Question

How do you divide $\dfrac{{4 + i}}{{8i}}?$

Answer 1

As we know that the above question consists of complex numbers. A complex number is of the form of $a + ib$, where $a,b$ are real numbers and $i$represents the imaginary unit. The term $i$is an imaginary number and it is called iota and it has the value of $\sqrt { - 1}$. These numbers are not really imaginary. In this question we will first rationalize the denominator and then by using algebraic and exponential, formulas we will solve it.

Complete step by step solution:
As per the given question we have $\left( {\dfrac{{4 + i}}{{8i}}} \right)$. To divide this fraction we have to convert the denominator to be a real number and not a complex number as given in the question.
Here we can see that the denominator is in the form of $8i$, so we multiply it with $i$to get the real number i.e. $8i \times i = 8{i^2}$. It gives us the real number $- 8$.
And we have multiply the same iota to the numerator to keep the fraction same which is $\dfrac{{i(4 + i)}}{{ - 8}} \Rightarrow \dfrac{{4i + {i^2}}}{{ - 8}}$.
The required new form is $\dfrac{{4i - 1}}{{ - 8}}$, It can also be written as by breaking the fractions $\dfrac{{4i}}{{ - 8}} - \dfrac{1}{{ - 8}} = - \dfrac{{1i}}{2} + \dfrac{1}{8}$.

Hence the required value is $\dfrac{1}{8} - \dfrac{1}{2}i$.

Note: We should be careful while solving this kind of questions and we need to be aware of the matrices, their properties and the exponential formulas. We should note that the value of ${i^2}$ is mistakenly taken as $1$, which is a completely wrong value and it may lead to the wrong answers. We should be careful with complex numbers. So ${i^4} = \sqrt { - 1} \times \sqrt { - 1} \times \sqrt { - 1} \times \sqrt { - 1} = 1$.