Question

How do you divide $\dfrac{{2i}}{{1 - i}}$?

Answer 1

Hint: This problem deals with simplifying the complex numbers. A complex number is a number that can be expressed in the form of \[a + ib\], where $a$ and $b$ are real numbers, and $i$ represents the imaginary unit, satisfying the equation ${i^2} = - 1$. Because no real number satisfies this equation, $i$ is called an imaginary number.

Complete step-by-step answer:
The given expression is $\dfrac{{2i}}{{1 - i}}$, we have to simplify the expression.
Now consider the given expression, as shown below:
$ \Rightarrow \dfrac{{2i}}{{1 - i}}$
Now multiply and divide the above expression with $1 + i$, that is multiplying with this expression in the numerator and the denominator, as shown below:
$ \Rightarrow \dfrac{{2i}}{{1 - i}} \times \dfrac{{1 + i}}{{1 + i}}$
Now simplifying this expression by solving the expressions in the numerator and the denominator.
Consider the numerator as shown:
$ \Rightarrow 2i\left( {1 + i} \right)$
Multiplying each term, as shown:
$ \Rightarrow 2i\left( {1 + i} \right) = 2i + 2{i^2}$
We know that ${i^2} = - 1$, substituting this in the above expression as shown:
$ \Rightarrow 2i\left( {1 + i} \right) = 2i - 2$
Now taking the number 2 common on the right hand side of the equation, as shown:
$\therefore 2i\left( {1 + i} \right) = 2\left( {i - 1} \right)$
Now consider the denominator as shown:
$ \Rightarrow \left( {1 - i} \right)\left( {1 + i} \right)$
Multiplying each term, as shown:
$ \Rightarrow \left( {1 - i} \right)\left( {1 + i} \right) = 1\left( {1 + i} \right) - i\left( {1 + i} \right)$
$ \Rightarrow \left( {1 - i} \right)\left( {1 + i} \right) = 1 + i - i - {i^2}$
Here $i$ and $ - i$ gets cancelled as shown below:
$ \Rightarrow \left( {1 - i} \right)\left( {1 + i} \right) = 1 - {i^2}$
We know that ${i^2} = - 1$, substituting this in the above expression as shown:
$ \Rightarrow \left( {1 - i} \right)\left( {1 + i} \right) = 1 - \left( { - 1} \right)$
$\therefore \left( {1 - i} \right)\left( {1 + i} \right) = 2$
Now replacing the numerator and the denominator with the obtained expressions, as shown:
$ \Rightarrow \dfrac{{2i}}{{1 - i}} \times \dfrac{{1 + i}}{{1 + i}} = \dfrac{{2\left( {i - 1} \right)}}{2}$
Now 2 gets cancelled in both the numerator and the denominator, as shown:
$ \Rightarrow \dfrac{{2i}}{{1 - i}} \times \dfrac{{1 + i}}{{1 + i}} = i - 1$
The simplification of the expression $\dfrac{{2i}}{{1 - i}}$ is given by:
$\therefore \dfrac{{2i}}{{1 - i}} = i - 1$

Note:
Please note that the backbone of this new number system is the number $i$, also known as the imaginary unit. So from here we can conclude that any imaginary number is also a complex number, and any real number is also a complex number.