Question

What is the value of ${{i}^{-1}}$ ?

Answer 1

The given question is solved using the concept of complex numbers. Firstly, we multiply and divide the given expression with the imaginary number $i$ . Then, we substitute the value of ${{i}^{2}}$to get the required result.

Complete step-by-step solution:
We are given an expression in the question and need to simplify it. We will be using the concept of complex numbers to simplify the expression.
Complex numbers, in mathematics, are represented in the form $p+iq$
Here,
p, q are the real numbers
$i$ is the unit imaginary number
The unit imaginary number $i$ is equal to the square root of minus one.
The value of $i$ is given by,
$\Rightarrow i=\sqrt{-1}$
Squaring the above equation on both sides, we get,
$\Rightarrow {{i}^{2}}=-1$
The inverse of the given number is the reverse of the number itself.
For example,
The inverse of the number $a$ is $\dfrac{1}{a}$
The value $\dfrac{1}{a}$ is also represented as ${{a}^{-1}}$
From the above,
$\Rightarrow {{a}^{-1}}=\dfrac{1}{a}$
Applying the same for ${{i}^{-1}}$ , we get,
$\Rightarrow {{i}^{-1}}=\dfrac{1}{i}$
Multiplying and dividing with an imaginary number $i$ on the right-hand side, we get,
$\Rightarrow {{i}^{-1}}=\dfrac{1}{i}\times \dfrac{i}{i}$
Multiplying the denominator on the right-hand side, we get,
$\Rightarrow {{i}^{-1}}=\dfrac{i}{{{i}^{2}}}$
From the above, we know that the value of ${{i}^{2}}=-1$
Substituting the value of ${{i}^{2}}$, we get,
$\Rightarrow {{i}^{-1}}=\dfrac{i}{\left( -1 \right)}$
Simplifying the above expression, we get,
$\therefore {{i}^{-1}}=-i$
The value of the inverse of the imaginary number $i$ is equal to the negative of the imaginary number $i$ or $-i$

Note: The question can also be solved using the concept of exponents as follows,
The expression ${{i}^{-1}}$ is multiplied and divided by ${{i}^{2}}$
$\Rightarrow {{i}^{-1}}\times \dfrac{{{i}^{2}}}{{{i}^{2}}}$
According to the rules of exponents,
$\Rightarrow {{a}^{m}}\times {{a}^{n}}={{a}^{\left( m+n \right)}}$
Applying the same, we get,
$\Rightarrow \dfrac{{{i}^{-1+2}}}{{{i}^{2}}}$
Simplifying the above expression,
$\Rightarrow \dfrac{i}{{{i}^{2}}}$
Substituting the value of ${{i}^{2}}$ ,
$\Rightarrow \dfrac{i}{\left( -1 \right)}$
$\therefore {{i}^{-1}}=-i$