# help me convince a friend that 0.333...=1 **URL:** \ **Category:** Great Debates\ **Created:** [April 13, 2004, 12:00pm UTC](https://boards.straightdope.com/t/help-me-convince-a-friend-that-0-333-1/239752 "2004-04-13T12:00:16Z")\ **Posts on this page:** 8\ **Page:** 1
**Author:** ![Takeo\_Jr](https://avatars.discourse-cdn.com/v4/letter/t/54ee81/32.png) [@Takeo\_Jr](https://boards.straightdope.com/u/Takeo_Jr)\ **Post date:** [April 13, 2004, 12:00pm UTC](https://boards.straightdope.com/t/help-me-convince-a-friend-that-0-333-1/239752/1 "2004-04-13T12:00:16Z")
on April 3rd, I posted this in GQ: > **[Does 0.999999..... equal 1?](https://boards.straightdope.com/sdmb/showthread.php?t=248963)** > > I got into a long discussion with a close friend of mine recently, it was very interesting. Here goes: 1/3 = 0.3333…, right? 3 X (1/3)= 1, right? 3 X 0.33333… = 0.9999…, right? so, 0.99999… = 1 RIGHT? He argued that 0.99999… gets VERY... My friend who I am arguing with 9and the whole reason I posted, to try and convince him) is still not convinced that 0.33333…=1 After seeing the post, he said: [quote] Hi. Have looked atthe linkvery briefly. First comment on “erislover” - 1/3+1/3+1/3=1/1=1 This is totally foolish because it is based on everybody accepting that 1/3=0.333…which is the actual problem. His whole way of trying to clear this up is to me very unscientific and illustrates the problem with getting info from the web. This: [But the truth of it, without calculus, is the method you show: .333… is just a representation of one-third.] THIS is the whole problem, I say that this is an insufficient representation of one-third for theoretical math. [end quote] BUNK!!! B-U-N-K, BUNK! Help me correct this guy! please! TJ I only posted this in GD as it was already a closed thread in GQ- but I need more ammo to shoot down my friend! ---
**Author:** ![Planet\_of\_the\_Shapes](https://avatars.discourse-cdn.com/v4/letter/p/b77776/32.png) [@Planet\_of\_the\_Shapes](https://boards.straightdope.com/u/Planet_of_the_Shapes)\ **Post date:** [April 13, 2004, 12:25pm UTC](https://boards.straightdope.com/t/help-me-convince-a-friend-that-0-333-1/239752/2 "2004-04-13T12:25:38Z")
The problem seems to be that he doesn’t accept that 1/3 is 0.3 recurring. Ask him what he thinks 1/3 is in decimal then? ---
**Author:** ![Telemark](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/telemark/32/372_2.png) [@Telemark](https://boards.straightdope.com/u/Telemark)\ **Post date:** [April 13, 2004, 1:41pm UTC](https://boards.straightdope.com/t/help-me-convince-a-friend-that-0-333-1/239752/3 "2004-04-13T13:41:15Z")
Your friend is correct. .3333… =! 1 .333… + .333… + .333… on the otherhand… ---
**Author:** ![Bytegeist](https://avatars.discourse-cdn.com/v4/letter/b/9dc877/32.png) [@Bytegeist](https://boards.straightdope.com/u/Bytegeist)\ **Post date:** [April 13, 2004, 2:30pm UTC](https://boards.straightdope.com/t/help-me-convince-a-friend-that-0-333-1/239752/4 "2004-04-13T14:30:53Z")
As **Telemark** hints, you probably meant to gather arguments that **0.9…** is equal to 1, not 0.3… Either that, or you’re impishly writing your numbers in base 4 without telling us. Many threads on this board have addressed the topic before — although, having only a “Guest” account, you might not have access to the search feature so as to find them all. Two decent Web pages though can be found [here](http://www.math.fau.edu/Richman/html/999.htm) and [here](http://db.uwaterloo.ca/~alopez-o/math-faq/node41.html). Google on “0.999 = 1” and you’ll find a bunch more. The real challenge isn’t to prove that 0.9… = 1. The challenge is to count _how many ways_ there are of proving it. ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [April 13, 2004, 3:44pm UTC](https://boards.straightdope.com/t/help-me-convince-a-friend-that-0-333-1/239752/5 "2004-04-13T15:44:54Z")
In base 4, .3… = 1. In base 5, .4… = 1. In base 16, .F… = 1. The reason is quite simple. 0.a[sub]1[/sub]a[sub]2[/sub]a[sub]3[/sub]…[sub]R[/sub] is defined to be the series whose nth term is given by a[sub]n[/sub]/R[sup]n[/sup]. With a[sub]i[/sub] = 1 for all i, this can be shown to converge (by means that you can find in an algebra 2 textbook) to 1. Does your friend know algebra 2? ---
**Author:** ![tdn](https://avatars.discourse-cdn.com/v4/letter/t/94ad74/32.png) [@tdn](https://boards.straightdope.com/u/tdn)\ **Post date:** [April 13, 2004, 9:34pm UTC](https://boards.straightdope.com/t/help-me-convince-a-friend-that-0-333-1/239752/6 "2004-04-13T21:34:59Z")
> [@Takeo Jr](#): > > I say that this is an insufficient representation of one-third for theoretical math. Your friend seems to think that a decimal representation is a matter of opinion or an alternate theory or some such. He doesn’t seem to get it that .333… is – by its very definition – one third. If he can’t get that very basic definition, I see no reason to multiply anything by 3. He refuses the premise, and that’s where you have to convince him. ---
**Author:** ![duality72](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/duality72/32/3910_2.png) [@duality72](https://boards.straightdope.com/u/duality72)\ **Post date:** [April 13, 2004, 9:56pm UTC](https://boards.straightdope.com/t/help-me-convince-a-friend-that-0-333-1/239752/7 "2004-04-13T21:56:55Z")
Here’s another simple algebra explanation, gleaned from [this site](http://mathforum.org/library/drmath/view/57041.html): x = .333… 10x = 3.333… 3.333… = 3 + .333… = 3 + x 10x = 3 + x 10x - x = 3 + x - x 9x = 3 x = 3/9 = 1/3 If your friend doesn’t understand simple algebra or still refuses to accept it, I suggest you stop attempting to teach pigs to sing. ---
**Author:** ![MaxTheVool](https://avatars.discourse-cdn.com/v4/letter/m/9fc29f/32.png) [@MaxTheVool](https://boards.straightdope.com/u/MaxTheVool)\ **Post date:** [April 13, 2004, 10:13pm UTC](https://boards.straightdope.com/t/help-me-convince-a-friend-that-0-333-1/239752/8 "2004-04-13T22:13:11Z")
> [@duality72](#): > > Here’s another simple algebra explanation, gleaned from [this site](http://mathforum.org/library/drmath/view/57041.html): > > x = .333… > 10x = 3.333… > 3.333… = 3 + .333… = 3 + x > 10x = 3 + x > 10x - x = 3 + x - x > 9x = 3 > x = 3/9 = 1/3 > > If your friend doesn’t understand simple algebra or still refuses to accept it, I suggest you stop attempting to teach pigs to sing. This proof works equally well for .9999… equalling 1, by the way.