Question

How do you rationalize imaginary numbers?

Answer 1

Hint: Problems like these are generally theoretical in nature and are very simple to solve. Complex numbers or imaginary numbers are those that have a real part as well as an imaginary part. Just like real functions are plotted on the coordinate plane, complex numbers are plotted on the complex plane or the argand plane. This argand plane consists of a real axis and an imaginary axis. Rationalization of complex numbers are possible if and only if the complex number has a numerator part and a denominator part. Rationalization is done by multiplying the conjugate of the complex number of the denominator, in both the numerator and the denominator.

Complete step by step solution:
Now, we start off with the solution to the given problem by writing that, the general representation of a complex number is \[a+ib\] . The conjugate of this general form of the complex number is given by \[a-ib\] . Now, suppose we have a complex number of the form \[\dfrac{c+id}{a+ib}\] .
Now, for the rationalization of this complex number, we multiply the numerator and the denominator with the conjugate of the complex number of the denominator \[a+ib\] , which is basically, \[a-ib\] . Doing so we get,
\[\Rightarrow \dfrac{\left( c+id \right)\left( a-ib \right)}{\left( a+ib \right)\left( a-ib \right)}\]
Now on multiplying the factors of the denominator we get,
\[\Rightarrow \dfrac{\left( c+id \right)\left( a-ib \right)}{{{a}^{2}}+{{b}^{2}}}\]

Note: The main thing that we must remember regarding the rationalization of complex numbers is that the denominator before rationalization should be an imaginary one and after should be real. We must always remember to multiply the conjugate on both the numerator and the denominator and also should recall the general form of the conjugate complex numbers. After we have formed the denominator of the complex number to be real, we must multiply the factors of the numerator, to get the proper answer.