Rationalize Denominator and Numerator

Apr 2015
684
7
Bronx, NY
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
Jun 2008
16,217
6,765
North Texas
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.
 
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Dec 2013
2,002
757
Colombia
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.
 
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Apr 2015
684
7
Bronx, NY
I always rationalize the denominator because it has become routine to do so.
 

HallsofIvy

MHF Helper
Apr 2005
20,249
7,909
It is easier to get common denominators to add fractions if they have "rational" denominators.
 
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Feb 2014
1,748
651
United States
I suspect that the original reason was to simplify computation.

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

MHF Hall of Honor
May 2006
12,028
6,341
Lexington, MA (USA)

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?

 
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Matt Westwood

MHF Hall of Honor
Jul 2008
1,281
426
Reading, UK
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.
 
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Apr 2015
684
7
Bronx, NY
I don't see what the big deal is, really.