Rationalize Denominator and Numerator

rapture

Why do math teachers stress the process of rationalizing numerator and denominator?
This is more a problem, I see, in terms of rationalizing the denominator.
Why can't fractions have a radical in the denominator? What's the big deal?

skeeter

MHF Helper
IMHO, there is no need to rationalize a denominator other than to learn the technique. Rationalizing the numerator comes into play in calculus, beginning with determining certain limits.

1 person

Archie

It is my opinion that conventions in rationalising, especially avoiding radicals in the denominator originated in a time when typesetting and printing technologies were considerably less advanced than they are today. Conventions grew up to make the typesetting easier.

Nowadays, I would encourage people to select the form that most clearly expresses the point they wish to make, the meaning they wish to convey, or the value the expression has.

1 person

rapture

I always rationalize the denominator because it has become routine to do so.

HallsofIvy

MHF Helper
It is easier to get common denominators to add fractions if they have "rational" denominators.

3 people

JeffM

I suspect that the original reason was to simplify computation.

$\dfrac{4}{\sqrt{3}}$ is far less easy to compute by hand than $\dfrac{4\sqrt{3}}{3}.$

1 person

Soroban

MHF Hall of Honor

I have shown my students these two tasks.

Without a calculator, evaluate: .$$\displaystyle (a)\;\frac{1}{\sqrt{2}} \qquad (b)\:\frac{\sqrt{2}}{2}$$

Code:
                ______________
(a) 1.414253562 ) 1.0000000000

____________
(b) 2 ) 1.414253562
Which division would you prefer?

1 person

Archie

I wouldn't compute either!

2 people

Matt Westwood

MHF Hall of Honor
If you are doing lots of compicklated mathematics and you end up with lots of messy irrational algebraic numbers at the end of it, it makes jolly good sense to convert them all into some sort of canonical form, otherwise you may find yourself not recognising that two numbers that look different are in fact the same.

Srslydudz, canonical forms of all sorts of objects are useful things to have around the shop. If it's difficult and hard work to get to them, then you need more practice.

2 people

rapture

I don't see what the big deal is, really.