# What is the use of rationalizing the denominator?

Mar 31, 2015

In ancient times (when I was a student ) we did arithmetic with pencil and paper.

$\sqrt{2} \approx 1.414$ For a decimal approximation would you rather do $\frac{1}{\sqrt{2}} = 1 \div 1.414$ or $\frac{\sqrt{2}}{2} = 1.414 \div 2$

Now that only hard-cores do arithmetic with pencil and paper, I think of it as a spelling rule. I think of reducing fractions as a spelling rule as well.

What is wrong with writing $\frac{77}{154}$?
Well, really the problem is, that's not how we spell $\frac{1}{2}$ If you and I are solving a problem and I write $\frac{77}{154}$ and you write $\frac{86}{746}$ did we get the same number or different numbers? (they are different)

Rationalizing denominators are similar at the algebra level.

For understanding calculus it becomes quite important to understand why $\frac{x - 9}{\sqrt{x} - 3} = \sqrt{x} + 3$.