Simplify the given expression $\dfrac{i}{{1 + 2i}}$?

Hint: In the given problem, we are required to simplify an expression involving complex numbers. For simplifying the given expression, we need to have a thorough knowledge of complex number sets and its applications in such questions. Algebraic rules and properties also play a significant role in simplification of such expressions.

Complete step by step answer:
In the question, we are given an expression which needs to be simplified using the knowledge of complex number sets.
For simplifying the given expression involving complex numbers, we need to first multiply the numerator and denominator with the conjugate of the complex number present in the denominator so as to obtain a real number in the denominator.
So, $\dfrac{i}{{1 + 2i}} = \dfrac{i}{{1 + 2i}} \times \dfrac{{1 - 2i}}{{1 - 2i}}$
Using the algebraic identity $\left( {a + b} \right)\left( {a - b} \right) = \left( {{a^2} - {b^2}} \right)$,
\[ = \dfrac{{i(1 - 2i)}}{{{{\left( 1 \right)}^2} - {{(2i)}^2}}}\]
\[ = \dfrac{{i - 2{i^2}}}{{{{\left( 1 \right)}^2} - 4{i^2}}}\]
We know that ${i^2} = - 1$. Hence, substituting ${i^2}$ as $ - 1$, we get,
\[ = \dfrac{{i - 2( - 1)}}{{\left( 1 \right) - ( - 4)}}\]
Opening brackets and simplifying further, we get,
\[ = \dfrac{{i + 2}}{{\left( 1 \right) + 4}}\]
\[ = \dfrac{{i + 2}}{5}\]
Distributing the denominator to both the terms, we get,
\[ = \dfrac{2}{5} + \dfrac{1}{5}i\]

Therefore, the given expression $\dfrac{i}{{1 + 2i}}$ can be simplified as: \[\left[ {\dfrac{2}{5} + \dfrac{1}{5}i} \right]\].

Note: The given problem revolves around the application of properties of complex numbers in questions. The question tells us about the wide ranging significance of the complex number set and its properties. The final answer can also be verified by working the solution backwards and getting back the given expression $\dfrac{i}{{1 + 2i}}$. Algebraic rules and properties also play a significant role in simplification of such expressions and we also need to have a thorough knowledge of complex number sets and its applications in such questions.