Question 1, Exercise 1.2

Solutions of Question 1 of Exercise 1.2 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan.

Show that for any complex number, $\operatorname{Re}(i z)=-\operatorname{Im}(z)$.

Solution. Suppose $$z=x+iy$$ $$\begin{align} iz&=i(x+iy)\\ &=ix-y\end{align}$$ Now $$\begin{align}Re(iz)&=-y\\ \implies Re(iz)&=-Im(z)\end{align}$$

Show that for any complex number, $\operatorname{Im}(i z)=\operatorname{Re}(z)$.

Solution. Suppose $$z=x+iy$$ $$\begin{align}iz&=i(x+iy)\\ &=ix-y\end{align}$$ Now $$\begin{align}Im(iz)&=x\\ \implies Im(iz)&=Re(z)\end{align}$$

Question 2>