Expert: Scotto - 1/21/2010

Question
My colleagues and I have been having a very animated discussion about why we still teach students to rationalize denominators.  We don't have a truly good answer only the simple response, "When I needed to divide using a slide rule..." and, "We rationalize so that there is a single accepted format for a simplified version."  

I remember a colleague saying that there were some Calculus BC questions which absolutely needed to be rationalized to be able to integrate.

Is there an absolute necessity for rationalizing (numerators or denominators) in higher mathematics?

Answer
I don't know of any rule, but all mathematicians I have met like to rationalize the denominator.
When they're looked at, 1/√2 makes no apparent sence like √2/2 does.
1/√2 requires the computation of finding the √ of a half.
√2/2 is 1.414/2 = 0.707.

Most math majors know to apprximate squareroots, we use
√2 is about 1.414, √3 is about 1.75, √5 is about 2.25, √6 is about 2.45, √7 is about 2.65, and so on, but if 1/√n is asked for any number, it has to be thought about for awhile.

Note that √2 is the only one carried out to 3 places.
The others are only carried out to 1 1/2 places, since they all end with a 5.

This is the case when an answer in the back of the book contains a fraction.
I don't know that I have ever seen a radical left in the denominator.

Also, what is someone asked, which is bigger - 14/√2 or 17/√3?
I can't say off the top of my head either.
However, multiply the first by √2/√2 and you get roughly 10.
Multiply the second by √3/√3 and you get roughly 29/3, which is less than the first one.

Nowadays, though, most have access to a spreadsheet.