And who decided it?
Nothing wrong with it, just a helpful technique from the days of paper-and-pencil arithmetic. Suppose you wanted a decimal approximation of 1/sqrt(2), and you had a table of square roots, but no calculator. Which long division problem would you rather do by hand: 1.414213562/2 or 1/1.414213562?
Other valid reasons, from a similar thread on the Math Forum:
There’s nothing wrong with it. People do it all the time.
Perhaps you’re thinking about the problems with having 0 as a denominator? That yields an indeterminate answer so it is prohibited. But irrationals are just fine.
Maybe you should post an example of what you were thinking of.
biqu already gave an example, and I was taught the same thing in algebra and geometry classes. When you reduce a fraction to lowest terms, you want to avoid having an irrational denominator, so you would multiply 1/sqrt(2) by sqrt(2)/sqrt(2) to yield sqrt(2)/2. I was taught that this was the “standard” way to write things like that, and the explanation given was the same as biqu’s: it’s a traditional holdover from the days of yore, when we didn’t have electronic calculators capable of doing the arithmetic for us.
How would you avoid having an irrational denominator in the case of 1/e?
e^ (-1) 
I was definitely taught the one over root two version. We learned tables of all the most “important” angles and their sines, cosines and tangents.
1/0! - 1/1! + 1/2! - 1/3! + 1/4! - …
Now cut that out. With a “thank you” to Jack Benny.
(1/e)/1 :smack:
I suspect this is the main reason. But in general (though not always, and not as much now that calculators are readily available) it’s preferable to have one’s denominators as simple as possible, even if it means having a yucky numerator. Having simple denominators helps when dividing, finding common denominators, etc.
And sometimes, in the process of rationalizing a denominator, you can do away with the denominator completely. Example:
6/sqrt(2) = 6sqrt(2)/2 = 3sqrt(2)
I think 3*sqrt(2) is noticeably simpler than 6 / sqrt(2), don’t you?
All this said, if your teacher requires you to rationalize your denominators, then what’s wrong with having an irrational number there is that you don’t get full credit.
And your teacher’s reason for doing this, besides all the stuff above, might be that they want to have only one right answer, rather than having to see 6/sqrt(2) and take the extra few seconds to realize that it’s also correct, even though it doesn’t match the answer key. Same reasoning applies to writing sqrt(18) as 3sqrt(2).
Perhaps it has to do with extension fields. The number 1/sqrt(6) is a real number but when the denomenator iss rationalized, it takes the form of an element of Q[sqrt(6)]
So was I. It was always 1/√2 when we learnt the important trig ratios off by heart. But when it came to doing calculations (using log tables i.e pre-calculators), √2/2 was much easier to use.
I thought about that. But if it’s equal to an element of Q[sqrt(6)], then it is an element of Q[sqrt(6)]. And I have to wonder how many high school teachers are thinking in terms of extensions of Q.
True, but you would never write sqrt(18) like that if working in Q[sqrt(2)]. Probably rationalizing denomenators started out the same way and the importance of writing elements of Q[sqrt(n)] as a + b[sqrt(n)] has be lost or at least is never taught to math teachers
I was also taught 1/√2. But then, if I needed to do any calculations, I was allowed to use a calculator. Normally I was taught to leave irrational numbers in the answer, and to simplify it as much as possible. 1/√2 was considered “simpler” than √2/2.
I have no preference about 1/√2 vs. √2/2 in general use, but with regard to trigonometry √2/2 is easier to remember:
sin 0° = √0/2
sin 30° = √1/2
sin 45° = √2/2
sin 60° = √3/2
sin 90° = √4/2