Why we rationalize, conjugate.

Question

My question is why we rationalise, conjugate any denominator containing irrational or imaginary quantity. What is the need to rationalize them?

Answer 1

As a tool to solve problems, sometimes it is desirable to do so. For instance,

$$\lim_{x\rightarrow 0} \frac x{\sqrt{x+1}-1} =\lim_{x\rightarrow 0} \frac {x(\sqrt{x+1}+1)}{(\sqrt{x+1}-1)(\sqrt{x+1}+1)} =\lim_{x\rightarrow 0} \frac {x(\sqrt{x+1}+1)}{x} =\lim_{x\rightarrow 0} \sqrt{x+1}+1 =2$$

Answer 2

Strictly speaking, there is no need to do this. It just looks a bit neater, and makes the numbers easier to handle. If someone were to ask you to put $\frac{8}{\sqrt{5}}$ on a number line, where would you put it? A lot of people find it easier to think of $\frac{8\sqrt{5}}{5}$, because we are used to dividing by natural numbers.

Answer 3

Often the reason is trivial. For example, we want to verify that $\frac{1}{i}$ is a complex number represented by $a+bi$ with real numbers $a,b$. Then it is helpful to write $$ \frac{1}{i}=\frac{i}{i^2}=-i. $$