Question

In complex numbers \[z\mathop z\limits^\_ = 0\] if and only if
(A) $\operatorname{Re} \left( z \right) = 0$
(B) $z = 0$
(C) $\operatorname{Im} (z) = 0$
(D) $\operatorname{Re} (z) = \operatorname{Im} (z)$

Answer 1

Hint:To solve this problem we have to assume $z = a + ib$ and after that we put this in \[z\mathop z\limits^\_ = 0\] and solve them.In $z = a + ib$, Real part of $z$ is $a$ and imaginary part of $z$ is $b$ And $\left| z \right| = \sqrt {{a^2} + {b^2}} $.Using these definitions and formulas we try to get the answer.

Complete step-by-step answer:
We assume that $z = a + ib$
And modulus of $z$
$\left| z \right| = \sqrt {{a^2} + {b^2}} $
So now given \[z\mathop z\limits^\_ = 0\]
As we know that \[z\mathop z\limits^\_ = {\left| z \right|^2}\]
Now so from above given equation we can write
\[z\mathop z\limits^\_ = {\left| z \right|^2} = 0\]
$\because \left| z \right| = \sqrt {{a^2} + {b^2}} $
So \[{\left| z \right|^2} = {a^2} + {b^2}\]
And we know that
\[{\left| z \right|^2} = 0\]
So \[{\left| z \right|^2} = {a^2} + {b^2} = 0\]
This condition is possible only when $a = 0$ and $b = 0$
So from this we can say $z = 0 + i0$
So from this we say that
 $\operatorname{Re} \left( z \right) = 0$ as well as $\operatorname{Im} (z) = 0$
And $z = 0$ because the real part and imaginary part are both zero.
Now as we see $\operatorname{Re} \left( z \right) = 0$ as well as $\operatorname{Im} (z) = 0$
From this we can say $\operatorname{Re} (z) = \operatorname{Im} (z) = 0$
So all four options are the correct answer.

So, the correct answer is “All options”.

Note:A complex number $z = x + iy$ is a purely real if its imaginary part is 0, i.e. $\operatorname{Im} (z)$ = 0 and purely imaginary if its real part is 0 i.e. $\operatorname{Re} (z)$ = 0.Two complex numbers ${z_1} = {x_1} + i{y_1}$ and ${z_2} = {x_2} + i{y_2}$ are equal, if ${x_1} = {x_2}$ and ${y_1} = {y_2}$ i.e. $\operatorname{Re} \left( {{z_1}} \right)$ = $\operatorname{Re} ({z_2})$ and $\operatorname{Im} ({z_1})$ = $\operatorname{Im} ({z_2})$.
Order relation “greater than’’ and “less than” are not defined for complex numbers.
Important identities:
1. Additive identity z + 0 = z = 0 + z
             Here, 0 is an additive identity.
2. Multiplicative identity: $z \times 1$ = $z$ = $1 \times z$
3. Conjugate of Complex Number: Let $z = x + iy$, if ‘i’ is replaced by ($ - i$), then said to be conjugate of the complex number z and it is denoted by $\mathop z\limits^\_ $, i.e. $\mathop z\limits^\_ = x - iy$.