Question

How do you perform the given division $ \dfrac{-1}{-9i} $ ?

Answer 1

Hint: We start solving the problem by equating the given division to a variable. We then multiply the numerator and denominator of the given division with the complex conjugate of $ -9i $ by using the fact that the complex conjugate of a complex number $ a+bi $ is $ a-bi $. We then make use of the fact that $ {{i}^{2}}=-1 $ to proceed through the problem. We then make the necessary calculations to get the required result of the division.

Complete step by step answer:
According to the problem, we are asked to find the result of the division $ \dfrac{-1}{-9i} $ .
Let us assume $ d=\dfrac{-1}{-9i} $ ---(1).
Now, let us multiply the numerator and denominator of the equation with the complex conjugate of $ -9i $.
We know that the complex conjugate of a complex number $ a+bi $ is $ a-bi $ . So, the complex conjugate of $ -9i $ is $ 9i $ . Let us multiply numerator and denominator with this complex number.
 $ \Rightarrow d=\dfrac{-1}{-9i}\times \dfrac{9i}{9i} $ .
 $ \Rightarrow d=\dfrac{-9i}{-81{{i}^{2}}} $ ---(2).
We know that $ {{i}^{2}}=-1 $ . Let us use this result in equation (2).
 $ \Rightarrow d=\dfrac{-9i}{-81\left( -1 \right)} $ .
 $ \Rightarrow d=\dfrac{-9i}{81} $ .
 $ \Rightarrow d=\dfrac{-i}{9} $ .
So, we have found the result of the given division $ \dfrac{-1}{-9i} $ as $ \dfrac{-i}{9} $ .
$ \therefore $ The result of the given division $ \dfrac{-1}{-9i} $ is $ \dfrac{-i}{9} $.

Note:
 Whenever we get this type of problem, we first multiply the denominator with its complex conjugate as the result must not have a complex number in its denominator. We should keep in mind that $ i $ is known as iota which is equal to $ \sqrt{-1} $ while solving this problem. Similarly, we can expect problems to find the result of the division $ \dfrac{4+i}{3-i} $.