Question

If \[z=\overline{z}\], then which of the following is correct?
(a) The real part of z is zero.
(b) The imaginary part of z is zero.
(c) The real part of z is equal to the imaginary part of z.
(d) The sum of real and imaginary parts of z is z.

Answer 1

Hint:At first take z as x + iy so \[\overline{z}\] will be x – iy then equate both of them to get that y = 0 and say about them which is an imaginary part of z.

Complete step-by-step answer:
In the question we are given \[z=\overline{z}\], we have to comment on the nature of z according to the options given.
Before doing so, we will learn what complex numbers are.
A complex number is a number that can be written in the form of a + bi, where a, b are real numbers and I is a solution of the equation \[{{x}^{2}}=-1\]. This is because no real value satisfies for equation \[{{x}^{2}}+1=0\] or \[{{x}^{2}}=-1\], hence it is called an imaginary number. For the complex number a + ib, a is considered as real part and b as imaginary part.
Despite the historical nomenclature “imaginary”, complex numbers are regarded in the mathematical sciences as just as “real” as real numbers and are fundamental in any aspect of scientific description of the natural world.
Here in the question, \[\overline{z}\] refers to the conjugate of complex number z which means that if z is a complex number a + ib the \[\overline{z}\] will be a – ib.
Now let if z = x + iy where x refers to the real part of complex number and y refers to the imaginary part of complex number.
The value of \[\overline{z}\] will be x – iy.
So we were given that \[z=\overline{z}\], so we can write it as x + iy = x – iy or 2iy = 0 or y = 0. As y was referred to as the imaginary part of z and y we got ‘0’. So, we can conclude that the imaginary part of z is 0.
So, the correct answer is (b).

Note: Students generally confuse themselves while writing the conjugate of a complex number. There is also a property related to a conjugate, that is a complex number is multiplied with its conjugate we will find real to modular value of that complex number.