Question

How do you divide \[\dfrac{{5 - i}}{{5 + i}}\] ?

Answer 1

Hint: Here, we have to find the quotient of two complex numbers. We will first multiply the numerator and the denominator by the complex conjugate. Then we will use the algebraic identity and Foil method to simplify the expression in numerator and denominator. We will simplify the equation using the basic mathematical operations to get the required answer.

Formula Used:
The difference between the square numbers is given by the algebraic identity \[{a^2} - {b^2} = \left( {a + b} \right)\left( {a - b} \right)\]

Complete Step by Step Solution:
We are given a complex number \[\dfrac{{5 - i}}{{5 + i}}\].
Now, we will multiply the numerator and the denominator by the complex conjugate. Therefore, we get
\[\dfrac{{5 - i}}{{5 + i}} = \dfrac{{5 - i}}{{5 + i}} \times \dfrac{{5 - i}}{{5 - i}}\]
\[ \Rightarrow \dfrac{{5 - i}}{{5 + i}} = \dfrac{{\left( {5 - i} \right)\left( {5 - i} \right)}}{{\left( {5 + i} \right)\left( {5 - i} \right)}}\]
The difference between the square numbers is given by the algebraic identity \[{a^2} - {b^2} = \left( {a + b} \right)\left( {a - b} \right)\]
Now, by multiplying the numerators by using the FOIL method and the denominators by using the algebraic identity, we get
\[ \Rightarrow \dfrac{{5 - i}}{{5 + i}} = \dfrac{{5\left( {5 - i} \right) - i\left( {5 - i} \right)}}{{{5^2} - {i^2}}}\]
Now, by multiplying each term in the expression, we get
\[ \Rightarrow \dfrac{{5 - i}}{{5 + i}} = \dfrac{{25 - 5i - 5i + {i^2}}}{{{5^2} - {i^2}}}\]
Adding and subtracting the like terms, we get
\[ \Rightarrow \dfrac{{5 - i}}{{5 + i}} = \dfrac{{25 - 10i + {i^2}}}{{25 - {i^2}}}\]
We know that \[{i^2} = - 1\]. So, we get
\[ \Rightarrow \dfrac{{5 - i}}{{5 + i}} = \dfrac{{25 - 10i + \left( { - 1} \right)}}{{25 - \left( { - 1} \right)}}\]
Again adding and subtracting the terms, we get
\[ \Rightarrow \dfrac{{5 - i}}{{5 + i}} = \dfrac{{24 - 10i}}{{26}}\]
Rewriting the expression, we get
\[ \Rightarrow \dfrac{{5 - i}}{{5 + i}} = \dfrac{{24}}{{26}} - \dfrac{{10i}}{{26}}\]
By simplifying the expression, we get
\[ \Rightarrow \dfrac{{5 - i}}{{5 + i}} = \dfrac{{12}}{{13}} - \dfrac{{5i}}{{13}}\]

Therefore, the quotient of the complex number \[\dfrac{{5 - i}}{{5 + i}}\] is\[\dfrac{{12}}{{13}} - \dfrac{{5i}}{{13}}\].

Note:
Here we need to keep in mind that the complex conjugate that is multiplied to both the numerators and the denominators should be the conjugate of the denominator. Also, if we are multiplying the complex conjugate, the conjugate should be taken only for the complex term. FOIL method is a method of multiplying the binomials by multiplying the first terms, then the outer terms, then the inner terms and at last the last terms. Thus, the product of two binomials is a trinomial.