Question

Express $2+i/2-i$

in the form a+ib.

Answer 1

$\large \bf \clubs \:  Given :-$

A Complex Number : $\dfrac{2 + i}{2 - i}$

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$\large \bf \clubs \:  Question :-$

To Express the Complex Number in General Form which is $\pmb{a + ib}$

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$\large \bf \clubs \:  Solution :-$

We Have,

$\dfrac{2 + i}{2 - i} \\ \\ = \dfrac{2 + i}{2 - i} \times \dfrac{2 + i}{2 + i} \\ \\ = \dfrac{(2 + i) {}^{2} }{2 {}^{2} - {i}^{2}} \\ \\ = \dfrac{4 + {i}^{2} + 4i }{4 - i {}^{2} } \\ \\ = \dfrac{4 + ( - 1) + 4i}{4 - ( - 1)} \: \: \: \: \: \{ \because \: \pmb { {i}^{2} = - 1} \} \\ \\ = \dfrac{4 - 1 + 4 i}{4 + 1} \\ \\ \dfrac{3+ 4 i}{5} \\ \\ = \dfrac{3}{5} \: + \: \frac{4}{5} i$

$\bold{Which \: is \: in \: the \: For m}\: \pmb{a + ib}$

$\Large\frak{ \text{W}here,}$

$\purple{ \Large \underline {\boxed{{\bf a= \frac{3}{5} \: \: and \: \: b = \frac{4}{5} } }}}$

$\Large\red{\mathfrak{  \text{W}hich \:\:is\:\: the\:\: required} }\\ \LARGE \red{\mathfrak{ \text{ A}nswer.}}$

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Answer 2

Answer:

$\underline{\purple{\ddot{\Mathsdude}}}$

♧♧Answered by Rohith kumar maths dude ;-

♧♧ Given:-

• 2+i/2-i (A complex number)

♧♧ To prove :-

• We should express the complex number in the general form which means a+ib form.

♧♧ Proof:-

We already have that,

= 2+i/2-i

♤Here we should rationalise the equation.

= 2+i/2-i×2-i/2-i

= (2+i)^2/2^2-(i)^2

= 4 +i^2+4i/4-(i)^2

♧♧We already know that,

i^2=-1

Applying on equation,

=4+(-1)+4 (-1)/4-(-1)^2

=4-1+4i/ 4+1

= 3+4i/5.

=3/5+4/5i

♧♧Now it is in the form of a+ib

♧♧Where,

●a=3/5 and b =4/5i

♧♧It is the required answer.

♧♧Hope it helps u mate.

♧♧Thank you .