Question

How do you simplify $\dfrac{{4 - 6i}}{i}$?

Answer 1

Hint: The question belongs to the simplification of complex numbers. Any fraction of complex numbers can be represented in only $\left( {a + ib} \right)$ form by doing some basic arithmetic operations. We know that the value of $${i^2} = - 1$ in the complex numbers. We can replace minus signs by simply replacing it with $${i^2}$$. To convert the given fraction into the simplest form, first we will look for the numerator of the fraction and we will write it into the simplest form by taking a common factor of it. We will write each of the terms inside the parenthesis separate in form of fraction.

Complete step by step solution:
Step: 1 the given complex number is,
$\dfrac{{4 - 6i}}{i}$
Multiply the denominator and numerator of the complex number with$$i$$ to simplify the given complex number.
$ \Rightarrow \dfrac{{i \times \left( {4 - 6i} \right)}}{{i \times i}}$
Multiply the $$i$$ inside the bracket of the given complex number.
$ \Rightarrow \dfrac{{i \times \left( {4 - 6i} \right)}}{{i \times i}} = \dfrac{{\left( {4i - 6{i^2}} \right)}}{{{i^2}}}$
We know that the value of ${i^2} = - 1$ in the complex number, so substitute ${i^2} = - 1$ in the given expression.
$ \Rightarrow \dfrac{{\left( {4i - 6{i^2}} \right)}}{{{i^2}}} = \dfrac{{4i - 6\left( { - 1} \right)}}{{\left( { - 1} \right)}}$
Now simplify the given express to write it into its simplest form.
$\dfrac{{4i - 6\left( { - 1} \right)}}{{\left( { - 1} \right)}} = \dfrac{{4i + 6}}{{ - 1}}$
Multiply by $\left( { - 1} \right)$ to both the numerator and denominator of complex number.
$ \Rightarrow \dfrac{{ - 1 \times \left( {4i + 6} \right)}}{{ - 1 \times - 1}}$
Simplify the number to get the final result.
$ \Rightarrow \dfrac{{ - 1 \times \left( {4i + 6} \right)}}{{ - 1 \times - 1}} = - \left( {6 + 4i} \right)$

Final Answer:
Therefore the simplest form of the given complex number is equal to $ - \left( {6 + 4i} \right)$.

Note:
Students are advised to remember the properties of complex numbers. They must use ${i^2} = - 1$ while solving the numbers. They should simplify the numbers with help of basic arithmetic operations as they do in normal. They must know that the simplest form of a given complex number is $a + ib$, where $a$ is the real part and $\left( {ib} \right)$ is the imaginary part of the complex number.