Question

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

Answer 1

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. $$$$