# Strange Calculator Discovery. **URL:** \ **Category:** Miscellaneous and Personal Stuff I Must Share\ **Created:** [September 1, 2014, 6:25am UTC](https://boards.straightdope.com/t/strange-calculator-discovery/697118 "2014-09-01T06:25:32Z")\ **Posts on this page:** 11\ **Page:** 1
**Author:** ![Jim\_B](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/jim_b/32/4338_2.png) [@Jim\_B](https://boards.straightdope.com/u/Jim_B)\ **Post date:** [September 1, 2014, 6:25am UTC](https://boards.straightdope.com/t/strange-calculator-discovery/697118/1 "2014-09-01T06:25:32Z")
I play around a lot with calculators. And I’m not ashamed to admit. You can learn a lot that way, believe it or not. I have even shared some of these things with the SDMB in the past. This next one is equally as bizarre as the rest. When you take the square root of .111111… you get .3333333… naturally, since the square root of one-ninth is one-third. But one time, just as a lark, I thought I’d square root .11 alone. Then .111 (again, only three digits), etc… Long story short, you get the following pattern: 0.33333333331666666666624999999998. As you can see, the .33333… pattern is followed by an intrusive 1666666… pattern, and a 2499999… pattern (presumably leading ultimately to 25, I assume). It happens with other numbers too. Take .44444… The square root of this repeating decimal is .66666…, two-thirds, naturally. But when you do the same thing, you get 0.66666666663333333333249999999996. A “333…” pattern emerges, and then again that “25” pattern. It doesn’t just happen with these. Consider .9999… That equals one, of course. But when you do the same, you get 0.99999999994999999999874999999994. Now, you get “5” and “75” as your hidden pattern. Also odd, is that these patterns are “put off” until infinity. Which I guess is permissible, even if they are never part of the actual number. What is the explanation for these strange patterns? (BTW, I put is in MPSIMS, because I assume I just sharing something, albeit odd and mathematical, with you all.) Well? :):)🙂 ---
**Author:** ![Derleth](https://avatars.discourse-cdn.com/v4/letter/d/b9e5f3/32.png) [@Derleth](https://boards.straightdope.com/u/Derleth)\ **Post date:** [September 1, 2014, 6:53am UTC](https://boards.straightdope.com/t/strange-calculator-discovery/697118/2 "2014-09-01T06:53:13Z")
The simple explanation is that calculators use a finite number of bits to represent numbers, and they use approximations of functions to get “good enough” answers to problems using a finite amount of hardware, a finite amount of software, and a finite amount of time. A full explanation of floating point numbers (how computers represent decimal fractions) would take not only a long thread, it would fill a university-level course, probably one called “Numerical Analysis”. ---
**Author:** ![The\_Great\_Unwashed](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/the_great_unwashed/32/4791_2.png) [@The\_Great\_Unwashed](https://boards.straightdope.com/u/The_Great_Unwashed)\ **Post date:** [September 1, 2014, 8:10am UTC](https://boards.straightdope.com/t/strange-calculator-discovery/697118/3 "2014-09-01T08:10:06Z")
There is the binomial approximation (1 - x)[suP]a[/suP] ≈ 1 - ax In the case of sqrt(0.111111) we have: sqrt(0.111111) = (1/9 - 1/9000000)[suP]1/2[/suP] =1/3(1 - 1/1000000))[suP]1/2[/suP] ≈ 1/3(1 - 1/20000000) = 0.3333333333… - 0.0000001666666666… = 0.3333331666666… So that explains the “1666” bit. The “24999” bit will be explained by the next term in the binomial expansion, and so on. ---
**Author:** ![Napier](https://avatars.discourse-cdn.com/v4/letter/n/ce73a5/32.png) [@Napier](https://boards.straightdope.com/u/Napier)\ **Post date:** [September 1, 2014, 12:38pm UTC](https://boards.straightdope.com/t/strange-calculator-discovery/697118/4 "2014-09-01T12:38:15Z")
> [@Derleth](#): > > The simple explanation is that calculators use a finite number of bits to represent numbers, and they use approximations of functions to get “good enough” answers to problems using a finite amount of hardware, a finite amount of software, and a finite amount of time. A full explanation of floating point numbers (how computers represent decimal fractions) would take not only a long thread, it would fill a university-level course, probably one called “Numerical Analysis”. I think this is it. I tried the Arbitrary Precision Calculator at [apfloat.appspot.com](http://apfloat.appspot.com) to calculate: sqrt(0.111000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) and the result it returned was: 0.333166624979153636711908687038596556216762938123024462459526991876854935570345245438171979288344173436908078192. [Edit after the fact: for some reason pasting numbers into SDMB put a space into one of the strings, but it doesn’t appear that way on the calculator web page.] ---
**Author:** ![dropzone](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/dropzone/32/7515_2.png) [@dropzone](https://boards.straightdope.com/u/dropzone)\ **Post date:** [September 1, 2014, 3:37pm UTC](https://boards.straightdope.com/t/strange-calculator-discovery/697118/5 "2014-09-01T15:37:57Z")
I believe the cause of the intrusive and repeated 666 shows that the End Times are upon us. Try it again and let us know when it becomes all sixes. ---
**Author:** ![BigT](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/bigt/32/12044_2.png) [@BigT](https://boards.straightdope.com/u/BigT)\ **Post date:** [September 1, 2014, 3:56pm UTC](https://boards.straightdope.com/t/strange-calculator-discovery/697118/6 "2014-09-01T15:56:21Z")
> [@dropzone](#): > > I believe the cause of the intrusive and repeated 666 shows that the End Times are upon us. Try it again and let us know when it becomes all sixes. He already got that with the square root of .44444444. (I guess, if you wanted to make sure there was no leading 0, you could do 44.4444444.) ---
**Author:** ![BigT](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/bigt/32/12044_2.png) [@BigT](https://boards.straightdope.com/u/BigT)\ **Post date:** [September 1, 2014, 4:07pm UTC](https://boards.straightdope.com/t/strange-calculator-discovery/697118/7 "2014-09-01T16:07:24Z")
> [@BigT](#): > > He already got that with the square root of .44444444. (I guess, if you wanted to make sure there was no leading 0, you could do 44.4444444.) Which is, of course, the square root of 1975.30863802469136, which is why the end times started April 24, 1975, at 4:19:34 AM. ---
**Author:** ![Jim\_B](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/jim_b/32/4338_2.png) [@Jim\_B](https://boards.straightdope.com/u/Jim_B)\ **Post date:** [November 11, 2014, 4:07am UTC](https://boards.straightdope.com/t/strange-calculator-discovery/697118/8 "2014-11-11T04:07:04Z")
I have to confess and add that there is another reason why this random discovery fascinates me. Even when you don’t square root multiple ones, even when you just square root .1, you still see the pattern: 0.31622776601683793319988935444327. And it clearly isn’t just the quirk of one calculator. It is a pattern built into the actual number. My question: could you use this to calculate irrational numbers, like √.1? Slide rules use an interesting method of calculating large and complex numbers. They replace massive amounts of multiplication with simple addition. Could this hold true for other numbers too? Here is the pattern again: √.11111=0.33333166666249997916653645742187. As you can see, there are even more patterns that follow the initial “16666…” and “25”. Could I be on to something here? And could I have discovered something amazing too? 🙂 ---
**Author:** ![njtt](https://avatars.discourse-cdn.com/v4/letter/n/ecd19e/32.png) [@njtt](https://boards.straightdope.com/u/njtt)\ **Post date:** [November 11, 2014, 4:13am UTC](https://boards.straightdope.com/t/strange-calculator-discovery/697118/9 "2014-11-11T04:13:56Z")
> [@Jim\_B](#): > > I have to confess and add that there is another reason why this random discovery fascinates me. Even when you don’t square root multiple ones, even when you just square root .1, you still see the pattern: 0.31622776601683793319988935444327. And it clearly isn’t just the quirk of one calculator. It is a pattern built into the actual number. No, it a rounding error built into the way all (or most) calculators work. > [@Jim\_B](#): > > My question: could you use this to calculate irrational numbers, like √.1? You can use your calculator to calculate approximations to irrationals, but it will not you a correct, fully precise answer. (Indeed, it could not represent it on its screen, even if it “knew”.) ---
**Author:** ![sisu](https://avatars.discourse-cdn.com/v4/letter/s/df705f/32.png) [@sisu](https://boards.straightdope.com/u/sisu)\ **Post date:** [November 11, 2014, 4:16am UTC](https://boards.straightdope.com/t/strange-calculator-discovery/697118/10 "2014-11-11T04:16:06Z")
I thought this was going to be about 55378008 but alas… ---
**Author:** ![Joey\_P](https://avatars.discourse-cdn.com/v4/letter/j/919ad9/32.png) [@Joey\_P](https://boards.straightdope.com/u/Joey_P)\ **Post date:** [November 11, 2014, 4:25am UTC](https://boards.straightdope.com/t/strange-calculator-discovery/697118/11 "2014-11-11T04:25:11Z")
> [@sisu](#): > > I thought this was going to be about 55378008 but alas… Ninja’d, by B minutes.