# Why doesn't .9999~ = 1? **URL:** \ **Category:** Factual Questions\ **Created:** [June 1, 2003, 11:12pm UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019 "2003-06-01T23:12:58Z")\ **Posts on this page:** 20\ **Page:** 1
**Author:** ![Lint6](https://avatars.discourse-cdn.com/v4/letter/l/c57346/32.png) [@Lint6](https://boards.straightdope.com/u/Lint6)\ **Post date:** [June 1, 2003, 11:12pm UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/1 "2003-06-01T23:12:58Z")
Someone asked this question on another message board I go to. I’m asking it here because the other message board is mostly about video games and I figured someone here would have a better idea. 1/3 = .3333~ 2/3 = .6666~ 3/3 = .9999~ But 3/3 = 1 So why doesn’t .9999~ = 1? ---
**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:** [June 1, 2003, 11:16pm UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/2 "2003-06-01T23:16:27Z")
[It does.](http://boards.straightdope.com/sdmb/showthread.php?s=&threadid=15863) ---
**Author:** ![Orbifold](https://avatars.discourse-cdn.com/v4/letter/o/779978/32.png) [@Orbifold](https://boards.straightdope.com/u/Orbifold)\ **Post date:** [June 1, 2003, 11:23pm UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/3 "2003-06-01T23:23:31Z")
See also [this thread](http://boards.straightdope.com/sdmb/showthread.php?s=&threadid=32760). And [this thread.](http://boards.straightdope.com/sdmb/showthread.php?s=&threadid=82064). (Curse you and your fast typing skills, **Q.E.D.**!!!) ---
**Author:** ![x-ray\_vision](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/x-ray_vision/32/351_2.png) [@x-ray\_vision](https://boards.straightdope.com/u/x-ray_vision)\ **Post date:** [June 1, 2003, 11:58pm UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/4 "2003-06-01T23:58:30Z")
[It doesn’t](http://mathforum.org/library/drmath/view/55748.html) ---
**Author:** ![Azael](https://avatars.discourse-cdn.com/v4/letter/a/a8b319/32.png) [@Azael](https://boards.straightdope.com/u/Azael)\ **Post date:** [June 2, 2003, 12:10am UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/5 "2003-06-02T00:10:20Z")
Read your link **x-ray vision** … It does. ---
**Author:** ![x-ray\_vision](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/x-ray_vision/32/351_2.png) [@x-ray\_vision](https://boards.straightdope.com/u/x-ray_vision)\ **Post date:** [June 2, 2003, 12:15am UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/6 "2003-06-02T00:15:11Z")
I sent the wrong link and I can’t find the one I meant to send, but .999… will always be smaller than 1. And 1/3 does not equal .333… ---
**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:** [June 2, 2003, 12:16am UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/7 "2003-06-02T00:16:00Z")
Might want to read your cites a bit more thoroughly, **x-ray vision** : > [@](#): > > You can show (using calculus or other methods) that with a large enough number of 9s in the expansion, you can get arbitrarily close to 1, and here’s the key: > > THERE IS NO OTHER NUMBER THAT THE SEQUENCE GETS ARBITRARILY CLOSE TO. > > Thus, if you are going to assign a value to .9999… (going on forever), the only sensible value is 1. ---
**Author:** ![Azael](https://avatars.discourse-cdn.com/v4/letter/a/a8b319/32.png) [@Azael](https://boards.straightdope.com/u/Azael)\ **Post date:** [June 2, 2003, 12:17am UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/8 "2003-06-02T00:17:42Z")
> [@](#): > > but .999… will always be smaller than 1. Really? By how much? ---
**Author:** ![TBone2](https://avatars.discourse-cdn.com/v4/letter/t/6a8cbe/32.png) [@TBone2](https://boards.straightdope.com/u/TBone2)\ **Post date:** [June 2, 2003, 12:19am UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/9 "2003-06-02T00:19:45Z")
> [@](#): > > \*Originally posted by Lint6 \* > \*\*3/3 = .9999~ > > But > 3/3 = 1\*\* Make up your mind. The first of your statements that I quoted is false, and can only be discussed using the principles of “The Calculus.” Your second quoted statement is true, following the basic rule of math that states that any number divided by itself equals unity, or 1. While 1/3 can be said to be .333333… and 2/3 can be said to be .6666666…, it is silly to simply multiply the infinite decimally-expressed value of 1/3 by 3 to arrive at a statement like “3/3 = .9999~.” ---
**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:** [June 2, 2003, 12:20am UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/10 "2003-06-02T00:20:11Z")
> [@](#): > > \*Originally posted by Azael \* > \*\*Really? By how much? \*\* By 1/infinity. 😃 ---
**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:** [June 2, 2003, 12:23am UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/11 "2003-06-02T00:23:56Z")
> [@](#): > > \*Originally posted by TBone2 \* > \*\*While 1/3 can be said to be .333333… and 2/3 can be said to be .6666666…, it is silly to simply multiply the infinite decimally-expressed value of 1/3 by 3 to arrive at a statement like “3/3 = .9999~.” \*\* No it’s not. You agree that 1/3 = .33333… and 2/3 = .66666. Ergo: Since 1/3 + 2/3 = 3/3 and .33333… + .66666… = .99999 Therefore 3/3 = .99999 Q.E.D. ---
**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:** [June 2, 2003, 12:25am UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/12 "2003-06-02T00:25:23Z")
Put … after both instances of .99999 in my above post. :smack: ---
**Author:** ![Achernar](https://avatars.discourse-cdn.com/v4/letter/a/e274bd/32.png) [@Achernar](https://boards.straightdope.com/u/Achernar)\ **Post date:** [June 2, 2003, 12:25am UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/13 "2003-06-02T00:25:29Z")
Anyone who thinks it is untrue that 0.999… = 1 does not understand limits. (Which is not a terrible thing; lots of Calculus students don’t understand limits.) Imagine you had a blackboard with 0.999 on it. And every second you added another 9 to the end. The number you would have at the “end of time” _would not_ be the same as 0.999… Also, it is quite possible for every entry in a sequence to be less than the limit of that sequence. That doesn’t mean that its limit is not its limit. **Q.E.D.** , please be a little more careful with your ellipses. They’re more important when discussing this problem than any other time, in my estimation. 😉 ---
**Author:** ![Jabba](https://avatars.discourse-cdn.com/v4/letter/j/b5e925/32.png) [@Jabba](https://boards.straightdope.com/u/Jabba)\ **Post date:** [June 2, 2003, 12:27am UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/14 "2003-06-02T00:27:12Z")
**TBone2** asserted: > [@](#): > > The first of your statements that I quoted is false, and can only be discussed using the principles of “The Calculus.” Your second quoted statement is true, following the basic rule of math that states that any number divided by itself equals unity, or 1. > > While 1/3 can be said to be .333333… and 2/3 can be said to be .6666666…, it is silly to simply multiply the infinite decimally-expressed value of 1/3 by 3 to arrive at a statement like “3/3 = .9999~.” Why is it silly? Using “The Calculus” ( or analysis, if you prefer;specifically, properties about infinite series): 0.333… = [symbol]S[/symbol]3/10[sup]i[/sup] = 1/3 ( sum of an infinite G.P.) and so 1 = 3/3 = 3\*1/3 = 3[symbol]S[/symbol]3/10[sup]i[/sup] = [symbol]S[/symbol]9/10[sup]i[/sup] = 0.999… ---
**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:** [June 2, 2003, 12:27am UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/15 "2003-06-02T00:27:22Z")
> [@](#): > > \*Originally posted by Achernar \* > \*\*\*\*Q.E.D.\*\*, please be a little more careful with your ellipses. They’re more important when discussing this problem than any other time, in my estimation. 😉 \*\* I’ve addressed that, schmuck! 😉 ---
**Author:** ![x-ray\_vision](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/x-ray_vision/32/351_2.png) [@x-ray\_vision](https://boards.straightdope.com/u/x-ray_vision)\ **Post date:** [June 2, 2003, 12:35am UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/16 "2003-06-02T00:35:10Z")
[here](http://www.math.lsa.umich.edu/mmss/courses/infinity/Equals1.shtml) is another web-page purporting that .999… doesn’t = 1, but check out the last two paragraphs (bolding mine). > [@](#): > > The numbers in the “Answer” column do indeed get closer and closer to some number, namely 1. **The fact that they never get there is irrelevant.** 1 is by definition the exact answer to the infinite addition problem. > > So in conclusion, Point-Nine-Repeating is indeed equal to one. In order to see it, we needed to know how to add up infinitely many small numbers. By the way, this knowledge itself is probably more important that the little fact that .999…=1. How the hell is the fact that they never get their irrelevant? The **fact** that they never get there is exactly why it doesn’t equal 1! ---
**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:** [June 2, 2003, 12:38am UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/17 "2003-06-02T00:38:40Z")
It’s an infinite series. When they talk about it never reaching one, they are referring to an increasing, but still finite, number of 9s being tacked onto the end. ---
**Author:** ![Achernar](https://avatars.discourse-cdn.com/v4/letter/a/e274bd/32.png) [@Achernar](https://boards.straightdope.com/u/Achernar)\ **Post date:** [June 2, 2003, 12:39am UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/18 "2003-06-02T00:39:53Z")
> [@](#): > > \*Originally posted by Q.E.D. \* > I’ve addressed that, schmuck! 😉 You’ll notice that the two posts in question are less than a minute apart. So nyah. And anyway, you missed one. The first 0.6666… ---
**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:** [June 2, 2003, 12:40am UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/19 "2003-06-02T00:40:46Z")
Those are just upside-down 9s. 😃 ---
**Author:** ![Jabba](https://avatars.discourse-cdn.com/v4/letter/j/b5e925/32.png) [@Jabba](https://boards.straightdope.com/u/Jabba)\ **Post date:** [June 2, 2003, 12:43am UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/20 "2003-06-02T00:43:31Z")
No, the paragraohs you quote are quite correct. It is an important feature of limits that they do noy have to reach their limiting value after any finite number of steps. For a ( usually) less controversial example, consider the sequence 1/n. Its terms are 1, 1/2, 1/3, 1/4, 1/5, … Most people would agree that this sequence tends to 0 as n tends to infinity, and they would be correct. The fact that no term of the sequence is actually 0 is wholly irrelevant. Similarly, when we say that 0.999… = 1 we mean that 1 is the ( unique) limit of the sequence 0.9, 0.99, 0.999, 0.9999, … [Next page](https://boards.straightdope.com/t/why-doesnt-9999-1/179019.md?page=2)