Question

If $z=\dfrac{4}{1-i}$ then bar z ($\bar{z}$) is equal to (where bar z is complex conjugate of z):
(a)2(1+i)
(b)1+i
(c)$\dfrac{2}{1-i}$
(d)$\dfrac{4}{1+i}$

Answer 1

The value of bar z ($\bar{z}$) is equal to$\bar{z}=x-iy$. Where bar z ($\bar{z}$) is the conjugate value of the complex term which is mentioned in the question as z. We will first assume our z = x + iy, with respect to which we will find the bar z ($\bar{z}$).

Complete step by step solution:
The value of z that has been mentioned the in the question is$z=\dfrac{4}{1-i}$, as we can see that the term that has been mentioned is a complex term as it has term (i) in it and we know that $\text{square root = }\sqrt{\text{positive number}}$, and we also know that square root will be a real number until the number is positive, i.e. $\text{square root = }\sqrt{\text{positive number}}$, and as we can see that the number inside the square root is negative i.e. $\text{square root = }\sqrt{\text{positive number}}$ we can easily say that the square root of this number will have imaginary roots and hence we will have a complex number that is mentioned in the question.
Now when we see the question we can see that the equation that has been mentioned in the question as $z=\dfrac{4}{1-i}$, we can also write it as z = x +iy, and as explained above that bar z ($\bar{z}$) is the conjugate of the complex number z, we can write the complex conjugate bar z ($\bar{z}$) as $\bar{z}=x-iy$
Now as we can see that the value of y in the complex number is negative, we will substitute the same in the complex conjugate of z which is bar z ($\bar{z}$) and we finally get the value of bar z ($\bar{z}$) as $\bar{z}=\dfrac{4}{1+i}$
From the options as stated above in the question we can easily say that option (d) is correct which is $\bar{z}=\dfrac{4}{1+i}$.

Note: The common mistake that is done in these type of questions is to how to calculate the value of bar z ($\bar{z}$), where the value of bar z ($\bar{z}$) is $\bar{z}=x-iy$ whenever the value of z = x +iy as we know that the bar z ($\bar{z}$) is the complex conjugate of z.