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Question: How do you divide \(\dfrac{{4 - 2i}}{{3 + 7i}}\) ?...

How do you divide 4−2i3+7i\dfrac{{4 - 2i}}{{3 + 7i}} ?

Explanation

Solution

In order to solve this sum, we multiply both the numerator and denominator with the conjugate term of the denominator. After that we express the denominator in a2−b2{a^2} - {b^2} form, and expand the numerator. We replace the imaginary number i2{i^2} with −1 - 1 , and simplify further to get our required answer.

Complete step by step solution:
In the given question, we need to divide 4−2i3+7i\dfrac{{4 - 2i}}{{3 + 7i}}, ii here represents an imaginary number: i=−1i = \sqrt { - 1}
A complex number is represented in the form of ‘a+bia + bi’, where ‘a’ is the real number and ‘bi’ is the imaginary number.
In order to divide the given number, we take the conjugate of the denominator. Conjugate means to take a similar complex number but opposite sign before the imaginary number.
Thus, conjugate of the denominator: 3−7i3 - 7i
Let us multiply both the denominator and numerator with the conjugate number:
⇒(4−2i)(3−7i)(3+7i)(3−7i)\Rightarrow \dfrac{{\left( {4 - 2i} \right)\left( {3 - 7i} \right)}}{{\left( {3 + 7i} \right)\left( {3 - 7i} \right)}}
Now we know that (a+b)(a−b)=a2−b2\left( {a + b} \right)\left( {a - b} \right) = {a^2} - {b^2}
Thus, we have:
⇒(4−2i)(3−7i)(32−(7i)2)\Rightarrow \dfrac{{\left( {4 - 2i} \right)\left( {3 - 7i} \right)}}{{\left( {{3^2} - {{\left( {7i} \right)}^2}} \right)}}
Simplifying the numerator and the denominator further, we get:
⇒12−28i−6i+14i29−49i2\Rightarrow \dfrac{{12 - 28i - 6i + 14{i^2}}}{{9 - 49{i^2}}}
Adding the like terms:
⇒12−34i+14i29−49i2\Rightarrow \dfrac{{12 - 34i + 14{i^2}}}{{9 - 49{i^2}}}
Now we know that i=−1i = \sqrt { - 1}
Therefore, i2=−1{i^2} = - 1
Placing this value in our required expression, we get:
⇒12−34i−149+49\Rightarrow \dfrac{{12 - 34i - 14}}{{9 + 49}}
On simplifying further, we get:
⇒−2−34i58\Rightarrow \dfrac{{ - 2 - 34i}}{{58}}
We can also write the above term as:
⇒−258−3458i\Rightarrow - \dfrac{2}{{58}} - \dfrac{{34}}{{58}}i
On reducing the numbers to the simplest form, we get:
⇒−129−1729i\Rightarrow - \dfrac{1}{{29}} - \dfrac{{17}}{{29}}i
Thus, we have our required answer.

Note: A complex number is a number that can be represented as a+bia + bi, where a and b are real numbers, and ii represents the imaginary unit, satisfying the equation i=−1i = - 1. Because no real number satisfies the equation, therefore ii is called an imaginary number. Some properties of complex numbers are:
When a, b, c and d are real numbers and a+b=c+da + b = c + d, then a=ca = c and b=db = d
The sum of two conjugate complex numbers is real. For example, if we have a number as z=a+ibz = a + ib , where aa and bb are real numbers, and the conjugate number z‾=a−ib\overline z = a - ib , then the sum of z+z‾z + \overline z is a real number.
The product of two conjugate complex numbers is real.