Question

How do you simplify $\dfrac{{2 + 3i}}{{1 + 2i}}$?

Answer 1

Hint:
In this question we are asked to simplify the expression which is of complex numbers, we will do this by multiplying and dividing the expression with the conjugate of the denominator, i.e, here it will be $1 - 2i$, and further multiplication and simplification the expression we will get the required result. And we have to remember that ${i^2} = - 1$, then combine all like terms i.e., combine real numbers with real numbers and imaginary numbers with imaginary numbers.

Complete step by step solution:
Given expression is $\dfrac{{2 + 3i}}{{1 + 2i}}$,
Now multiplying and dividing the expression with the conjugate of the denominator, so here the conjugate of the denominator$1 + 2i$will be $1 - 2i$, we get,
$ \Rightarrow \dfrac{{2 + 3i}}{{1 + 2i}} \times \dfrac{{1 - 2i}}{{1 - 2i}}$,
Now multiplying we get,
$ \Rightarrow \dfrac{{\left( {2 + 3i} \right)\left( {1 - 2i} \right)}}{{\left( {1 + 2i} \right)\left( {1 - 2i} \right)}}$,
Now multiplying the terms in the numerator, and applying the identity $\left( {a - b} \right)\left( {a + b} \right) = {a^2} - {b^2}$, we get,
$ \Rightarrow \dfrac{{2 \times 1 + 2 \times \left( { - 2i} \right) + \left( {3i} \right)\left( 1 \right) + \left( {3i} \right)\left( { - 2i} \right)}}{{\left( {{1^2}} \right) - {{\left( { - 2i} \right)}^2}}}$,
Now simplifying we get,
$ \Rightarrow \dfrac{{2 - 4i + 3i - 6{i^2}}}{{1 - \left( {4{i^2}} \right)}}$,
Now combining like terms we get and we know that ${i^2} = - 1$, substituting the value in the expression, we get,
$ \Rightarrow \dfrac{{2 - i + 6\left( { - 1} \right)}}{{1 - \left( {4\left( { - 1} \right)} \right)}}$,
Now simplifying we get,
$ \Rightarrow \dfrac{{2 - i - 6}}{{1 + 4}}$,
Now adding and subtracting the like terms we get,
$ \Rightarrow \dfrac{{ - 4 - i}}{5}$,
So the simplified form of the expression is $\dfrac{{ - 4 - i}}{5}$.

$\therefore $The simplified form of the expression when we divide $\dfrac{{2 + 3i}}{{1 + 2i}}$ is $\dfrac{{ - 4 - i}}{5}$.

Note:
Complex numbers are combination of two types of numbers i.e., real numbers and imaginary numbers, and they are defined by the inclusion of $i$ term, whose value is defined by$i = \sqrt { - 1} $, the general form for a complex number is defined by,
$z = a + ib$, where $z$ is the complex number, $a$ is any real number and $b$ is the imaginary part of the complex number, both of which can be positive or negative. The complex numbers cannot be marked on a number line.