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

This is the second part of the multi part Q1.

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 ,
• $\bar{z}=a-ib$
• $\mid z\mid=\sqrt{a^2+b^2}$
• $z^{-1}=\large\frac{a-ib}{a^2+b^2}$
• Also $Re(z)=a,Im(z)=b$
Step 1:
$\large\frac{(1+i)(1-2i)}{1+3i}=\large\frac{1+i-2i-2i^2}{1+3i}$
$\qquad\qquad\quad=\large\frac{1-i+2}{1+3i}$
$\qquad\qquad\quad=\large\frac{3-i}{1+3i}$
Step 2:
Multiply $\large\frac{1-3i}{1-3i}$ on RHS we get
$\qquad\qquad\quad=\large\frac{3-i}{1+3i}\frac{1-3i}{1-3i}$
$\qquad\qquad\quad=\large\frac{3-9i-i+3i^2}{1+9}$
$\qquad\qquad\quad=\large\frac{3-10i-3}{10}$
$\qquad\qquad\quad=-i$
$\qquad\qquad\quad=0-i$