Question

Simplify $\left( {2 - 3i} \right) \div \left( {1 + 5i} \right)$?

Answer 1

Hint: As it is given that the expression is in fraction and is of a complex number. First, multiply the expression by the conjugate of the denominator and apply the fact ${i^2} = 1$ to rationalize the denominator. After that simplify the numerator and separate the real and imaginary parts to get the desired result.

Complete step by step answer:
The given expression $\left( {2 - 3i} \right) \div \left( {1 + 5i} \right)$ is the fraction of complex numbers.
First, we'll learn what complex numbers are before doing so.
A complex number is a number that can be written in the form of a+ bi, where a, b are real numbers, and i is a solution to the equation ${x^2} = - 1$. This is because no real value of the equation fulfils ${x^2} + 1 = 0$. Therefore, i is called the imaginary number.
A is known as the real part of the complex number a + ib, and b as the imaginary part. Despite the historical nomenclature, "imaginary" complex numbers are considered as "real" as real numbers in mathematical sciences and are fundamental in any aspect of the natural world's scientific description.
Now multiply the expression with the conjugate of the denominator,
$ \Rightarrow \dfrac{{2 - 3i}}{{1 + 5i}} \times \dfrac{{1 - 5i}}{{1 - 5i}}$
Simplify the terms,
$ \Rightarrow \dfrac{{2 - 3i - 10i + 15{i^2}}}{{1 - 25{i^2}}}$
Substitute ${i^2} = - 1$,
$ \Rightarrow \dfrac{{2 - 3i - 10i - 15}}{{1 + 25}}$
On rearranging and simplifying, we get
$ \Rightarrow \dfrac{{ - 13 - 13i}}{{26}}$
Take 13 commons from the numerator,
$ \Rightarrow \dfrac{{13\left( { - 1 - i} \right)}}{{26}}$
Cancel out the common factors,
$ \Rightarrow \dfrac{{ - 1 - i}}{2}$
Hence, the simplified form is $ - \dfrac{1}{2} - \dfrac{i}{2}$.

Note: There is a rule that complex numbers should not be there in the fraction denominator, so if it is there, we will rationalize it by multiplying it with its conjugate, such as if $a + ib$ is there in the numerator, then we should multiply with $a - ib$.