Why is it so important to rationalize radicals in the denominator?

[harvellt]

Even in my second semester of calc I have yet to see a situation where the extra step made any sense why is it important to write $$\frac{3\sqrt{13}}{13}$$ instead of leaving $$\frac{3}{\sqrt{13}}$$. Its not a big deal but even my profs say its not that important so it has peaked my curiosity.

 

[yenchin]

For me I don't see it as that important, except perhaps it allows you to see the approximate value easier in *some* cases. For example, we know that sqrt(2) is about 1.4142. So the reciprocal 1/sqrt(2) will be 1/1.4142, not that easy to see what the value is compared to sqrt(2)/2 which is about 1.4142/2 = 0.7071.

We have a discussion at https://www.physicsforums.com/showthread.php?t=130776

 

[CRGreathouse]

Having a standard form makes it easier to see when two numbers are equal. It's not terribly important what that form is. Standardizing it so that all radicals appear in the denominator would work, too. But without this you'd have sqrt(2)/2 and 1/sqrt(2) which are equal as real numbers but unequal as strings.

The *process* of rationalizing the denominator is important regardless of form, though. It's required for sensible multiplication of radicals (and even for addition in complex arithmetic).

 

[waht]

It was a good exercise though, back in algebra.