(iii) i1+i\frac{i}{1 + i}1+ii Solution Multiply numerator and denominator by conjugate (1−i):i1+i⋅1−i1−i=i(1−i)(1+i)(1−i)Simplify numerator:i(1−i)=i−i2=i−(−1)=1+iSimplify denominator:(1+i)(1−i)=12+12=2Write in a+ib form:1+i2=12+12i\begin{aligned} & \boxed{\text{Multiply numerator and denominator by conjugate } (1-i):} \\ \\ & \frac{i}{1+i} \cdot \frac{1-i}{1-i} = \frac{i(1-i)}{(1+i)(1-i)} \\ \\ & \boxed{\text{Simplify numerator:}} \\ \\ & i(1-i)=i-i^2=i-(-1)=1+i \\ \\ & \boxed{\text{Simplify denominator:}} \\ \\ & (1+i)(1-i)=1^2+1^2=2 \\ \\ & \boxed{\text{Write in } a+ib \text{ form:}} \\ \\ & \frac{1+i}{2} = \frac{1}{2} + \frac{1}{2}i \end{aligned}Multiply numerator and denominator by conjugate (1−i):1+ii⋅1−i1−i=(1+i)(1−i)i(1−i)Simplify numerator:i(1−i)=i−i2=i−(−1)=1+iSimplify denominator:(1+i)(1−i)=12+12=2Write in a+ib form:21+i=21+21i