Question

i) iota/1+iota
ii) 1-iota/1+iota​

Answer 1

Question:

Express the following numbers in the form a + ib, a, b ∈ R:

i) $\displaystyle{\sf\:\dfrac{i}{1\:+\:i}}$

ii) $\displaystyle{\sf\:\dfrac{1\:-\:i}{1\:+\:i}}$

Answer:

i) $\displaystyle{\boxed{\red{\sf\:\dfrac{i}{1\:+\:i}\:=\:\dfrac{1}{2}\:+\:\dfrac{1}{2}\:i}}}$

ii) $\displaystyle{\boxed{\pink{\sf\:\dfrac{1\:-\:i}{1\:+\:i}\:=\:0\:+\:(\:-\:1\:)\:i}}}$

Step-by-step-explanation:

i)

$\displaystyle{\sf\:\dfrac{i}{1\:+\:i}}$

By multiplying and dividing by 1 - i, we get,

$\displaystyle{\implies\sf\:\dfrac{i}{1\:+\:i}\:\times\:\dfrac{1\:-\:i}{1\:-\:i}}$

$\displaystyle{\implies\sf\:\dfrac{i\:(\:1\:-\:i\:)}{(\:1\:+\:i\:)\:(\:1\:-\:i\:)}}$

$\displaystyle{\implies\sf\:\dfrac{i\:-\:i^2}{1^2\:-\:i^2}}$

$\displaystyle{\implies\sf\:\dfrac{i\:-\:(\:-\:1\:)}{1\:-\:(\:-\:1\:)}}$

$\displaystyle{\implies\sf\:\dfrac{i\:+\:1}{1\:+\:1}}$

$\displaystyle{\implies\sf\:\dfrac{i\:+\:1}{2}}$

$\displaystyle{\implies\sf\:\dfrac{i}{2}\:+\:\dfrac{1}{2}}$

$\displaystyle{\implies\sf\:\dfrac{1}{2}\:+\:\dfrac{1}{2}\:i}$

$\displaystyle{\therefore\:\underline{\boxed{\red{\sf\:\dfrac{i}{1\:+\:i}\:=\:\dfrac{1}{2}\:+\:\dfrac{1}{2}\:i}}}}$

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ii)

Now,

$\displaystyle{\sf\:\dfrac{1\:-\:i}{1\:+\:i}}$

By multiplying and dividing by 1 - i, we get,

$\displaystyle{\implies\sf\:\dfrac{1\:-\:i}{1\:+\:i}\:\times\:\dfrac{1\:-\:i}{1\:-\:i}}$

$\displaystyle{\implies\sf\:\dfrac{(\:1\:-\:i\:)\:(\:1\:-\:i\:)}{(\:1\:+\:i\:)\:(\:1\:-\:i\:)}}$

$\displaystyle{\implies\sf\:\dfrac{(\:1\:-\:i\:)^2}{1^2\:-\:i^2}}$

$\displaystyle{\implies\sf\:\dfrac{1^2\:-\:2\:\times\:1\:\times\:i\:+\:i^2}{1\:-\:(\:-\:1\:)}}$

$\displaystyle{\implies\sf\:\dfrac{1\:-\:2i\:+\:(\:-\:1\:)}{1\:+\:1}}$

$\displaystyle{\implies\sf\:\dfrac{1\:-\:2i\:-\:1}{2}}$

$\displaystyle{\implies\sf\:\dfrac{-\:\cancel{2}\:i}{\cancel{2}}}$

$\displaystyle{\implies\sf\:-\:i}$

$\displaystyle{\implies\sf\:0\:+\:(\:-\:1\:)\:i}$

$\displaystyle{\therefore\:\underline{\boxed{\pink{\sf\:\dfrac{1\:-\:i}{1\:+\:i}\:=\:0\:+\:(\:-\:1\:)\:i}}}}$

Answer 2

Answer:

Required Answer:

$\sf{ i)\boxed{ \red{ \frac{i}{1 + i} }}}$

$\sf{ i)\boxed{ \red{ \frac{1 - i}{1 + i} }}}$

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i.

First, multiplying and dividing by 1 - i , we get ,

$\tt{ \implies \: \frac{i}{1 + i} \times \frac{1 - 1}{1 - i} }$

$\tt{ \implies \: \frac{i(1 - i)}{(1 + i)(1 - i)} }$

$\tt{ \implies \: \frac{i - {i}^{2} }{ {1}^{2} - {i}^{2} } }$

$\tt{ \implies \: \frac{i - ( - 1)}{1 - ( - 1)} }$

$\tt{ \implies \: \frac{i + 1}{1 + 1} }$

$\tt{ \implies \: \frac{i + 1}{2} }$

$\tt{ \implies \: \frac{i}{2} + \frac{1}{2} }$

$\tt{ \implies \: \frac{1}{2} + \frac{1}{2} i}$