Question

Consider the following statements.
$\left(i\right)z_1-(z_2+z_3)=(z_1-z_2)+z_3$
$\left(ii\right)$The value of all $i^4,i^6,i^{68}\&i^{222}$ are all integers.
$\left(iii\right)$Adding a complex number $\frac{4}{i}+\frac{3}{i^3}$ to its additive inverse results in the additive identity of that complex number.
$\left(iv\right)$If $z_1=4+2i,z_2=3+7i,$ then $2z_1-z_2=5+3i.$

Which of the above statements is/are NOT correct ?

• A. $\left(i\right)$
• B. $\left(iii\right)$
  
• C. $\left(ii\right)$
• D. $\left(iv\right)$

Answer 1

Answer: (A), (D)

$\left(iv\right)$

$\left(i\right)z_1-(z_2+z_3)=z_1-z_2-z_3=(z_1-z_2)-z_3$ which isn’t equal to $(z_1-z_2)+z_3.$
Therefore, statement $\left(i\right)$ is ruled out.

$\left(ii\right)$ The value of $i^4=1,i^6=i^{4+2}=i^2=-1,i^{68}=i^{4\left(17\right)}=1$ and $i^{222}=i^{4\left(55\right)+2}=i^2=-1.$
Thus, this statement is correct.

$\left(iii\right)$ let's simplify the complex number $\frac{4}{i}+\frac{3}{i^3}$ in the form of $Re\left(z\right)+iIm\left(z\right)$.

So,
$\frac{4}{i}+\frac{3}{i^3}=\frac{4}{i}\times \frac{i}{i}+\frac{3}{i^3}\times \frac{i}{i}=\frac{4i}{i^2}+\frac{3i}{i^4}=\frac{4i}{-1}+\frac{3i}{1}=-i.$

The additive inverse of this number will be $i$.
Adding $-i$ and $i$ , we get $0$ or $0i$, which is nothing but the additive identity of complex numbers. So this statement is also correct.

$\left(iv\right)z_1=4+2i$
$z_2=3+7i$
$2z_1-z_2=5-3i\ne 5+3i$
Therefore this is also incorrect.

Therefore we can say that statement $\left(ii\right)$ and statement $\left(iii\right)$ are correct.