Question

If z = x+iy and z1 is the complex conjugate of z, then the imaginary part of the complex function f(z) =z + z1 is
Select one:
O a.o
O
b. 2x
O
c. 2y
0
O
d. 2y
Activat​

Answer 1

Complex Numbers.

Definition of Complex Numbers:

A complex number z is defined to be an orderee pair of real numbers (a, b) that satisfies the following condition (i) and the following laws of operations (ii) and (iii)

• (i) (a, b) = (c, d) iff a = c, b = d
• (ii) (a, b) + (c, d) = (a + c, b + d)
• (iii) (a, b) . (c, d) = (ac - bd, ad + bc)

Conjugate of a Complex Number:

Let z = a + bi be a complex number. The conjugate of z, denoted by z1, is defined to be the complex number a - bi.

Now we move to solve the given problem:

Here the given imaginary number is

z = x + iy

Given that z1 is the complex conjugate of z. Then z1 = x - iy

Now, f (z) = z + z1

= x + iy + x - iy

= 2x

i.e., f (z) = 2x, a real value.

We write: f (z) = 2x + i.0

This shows that the imaginary part is 0.

Option (a) = 0 is correct.

Answer 2

The imaginary part of the complex function f(z) = z + z1 is a. 0

Step-by-step explanation:

A complex number 'z' is given as:

z = x + iy

The conjugate of the complex number 'z' is given as:

z1 = x - iy

Now, the complex function is:

f(z) = z + z1

f(z) = x + iy + x - iy

f(z) = 2x + 0

∴ f(z) = 2x + i0

In the above function, the real part is '2' and the imaginary part is '0'.