Question

If $z$ is a complex number such that Re$\left( z \right)$ = Im$\left( z \right)$, then
A) Re$\left( {{z^2}} \right) = 0$
B) Im$\left( {{z^2}} \right) = 0$
C) Re$\left( {{z^2}} \right) = 0$ = Im$\left( {{z^2}} \right)$
D) Re$\left( {{z^2}} \right) = -$Im$\left( {{z^2}} \right)$

Answer 1

We will consider a complex number. Using the given information, we will find the relation between the real and imaginary part of the given complex number.
Then, we will find the value of ${z^2}$.
Using the relation between the real and imaginary part we can find the value of the real and imaginary part of ${z^2}$. Hence, we will find the correct option.

Complete step-by-step answer:
It is given that; $z$ is a complex number such that Re$\left( z \right)$ = Im$\left( z \right)$
We have to find the correct option among the given options.
Let us consider,$z = x + iy$ where $x,y$ are real numbers.
Since, Re$\left( z \right)$ = Im$\left( z \right)$
So, we have, $x = y$
Now, ${z^2} = {(x + iy)^2}$
Applying the formula, we get,
$= {x^2} + {i^2}{y^2} + 2ixy$
We know that, ${i^2} = - 1$
Simplifying we get,
$= {x^2} - {y^2} + 2ixy$
Since, $x = y \Rightarrow {x^2} = {y^2}$
Substitute the values we get,
Simplifying again we get,
$= 2i{x^2}$
So, we have,
Re$\left( {{z^2}} \right) = 0$ and Im$\left( {{z^2}} \right) = 2{x^2}$

Hence, the correct option is A) Re$\left( {{z^2}} \right) = 0$

Note: Imaginary numbers are the numbers when squared it gives the negative result.
In other words, imaginary numbers are defined as the square root of the negative numbers where it does not have a definite value. It is mostly written in the form of real numbers multiplied by the imaginary unit called $i$.
Complex numbers are the combination of both real numbers and imaginary numbers. The complex number is of the standard form: $a + ib$
Where
$a$ and $b$ are real numbers.
$i$ is an imaginary unit.