{{i - 1}}\]
Rearrange the denominator, we will get,
\[\overline z = \dfrac{1}{{ - 1 + i}}\]
The complex number will be,
\[\Rightarrow z = \dfrac{1}{{ - 1 - i}}\]
\[\Rightarrow z = \dfrac{1}{{ - (1 + i)}}\]
\[\Rightarrow z= \dfrac{{ - 1}}{{1 + i}} \\
\therefore z= \dfrac{{ - 1}}{{i + 1}} \]

Hence, the correct answer is option D.

Note:The main difference between a complex number and its conjugate is the sign of an imaginary number only. The complex number is of the form a+ib, where real number a is called the real part and the real number b is called the imaginary part. Here they have given a conjugate of a complex number and we need to find the complex number. You can solve by taking the given conjugate number as a complex number and solve accordingly. But since you will get the correct answer yet your answer will be counted as wrong. Because the symbols used for both complex numbers and conjugates are different.