Browse Questions

# Find the real and imaginary parts of the following complex numbers: $\large\frac{2+5i}{4-3i}$

This is the second part of the multi-part question Q2.

Toolbox:
• If $z=a+ib$ then ,
• $\bar{z}=a-ib$
• $\mid z\mid=\sqrt{a^2+b^2}$
• $z^{-1}=\large\frac{a-ib}{a^2+b^2}$
• $z\bar{z}=a^2+b^2$
• Also $Re(z)=a,Im(z)=b$
• If $z_1=a+ib,z_2=c+id$
• $z_1z_2=(a+ib)(c+id)=(ac-bd)+i(ad+bc)$
• $\mid z_1z_2\mid=\mid z_1\mid\mid z_2\mid$
$z=\large\frac{2+5i}{4-3i}$
$\large\frac{2+5i}{4-3i}=\large\frac{2+5i}{4-3i}\large\frac{4+3i}{4+3i}$
$\quad\quad\;\;=\large\frac{(8-15)+i(20+6)}{16+9}$
$\quad\quad\;\;=\large\frac{-7+26i}{25}$
$Re(z)=\large\frac{-7}{25}$
$Im(z)=\large\frac{26}{25}$