# What's wrong with having an irrational number in the denominator? **URL:** \ **Category:** Factual Questions\ **Created:** [March 20, 2006, 9:40pm UTC](https://boards.straightdope.com/t/whats-wrong-with-having-an-irrational-number-in-the-denominator/349201 "2006-03-20T21:40:06Z")\ **Posts on this page:** 19\ **Page:** 1
**Author:** ![Strinka](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/strinka/32/512_2.png) [@Strinka](https://boards.straightdope.com/u/Strinka)\ **Post date:** [March 20, 2006, 9:40pm UTC](https://boards.straightdope.com/t/whats-wrong-with-having-an-irrational-number-in-the-denominator/349201/1 "2006-03-20T21:40:06Z")
And who decided it? ---
**Author:** ![biqu](https://avatars.discourse-cdn.com/v4/letter/b/7feea3/32.png) [@biqu](https://boards.straightdope.com/u/biqu)\ **Post date:** [March 20, 2006, 9:59pm UTC](https://boards.straightdope.com/t/whats-wrong-with-having-an-irrational-number-in-the-denominator/349201/2 "2006-03-20T21:59:31Z")
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](http://mathforum.org/kb/thread.jspa?forumID=63&threadID=1288309&messageID=4065962#4065962) on the Math Forum: > [@](#): > > 1. Tradition: We old folks have always done it this way. > > 2. SAT, ACT or other kinds of MC tests: The answers might appear only this > way and students would have to know how to get the same form. > > 3. In trig class no one ever writes sin(PI/4)=1/sqrt(2). Everyone writes > sin(PI/4)=sqrt(2)/2 [Of course this is just a variant of number 1., i.e. > tradition.] ---
**Author:** ![Exapno\_Mapcase](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/exapno_mapcase/32/1051_2.png) [@Exapno\_Mapcase](https://boards.straightdope.com/u/Exapno_Mapcase)\ **Post date:** [March 20, 2006, 10:02pm UTC](https://boards.straightdope.com/t/whats-wrong-with-having-an-irrational-number-in-the-denominator/349201/3 "2006-03-20T22:02:19Z")
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. ---
**Author:** ![Excalibre](https://avatars.discourse-cdn.com/v4/letter/e/898d66/32.png) [@Excalibre](https://boards.straightdope.com/u/Excalibre)\ **Post date:** [March 20, 2006, 10:44pm UTC](https://boards.straightdope.com/t/whats-wrong-with-having-an-irrational-number-in-the-denominator/349201/4 "2006-03-20T22:44:40Z")
> [@Exapno Mapcase](#): > > 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. ---
**Author:** ![David\_Simmons](https://avatars.discourse-cdn.com/v4/letter/d/9de053/32.png) [@David\_Simmons](https://boards.straightdope.com/u/David_Simmons)\ **Post date:** [March 20, 2006, 11:10pm UTC](https://boards.straightdope.com/t/whats-wrong-with-having-an-irrational-number-in-the-denominator/349201/5 "2006-03-20T23:10:22Z")
> [@Excalibre](#): > > **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? ---
**Author:** ![John\_Mace](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/john_mace/32/185_2.png) [@John\_Mace](https://boards.straightdope.com/u/John_Mace)\ **Post date:** [March 20, 2006, 11:13pm UTC](https://boards.straightdope.com/t/whats-wrong-with-having-an-irrational-number-in-the-denominator/349201/6 "2006-03-20T23:13:05Z")
> [@David Simmons](#): > > How would you avoid having an irrational denominator in the case of 1/e? e^ (-1) 🙂 ---
**Author:** ![Capt.Ridley\_s\_Shooting\_Party](https://avatars.discourse-cdn.com/v4/letter/c/cc9497/32.png) [@Capt.Ridley\_s\_Shooting\_Party](https://boards.straightdope.com/u/Capt.Ridley_s_Shooting_Party)\ **Post date:** [March 20, 2006, 11:19pm UTC](https://boards.straightdope.com/t/whats-wrong-with-having-an-irrational-number-in-the-denominator/349201/7 "2006-03-20T23:19:32Z")
> [@](#): > > 1. In trig class no one ever writes sin(PI/4)=1/sqrt(2). Everyone writes > sin(PI/4)=sqrt(2)/2 [Of course this is just a variant of number 1., i.e. > tradition.] 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. ---
**Author:** ![Q.E.D](https://avatars.discourse-cdn.com/v4/letter/q/51bf81/32.png) [@Q.E.D](https://boards.straightdope.com/u/Q.E.D)\ **Post date:** [March 20, 2006, 11:22pm UTC](https://boards.straightdope.com/t/whats-wrong-with-having-an-irrational-number-in-the-denominator/349201/8 "2006-03-20T23:22:58Z")
> [@David Simmons](#): > > How would you avoid having an irrational denominator in the case of 1/e? 1/0! - 1/1! + 1/2! - 1/3! + 1/4! - … 😃 ---
**Author:** ![David\_Simmons](https://avatars.discourse-cdn.com/v4/letter/d/9de053/32.png) [@David\_Simmons](https://boards.straightdope.com/u/David_Simmons)\ **Post date:** [March 20, 2006, 11:30pm UTC](https://boards.straightdope.com/t/whats-wrong-with-having-an-irrational-number-in-the-denominator/349201/9 "2006-03-20T23:30:50Z")
> [@John Mace](#): > > e^ (-1) 🙂 > [@Q.E.D.](#): > > 1/0! - 1/1! + 1/2! - 1/3! + 1/4! - … > > 😃 Now cut that out. With a “thank you” to Jack Benny. ---
**Author:** ![jawdirk](https://avatars.discourse-cdn.com/v4/letter/j/df705f/32.png) [@jawdirk](https://boards.straightdope.com/u/jawdirk)\ **Post date:** [March 21, 2006, 1:33am UTC](https://boards.straightdope.com/t/whats-wrong-with-having-an-irrational-number-in-the-denominator/349201/10 "2006-03-21T01:33:35Z")
> [@David Simmons](#): > > Now cut that out. With a “thank you” to Jack Benny. (1/e)/1 :smack: ---
**Author:** ![Thudlow\_Boink](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/thudlow_boink/32/320_2.png) [@Thudlow\_Boink](https://boards.straightdope.com/u/Thudlow_Boink)\ **Post date:** [March 21, 2006, 1:49am UTC](https://boards.straightdope.com/t/whats-wrong-with-having-an-irrational-number-in-the-denominator/349201/11 "2006-03-21T01:49:36Z")
> [@biqu](#): > > 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? 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) = 6_sqrt(2)/2 = 3_sqrt(2) I think 3\*sqrt(2) is noticeably simpler than 6 / sqrt(2), don’t you? ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [March 21, 2006, 2:04am UTC](https://boards.straightdope.com/t/whats-wrong-with-having-an-irrational-number-in-the-denominator/349201/12 "2006-03-21T02:04:35Z")
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. ---
**Author:** ![TJdude825](https://avatars.discourse-cdn.com/v4/letter/t/dec6dc/32.png) [@TJdude825](https://boards.straightdope.com/u/TJdude825)\ **Post date:** [March 21, 2006, 3:33am UTC](https://boards.straightdope.com/t/whats-wrong-with-having-an-irrational-number-in-the-denominator/349201/13 "2006-03-21T03:33:44Z")
> [@ultrafilter](#): > > 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). ---
**Author:** ![Saint\_Cad](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/saint_cad/32/18907_2.png) [@Saint\_Cad](https://boards.straightdope.com/u/Saint_Cad)\ **Post date:** [March 21, 2006, 5:07am UTC](https://boards.straightdope.com/t/whats-wrong-with-having-an-irrational-number-in-the-denominator/349201/14 "2006-03-21T05:07:56Z")
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)] ---
**Author:** ![Cunctator](https://avatars.discourse-cdn.com/v4/letter/c/43a26b/32.png) [@Cunctator](https://boards.straightdope.com/u/Cunctator)\ **Post date:** [March 21, 2006, 5:17am UTC](https://boards.straightdope.com/t/whats-wrong-with-having-an-irrational-number-in-the-denominator/349201/15 "2006-03-21T05:17:50Z")
> [@Dominic Mulligan](#): > > 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. 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. ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [March 21, 2006, 5:18am UTC](https://boards.straightdope.com/t/whats-wrong-with-having-an-irrational-number-in-the-denominator/349201/16 "2006-03-21T05:18:18Z")
> [@SaintCad](#): > > 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)] 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**. ---
**Author:** ![Saint\_Cad](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/saint_cad/32/18907_2.png) [@Saint\_Cad](https://boards.straightdope.com/u/Saint_Cad)\ **Post date:** [March 21, 2006, 6:31am UTC](https://boards.straightdope.com/t/whats-wrong-with-having-an-irrational-number-in-the-denominator/349201/17 "2006-03-21T06:31:45Z")
> [@ultrafilter](#): > > 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 ---
**Author:** ![Fromage\_A\_Trois](https://avatars.discourse-cdn.com/v4/letter/f/a88e57/32.png) [@Fromage\_A\_Trois](https://boards.straightdope.com/u/Fromage_A_Trois)\ **Post date:** [March 21, 2006, 12:17pm UTC](https://boards.straightdope.com/t/whats-wrong-with-having-an-irrational-number-in-the-denominator/349201/18 "2006-03-21T12:17:56Z")
> [@Cunctator](#): > > > [@Dominic Mulligan](#): > > > > 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. > > 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 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. ---
**Author:** ![MartinL](https://avatars.discourse-cdn.com/v4/letter/m/ebca7d/32.png) [@MartinL](https://boards.straightdope.com/u/MartinL)\ **Post date:** [March 21, 2006, 3:18pm UTC](https://boards.straightdope.com/t/whats-wrong-with-having-an-irrational-number-in-the-denominator/349201/19 "2006-03-21T15:18:06Z")
> [@Cunctator](#): > > 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 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