# Why is the square root of 2 1.414? **URL:** \ **Category:** Factual Questions\ **Created:** [January 10, 2003, 10:30am UTC](https://boards.straightdope.com/t/why-is-the-square-root-of-2-1-414/147714 "2003-01-10T10:30:53Z")\ **Posts on this page:** 20\ **Page:** 2
**Author:** ![Liberal](https://avatars.discourse-cdn.com/v4/letter/l/848f3c/32.png) [@Liberal](https://boards.straightdope.com/u/Liberal)\ **Post date:** [January 10, 2003, 6:34pm UTC](https://boards.straightdope.com/t/why-is-the-square-root-of-2-1-414/147714/21 "2003-01-10T18:34:46Z")
> [@](#): > > \*Originally posted by ultrafilter \* > \*\*For some reason, a lot of people don’t like this proof. I’ve seen a couple threads here go to four pages while people debate whether .9… is _really_ equal to 1. \*\* One of those was mine. I ended up conceding the point and changing my mind. ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [January 10, 2003, 6:38pm UTC](https://boards.straightdope.com/t/why-is-the-square-root-of-2-1-414/147714/22 "2003-01-10T18:38:54Z")
Was it you, **Lib**? I don’t remember your participation. Maybe you were just completely overshadowed by those who needed more convincing. ---
**Author:** ![Hari\_Seldon](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/hari_seldon/32/5173_2.png) [@Hari\_Seldon](https://boards.straightdope.com/u/Hari_Seldon)\ **Post date:** [January 10, 2003, 6:49pm UTC](https://boards.straightdope.com/t/why-is-the-square-root-of-2-1-414/147714/23 "2003-01-10T18:49:55Z")
If you try to subtract .999999… from 1.000000… using the standard rules for subtraction of infinite decimals (which are somewhat less than obvious if you think about it), you get 0.00000000… and if you don’t accept that that is 0, then there is nothing I can do for you. But it is inherent in what you mean by an infinite decimal. And yes, if you accept that 1/3 = .333333…, then 3 \* 1/3 = .99999999…, again according to the rules for multiplying infinite decimals, which again are far from evident. I once heard an hour lecture whose entire content was that you could carry out arithmetic operations with infinite decimals, at least in principle. ---
**Author:** ![William\_Ashbless](https://avatars.discourse-cdn.com/v4/letter/w/a87d85/32.png) [@William\_Ashbless](https://boards.straightdope.com/u/William_Ashbless)\ **Post date:** [January 10, 2003, 8:05pm UTC](https://boards.straightdope.com/t/why-is-the-square-root-of-2-1-414/147714/24 "2003-01-10T20:05:55Z")
How about a proof that the square root of 2 is a nonterminating nonrepeating decimal (cannot be represented by fractions) or in other words is not rational? Assume sqrt(2) is rational. Its rational so it can be represented by some two positive integers a and b (b!=0) as: sqrt(2) = a/b such that a/b is in lowest terms (that is, there are no common integer factors other than 1 between them). If it is not in lowest terms, divide out the common factors until it is in lowest terms, and use that a and b. Since there are no common integer factors other than 1, then specifically 2 is not an integer factor common between them. What this means is that they cannot both be even numbers, or you’d be able to divide both by two and still have integers on the top and bottom. Then, 2 = a^2 / b^2 Keep in mind that since a and b were integers, a^2 and b^2 are also integers. 2b^2 = a^2 This says that a^2 is twice b^2. But if you double any number (b^2 in this case) then it becomes even (because it has a factor of 2). So a^2 is even. But if a^2 is even, a must be even, because any odd integer multiplied by itself is odd (since it has no even factors to square). So a is even. Since both a and b cannot be even, b must be odd. b is odd, so b^2 is also odd. However, b^2 = a^2 / 2. Since a is even, it means it can be represented by 2k for some integer k. b^2 = (2k)^2 / 2 = 4k^2 / 2 = 2k^2 It doesn’t matter if k^2 is odd or even at this point – we multiply it by 2, so 2k^2 is obviously even. But this implies that b^2 is even. We’ve already proven it must be odd. We’ve arrived at a contradiction. This means one of our assumptions must be wrong. The only assumption we’ve made is that sqrt(2) is rational. Therefore it is not rational, and cannot be represented as a fraction a/b for integers a and b. So it is an irrational number between 1 and 2 that cannot be represented as a simple fraction and therefore has no simple repeating decimal form. I just love this proof for some reason. 🙂 ---
**Author:** ![Achernar](https://avatars.discourse-cdn.com/v4/letter/a/e274bd/32.png) [@Achernar](https://boards.straightdope.com/u/Achernar)\ **Post date:** [January 10, 2003, 8:26pm UTC](https://boards.straightdope.com/t/why-is-the-square-root-of-2-1-414/147714/25 "2003-01-10T20:26:20Z")
> [@](#): > > \*Originally posted by ultrafilter \* > For some reason, a lot of people don’t like this proof. I’m one of them. But I am, of course, fully aware that 0.99… = 1. In fact, it’s what I would have guessed, and seeing a proof of it was not surprising at all to me. The reason I dislike the proof is that it blindly applies algebra to something that is, in the end, based on calculus. I certainly wouldn’t put it among the classic proofs of mathematics, like the one that **William\_Ashbless** has graced us with. Speaking of which, if I had done it, I would have explicitly invoked the Fundamental Theorem of Arithmetic. I like that theorem, because non-mathematicians are often so shocked that Arithmetic has theorems. 🙂 ---
**Author:** ![William\_Ashbless](https://avatars.discourse-cdn.com/v4/letter/w/a87d85/32.png) [@William\_Ashbless](https://boards.straightdope.com/u/William_Ashbless)\ **Post date:** [January 10, 2003, 8:27pm UTC](https://boards.straightdope.com/t/why-is-the-square-root-of-2-1-414/147714/26 "2003-01-10T20:27:36Z")
Please so grace us. I’d love to hear an alternate proof. 😃 ---
**Author:** ![KarmaComa](https://avatars.discourse-cdn.com/v4/letter/k/e36b37/32.png) [@KarmaComa](https://boards.straightdope.com/u/KarmaComa)\ **Post date:** [January 10, 2003, 8:33pm UTC](https://boards.straightdope.com/t/why-is-the-square-root-of-2-1-414/147714/27 "2003-01-10T20:33:17Z")
> [@](#): > > \*Originally posted by whuckfistle \* > \*\*Let`s go for FIVE pages here!!! \*\* Thank you! I thought I was the only one driven crazy by the hordes of penis flexers who jump in for their moment of glory every time a super-triv math question arises. :smack: ---
**Author:** ![William\_Ashbless](https://avatars.discourse-cdn.com/v4/letter/w/a87d85/32.png) [@William\_Ashbless](https://boards.straightdope.com/u/William_Ashbless)\ **Post date:** [January 10, 2003, 8:44pm UTC](https://boards.straightdope.com/t/why-is-the-square-root-of-2-1-414/147714/28 "2003-01-10T20:44:46Z")
We’re here to educate, discuss, and hopefully manage to correct ignorance in ourselves and others. Sometimes it’s not easy to come to the ‘truth’ since zealous belief, stubborn philosophies, wildly differing opinions, and occasionally a lack of available data cloud the issue. At least with math we have some not inconsiderable tools to work with. I’ll be a ‘penis-flexer’ if it helps educate you (or me), sir. Pardon the disturbing unintended image that creates. 😃 Mayhaps you shouldn’t open any more math discussions? ---
**Author:** ![Achernar](https://avatars.discourse-cdn.com/v4/letter/a/e274bd/32.png) [@Achernar](https://boards.straightdope.com/u/Achernar)\ **Post date:** [January 10, 2003, 9:16pm UTC](https://boards.straightdope.com/t/why-is-the-square-root-of-2-1-414/147714/29 "2003-01-10T21:16:42Z")
There’s no alternate proof; the Fundamental Theorem is implicit in your designation of integers as exclusively “even” or “odd”. Pretty fundamental, I know. Hence the name. 🙂 **KarmaComa** : Maybe we’re not trying to impress each other; maybe we just think it’s cool. ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [January 10, 2003, 9:18pm UTC](https://boards.straightdope.com/t/why-is-the-square-root-of-2-1-414/147714/30 "2003-01-10T21:18:48Z")
> [@](#): > > \*Originally posted by KarmaComa \* > \*\*Thank you! I thought I was the only one driven crazy by the hordes of penis flexers who jump in for their moment of glory every time a super-triv math question arises. :smack: \*\* You’re upset about people answering questions on a messageboard whose major purpose is to disseminate information? That’s…fascinating. ---
**Author:** ![William\_Ashbless](https://avatars.discourse-cdn.com/v4/letter/w/a87d85/32.png) [@William\_Ashbless](https://boards.straightdope.com/u/William_Ashbless)\ **Post date:** [January 10, 2003, 9:24pm UTC](https://boards.straightdope.com/t/why-is-the-square-root-of-2-1-414/147714/31 "2003-01-10T21:24:23Z")
Actually, I’ve found two alternate proofs at least by doing a search on ‘root of 2 is irrational’ on search engines… [http://www.maths.soton.ac.uk/~gan/MA111/Slides/sld049.htm](http://www.maths.soton.ac.uk/~gan/MA111/Slides/sld049.htm) [http://www.maths.soton.ac.uk/~gan/MA111/Slides/sld051.htm](http://www.maths.soton.ac.uk/~gan/MA111/Slides/sld051.htm) Those two from the same site. There’s another more direct proof based on some sort of series, but I can’t find it. ---
**Author:** ![Jabba](https://avatars.discourse-cdn.com/v4/letter/j/b5e925/32.png) [@Jabba](https://boards.straightdope.com/u/Jabba)\ **Post date:** [January 10, 2003, 10:59pm UTC](https://boards.straightdope.com/t/why-is-the-square-root-of-2-1-414/147714/32 "2003-01-10T22:59:43Z")
Since we’re giving irrationality proofs, how about the more general THEOREM: Suppose that a[sub]0[/sub], …, a[sub]n-1[/sub] are integers and that x is a root of x[sup]n[/sup] + a[sub]n-1[/sub]x[sup]n-1[/sup] + … + a[sub]0[/sub] = 0. Then x is either an integer or irrational. Note that since sqrt(2) is a root of x[sup]2[/sup] - 2 = 0 and is certainly not an integer, this implies that sqrt(2) is irrational. Proof of theorem: Suppose x = b/a, where a and b are integers with no common factor. Then b[sup]n[/sup]/a[sup]n[/sup] + a[sub]n-1[/sub]b[sup]n-1[/sup]/a[sup]n-1[/sup] + … + a[sub]0[/sub] = 0. Multiply throughout by a[sup]n[/sup]: -b[sup]n[/sup] = a[sub]n-1[/sub]b[sup]n-1[/sup]a + … + ba[sup]n-1[/sup] + a[sub]0[/sub]a[sup]n[/sup] Now a divides each term on the right-hand side, so a divides the left-hand side, and a divides b[sup]n[/sup]. Since a and b are coprime, this implies that a = 1 ( again this uses the fundamental theorem of arithmetic). We have shown that if x is rational then it is an integer, and this proves the theorem. ---
**Author:** ![manhattan](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/manhattan/32/7_2.png) [@manhattan](https://boards.straightdope.com/u/manhattan)\ **Post date:** [January 11, 2003, 12:32am UTC](https://boards.straightdope.com/t/why-is-the-square-root-of-2-1-414/147714/33 "2003-01-11T00:32:54Z")
> [@](#): > > \*Originally posted by KarmaComa \* > \*\*the hordes of penis flexers… \*\* I’m afraid you misspelled “our best customers.” Although I have complete confidence that some of them have also flexed a penis, their own or that of a loved one, from time to time. ---
**Author:** ![sturmhauke](https://avatars.discourse-cdn.com/v4/letter/s/e47c2d/32.png) [@sturmhauke](https://boards.straightdope.com/u/sturmhauke)\ **Post date:** [January 11, 2003, 1:55am UTC](https://boards.straightdope.com/t/why-is-the-square-root-of-2-1-414/147714/34 "2003-01-11T01:55:38Z")
> [@](#): > > \*Originally posted by Hari Seldon \* > \*\*There is nothing to add to the above except to confirm that sqrt(2) is not a repeating decimal (like, say, 1/7 = .142857142857…), but just goes on forever with no evident pattern. This was discovered by the Pythagoreans at least 2500 years ago (and their proof is still used today) and it sort of destroyed their philosophy that everything could be done with whole numbers and fractions thereof. Interestingly, sqrt(2), like all quadratic roots, can be expressed as a periodic continued fraction, but I will not attempt to illustrate a continued fraction in this medium. Once I did, but the formatter completely destroyed it, taking out all the carriage returns and spaces I had so carefully entered. In words, the square root of 2 is 1 plus 1 over (2 plus 1 over (1 plus 2 over (1 plus 2 over 1 (plus…)))…) where the 1s and 2s, not to mention the parentheses, go on forever. \*\* Use the [code] [/code] tag (minus the internal spaces of course): ```auto 1 + 1 --------- 2 + 1 ----- 2 + 1 - ... 2 ``` ---
**Author:** ![JRDelirious](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/jrdelirious/32/9531_2.png) [@JRDelirious](https://boards.straightdope.com/u/JRDelirious)\ **Post date:** [January 11, 2003, 2:04am UTC](https://boards.straightdope.com/t/why-is-the-square-root-of-2-1-414/147714/35 "2003-01-11T02:04:44Z")
Hey, I enjoy a good display of intellectual penis-flexing 😃 [sub]But what is the distaff equivalent? Intellectual boob-flashing?[/sub] ---
**Author:** ![William\_Ashbless](https://avatars.discourse-cdn.com/v4/letter/w/a87d85/32.png) [@William\_Ashbless](https://boards.straightdope.com/u/William_Ashbless)\ **Post date:** [January 11, 2003, 2:16am UTC](https://boards.straightdope.com/t/why-is-the-square-root-of-2-1-414/147714/36 "2003-01-11T02:16:29Z")
Can anyone demonstrate a reasonable proof that that continued fraction equals root 2? I’m fascinated. (Much as I took some algebra and calculus courses, I don’t remember any proof techniques involving continued fractions…) ---
**Author:** ![Achernar](https://avatars.discourse-cdn.com/v4/letter/a/e274bd/32.png) [@Achernar](https://boards.straightdope.com/u/Achernar)\ **Post date:** [January 11, 2003, 2:50am UTC](https://boards.straightdope.com/t/why-is-the-square-root-of-2-1-414/147714/37 "2003-01-11T02:50:50Z")
Sure. Add 1 to it and call your result x. Note, by direct substitution, that: x = 2 + 1 / x Or: x[sup]2[/sup] - 2x - 1 = 0 So x = 1 - sqrt(2), or x = 1 + sqrt(2). But x is clearly positive. So there you go. ---
**Author:** ![Jabba](https://avatars.discourse-cdn.com/v4/letter/j/b5e925/32.png) [@Jabba](https://boards.straightdope.com/u/Jabba)\ **Post date:** [January 11, 2003, 2:52am UTC](https://boards.straightdope.com/t/why-is-the-square-root-of-2-1-414/147714/38 "2003-01-11T02:52:33Z")
Well, we can certainly make the result intuitively plausible. Write u = sqrt(2), for typographical convenience. Note that 1 \< u \< 2, and that (u-1)(u+1) = 1. Thus u = 1 + (u-1) = 1 + 1/(u+1) = 1 + 1/(2 + u-1) = … ---
**Author:** ![jackelope](https://avatars.discourse-cdn.com/v4/letter/j/a6a055/32.png) [@jackelope](https://boards.straightdope.com/u/jackelope)\ **Post date:** [January 11, 2003, 2:53am UTC](https://boards.straightdope.com/t/why-is-the-square-root-of-2-1-414/147714/39 "2003-01-11T02:53:18Z")
**William\_Ashbless** , thanks for the “sqrt 2 is irrational” proof above; that’s fascinating. I’m more of a mathematics enthusiast than an actual mathematician, but I was able to follow that clearly. _And (my) ignorance retreats another step…_ ---
**Author:** ![Jabba](https://avatars.discourse-cdn.com/v4/letter/j/b5e925/32.png) [@Jabba](https://boards.straightdope.com/u/Jabba)\ **Post date:** [January 11, 2003, 2:57am UTC](https://boards.straightdope.com/t/why-is-the-square-root-of-2-1-414/147714/40 "2003-01-11T02:57:18Z")
I should say that both my proof and **Achernar’s** have the same failing, namely that they assume that the infinite continued fraction converges. Strictly, all we have shown is that if the fraction converges, then it converges to sqrt(2). In fact, the proof that all simple continued fractions converge is quite simple, but it involves a lot of fractions and subscripts, which are a real pain to type out in vB. If you want to see it, I’ll consider it tomorrow ( if no-one does it in the meantime). [Previous page](https://boards.straightdope.com/t/why-is-the-square-root-of-2-1-414/147714.md?page=1) [Next page](https://boards.straightdope.com/t/why-is-the-square-root-of-2-1-414/147714.md?page=3)