Rationalization Of Real Numbers


Definition:

Rationalization is explained as a process of simplifying an expression by eliminating the radicals without changing the value of the expression.

It is a process that is undertaken to make the fraction denominator real. Especially in the cases, where we might find complex numbers in the denominator. Dividing any number by a complex number is not likely to be possible, as we do not know exact value of imaginary number. In case, we are having complex numbers in the fraction, although it is possible for us to find the fraction value without rationalization, calculations become tough as compared to division when done using real numbers.

Rationalize the Denominator

Rationalizing the denominator is a process of eliminating the roots in the fractions.

There are different methods to rationalize a denominator are demonstrated below:

Case 1: Where there is only a monomial denominator (Single square root).

Multiply both the numerator and the denominator by whatever makes the denominator an expression so that, it contains no longer a radical. If it simplifies the expression, you can even multiply the numerator and denominator by it.

Case 2: If the denominator is of the type a + b√b, then multiply and divide with the conjugate and then, apply the formula. 

(a - b)(a + b) = (a22 - b22) for the denominator. Then, the irrational denominator becomes rational. This is the process to rationalize the denominator of this kind.

Example:

Simplify 5/√6

Solution:

 ( 5/√6) x (√6/√6)

= (5√6) / 6

Rationalize the Numerator:

Rationalizing numerator is similar to rationalizing the denominator. In this method, we multiply numerator and denominator by a radical that will get rid of the radical in the numerator.

E.g.  √( 5/6) , Rationalize the numerator

Solution:

√ (5/6) x (√5/√5)

= 5/√30