Question

How do you write the following quotient in standard form \[\dfrac{{2 + i}}{{2 - i}}\] ?

Answer 1

Hint:In order to convert the above question into standard we have to multiply both denominator and numerator with the complex conjugate of the denominator.

Formula:
 \[{(A + B)^2} = {A^2} + {B^2} + 2 \times A \times B\]
 $ (A + B)(A - B) = {A^2} - {B^2} $

Complete step by step solution:
Given a complex fraction \[\dfrac{{2 + i}}{{2 - i}}\] .let it be z
Here i is the imaginary number

First we’ll find out the complex conjugate of denominator of z i.e. \[2 - i\] which will be \[2 + i\]

Now dividing both numerator and denominator with the complex conjugate of denominator
 \[ = \dfrac{{2 + i}}{{2 - i}} \times \dfrac{{2 + i}}{{2 + i}}\]

Using formula $ (A + B)(A - B) = {A^2} - {B^2} $
 \[ = \dfrac{{{{\left( {2 + i} \right)}^2}}}{{{2^2} - {i^2}}}\]
Using formula \[{(A + B)^2} = {A^2} + {B^2} + 2 \times A \times B\]
 \[ = \dfrac{{4 + 4i + {i^2}}}{{4 - {i^2}}}\]

Replacing $ {i^2}\, $ with $ - 1 $
 \[
= \dfrac{{4 + 4i + ( - 1)}}{{4 - ( - 1)}} \\
= \dfrac{{4 + 4i - 1}}{{4 + 1}} \\
= \dfrac{{3 + 4i}}{5} \\
\]

Converting the above question into standard form by comparing it with standard form $ a + ib $
 \[ = \dfrac{3}{5} + \dfrac{{4i}}{5}\]

Where Real number is \[\dfrac{3}{5}\] and imaginary number is \[\dfrac{{4i}}{5}\]

Therefore ,our required answer is \[\dfrac{3}{5} + \dfrac{{4i}}{5}\]

Note:
1. Real Number: Any number which is available in a number system, for example, positive, negative,
zero, whole number, discerning, unreasonable, parts, and so forth are Real numbers. For instance: 12, - 45, 0, 1/7, 2.8, √5, and so forth, are all the real numbers.

2. A Complex number is a number which are expressed in the form $ a + ib $ where $ ib $ is the imaginary part and $ a $ is the real number .i is generally known by the name iota. \[\]