Question

Express the complex number $\frac{1}{(1+i)}$ in a + ib form where a and b are real. Find the value of $a^2+b^2$.

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Answer 1

Solution: Let z = $\frac{1}{1+i}$

Rationalize the given complex numberby multiplying 1 - i to mumerator and denominator

= $\frac{1}{1+i}$ $\times$ $\frac{1-i}{1-i}$

= $\frac{1-i}{1-i^2}$ = $\frac{1-i}{1+1}$ = $\frac{1-i}{2}$

a = $\frac{1}{2}$, b = $\frac{-1}{2}$

So, $a^2$ + $b^2$ = ($\frac{1}{4}$) + ($\frac{1}{4}$) = $\frac{1}{2}$