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Find the real and imaginary parts of the following complex numbers: $\large\frac{1}{1+i}$

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

1 Answer

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 $z_1=a+ib,z_2=c+id$
  • $z_1z_2=(a+ib)(c+id)=(ac-bd)+i(ad+bc)$
  • $\mid z_1z_2\mid=\mid z_1\mid\mid z_2\mid$
$z=\large\frac{1}{1+i}$
$\large\frac{1}{1+i}$$=(1+i)^{-1}$
$\quad\quad=\large\frac{1-i}{1+1}$
$\quad\quad=\large\frac{1}{2}-\large\frac{i}{2}$
$Re(z)=\large\frac{1}{2}$
$Im(z)=\large\frac{-1}{2}$
answered Jun 7, 2013 by sreemathi.v
edited Jun 7, 2013 by sreemathi.v
 

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