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

Hint: 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.