Browse Questions

# Express the following in the standard form $a + ib$: $\large\frac{2\left ( i-3 \right )}{\left ( 1+i \right )^{2}}$

This is the first part of the Q1 multipart question.

Toolbox:
• $i^2=-1,i^3=-i,i^4=1$.
• In general,$i^{4n-3}=i$,$i^{4n-2}=-1$,$i^{4n}=1$
• If $z=a+ib$ then ,
• $\bar{z}=a-ib$
• $\mid z\mid=\sqrt{a^2+b^2}$
• $z^{-1}=\large\frac{a-ib}{a^2+b^2}$
• $zz^{-1}=a^2+b^2$
• Also $Re(z)=a,Im(z)=b$
$\large\frac{2(i-3)}{(1+i)^2}=\large\frac{2(i-3)}{1+2i+i^2}$
$\qquad\quad=\large\frac{2(i-3)}{1+2i-1}$
$\qquad\quad=\large\frac{i-3}{1}$
$\qquad\quad=\large\frac{(i-3)i}{i^2}$
$\qquad\quad=\large\frac{i^2-3i}{-1}$
$\qquad\quad=\large\frac{-1-3i}{-1}$
$\qquad\quad=1+3i$