# .99999999 equal to 1 ???

**URL:** <https://boards.straightdope.com/t/99999999-equal-to-1/76468>\
**Category:** Factual Questions\
**Created:** [August 15, 2001, 11:42pm UTC](https://boards.straightdope.com/t/99999999-equal-to-1/76468 "2001-08-15T23:42:36Z")\
**Posts on this page:** 20\
**Page:** 4

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**Author:** ![WordMan](https://avatars.discourse-cdn.com/v4/letter/w/e8c25b/32.png) [@WordMan](https://boards.straightdope.com/u/WordMan)\
**Post date:** [August 16, 2001, 2:17pm UTC](https://boards.straightdope.com/t/99999999-equal-to-1/76468/61 "2001-08-16T14:17:18Z")

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> [@](#):
>
> \*Originally posted by China Girl \*  
> \*\*OH, PLEASE!!!
> 
> ## quote:
> 
> ## Evno gets into the act with I’m not a math person, but I play one on T.V. And I have a question: If .99999999… = 1, then does -.11111111… = 0?
> 
> Surely we can see not many math people here!  
> I know .9rep=1 since I was in elementary school. Don’t you people see the proofs.  
> Thanks to those people who still have clear heads! JC Princeton, it’s effortless to convience those “not a math person” a math truth. \*\*

Uh, excuse me China Girl, while I agree with the math point you are frustrated that people don’t understand, isn’t the point of the SDMB to bring the curious together with the informed to discuss issues? The fact that sometimes people need to “pressure test” their ideas (repeatedly at times) and maybe not come around to expert’s point of view is just part of the process.

No need to be dismissive and harsh.

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**Author:** ![WordMan](https://avatars.discourse-cdn.com/v4/letter/w/e8c25b/32.png) [@WordMan](https://boards.straightdope.com/u/WordMan)\
**Post date:** [August 16, 2001, 2:22pm UTC](https://boards.straightdope.com/t/99999999-equal-to-1/76468/62 "2001-08-16T14:22:16Z")

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> [@](#):
>
> \*Originally posted by China Girl \*  
> \*\*JC Princeton, it’s effortless to convience those “not a math person” a math truth. \*\*

BTW, it would be “pointless” not “effortless” - if it were effortless, you wouldn’t have felt the need to get feisty.

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**Author:** ![douglips](https://avatars.discourse-cdn.com/v4/letter/d/e68b1a/32.png) [@douglips](https://boards.straightdope.com/u/douglips)\
**Post date:** [August 16, 2001, 2:24pm UTC](https://boards.straightdope.com/t/99999999-equal-to-1/76468/63 "2001-08-16T14:24:08Z")

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> [@](#):
>
> \*Originally posted by kinoons \*  
> \*\*okay, you want to prove to me that 0.999… is equal to 1.000… then do this for me…  
> draw a line. Label one point on the line 0.9. one inch later on that line mark a point 1.000… (add as many 0’s as you would like.
> 
> now, draw a line for 0.99. Now draw a line for 0.999. Now draw a line for 0.9999. Now one for 0.99999. Continue to keep drawing lines please.
> 
> Call me when you draw an additional line on 1.000 \*\*

Kinoons, you’ve just accidentally disproved your point, and proved that .9999(infinite) is 1.0.  
You say “Call me when you draw a line on 1.0.” I say, “call me when you draw a line on .9999999999(infinite)”. Do you see that? Because we are dealing with infinity, I can never draw a line on .99999999… unless I am also drawing it on 1.0.

In other words, before I can satisfy your test, you must satisfy mine. First, you draw a line on .9999… and then I’ll show you the line on 1.0.

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**Author:** ![John\_Kentzel-Griffin](https://avatars.discourse-cdn.com/v4/letter/j/ccd318/32.png) [@John\_Kentzel-Griffin](https://boards.straightdope.com/u/John_Kentzel-Griffin)\
**Post date:** [August 16, 2001, 4:00pm UTC](https://boards.straightdope.com/t/99999999-equal-to-1/76468/64 "2001-08-16T16:00:11Z")

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:: pats **China Girl** on the back ::

There, there. I feel your pain. This question has come up before. And it always seems to go just like this thread. I know how frustrating it is, beleive me. Let’s see if we can find some other thread that won’t be so annoying. Oh, look! Here is a thread about [Xeno’s Paradox](http://boards.straightdope.com/sdmb/showthread.php?threadid=81905).

\<g d & r\>

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**Author:** ![lucwarm](https://avatars.discourse-cdn.com/v4/letter/l/e19adc/32.png) [@lucwarm](https://boards.straightdope.com/u/lucwarm)\
**Post date:** [August 16, 2001, 4:31pm UTC](https://boards.straightdope.com/t/99999999-equal-to-1/76468/65 "2001-08-16T16:31:10Z")

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I would add to the above discussion that the answer to the question depends on the definitions you choose.

It is simple and consistent to equate (0.999 . . .) and 1, but by no means required. But be advised that if you choose not to equate the two, you will end up with some unpleasant little holes/limitations/undefined areas in your mathematical theory.

(Note also that conventional mathematics leaves certain expressions and operations undefined. For example, consider the following proof:

Let S = 1 + 2 + 4 + 8 . . .

Then S - 1 = 2 + 4 + 8 . . .

Then (S - 1) / 2 = 1 + 2 + 4 + 8 . . .

Thus, (S - 1) / 2 = S

So S - 1 = 2S, and S = -1

Therefore 1 + 2 + 4 + 8 . . . = -1 )

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**Author:** ![WordMan](https://avatars.discourse-cdn.com/v4/letter/w/e8c25b/32.png) [@WordMan](https://boards.straightdope.com/u/WordMan)\
**Post date:** [August 16, 2001, 4:36pm UTC](https://boards.straightdope.com/t/99999999-equal-to-1/76468/66 "2001-08-16T16:36:52Z")

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> [@](#):
>
> \*Originally posted by lucwarm \*  
> \*\*I would add to the above discussion that the answer to the question depends on the definitions you choose.
> 
> It is simple and consistent to equate (0.999 . . .) and 1, but by no means required. But be advised that if you choose not to equate the two, you will end up with some unpleasant little holes/limitations/undefined areas in your mathematical theory.
> 
> (Note also that conventional mathematics leaves certain expressions and operations undefined. For example, consider the following proof:
> 
> Let S = 1 + 2 + 4 + 8 . . .
> 
> Then S - 1 = 2 + 4 + 8 . . .
> 
> Then (S - 1) / 2 = 1 + 2 + 4 + 8 . . .
> 
> Thus, (S - 1) / 2 = S
> 
> So S - 1 = 2S, and S = -1
> 
> Therefore 1 + 2 + 4 + 8 . . . = -1 ) \*\*

Okay - how the heck did THAT happen? What type of mathematical/logical fallacy did you illustrate here?!

(or, to put it more simply - don’t DO that to me - my little brain can’t take it…:D)

Word(obviously not Math)Man

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**Author:** ![kinoons](https://avatars.discourse-cdn.com/v4/letter/k/a698b9/32.png) [@kinoons](https://boards.straightdope.com/u/kinoons)\
**Post date:** [August 16, 2001, 4:45pm UTC](https://boards.straightdope.com/t/99999999-equal-to-1/76468/67 "2001-08-16T16:45:28Z")

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okay, call me ignorant, stupid, so on and so forth – No matter how you manipulate the math to make the proofs work, .0999… cannot equal to one. It can become infinitely close, but it cannot equal one. The only thing that can equal one is one. It is truly less a mathematical debate than a logical one. Two numbers cannot occupy the same point on the number line. It’s that simple. I’ve understood that since elementary school as well.

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**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\
**Post date:** [August 16, 2001, 4:51pm UTC](https://boards.straightdope.com/t/99999999-equal-to-1/76468/68 "2001-08-16T16:51:53Z")

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**Wordman** : the series is divergent, and therefore it doesn’t have a defined sum. When you try to treat divergent series as if they were convergent, you get odd results. For a less technical understanding, let’s pretend that S is equal to infinity. Infinity minus one is still infinite, and infinity over two is also infinite. It shouldn’t be a problem.

**Kinoons** : I invite you to look at any 7th or 8th grade algebra textbook for a proof of this. You’re missing a clause from your principle; it should read “Two numbers can’t occupy the same point on the number line **unless** they’re equal.” .999… is a different representation for the number 1, and two different representations for the same number point to the same point on the number line. My previous post on this topic, btw, is a good rigorous proof that .999… and 1 are the same number. Why don’t you explain why that one’s wrong?

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**Author:** ![John\_Kentzel-Griffin](https://avatars.discourse-cdn.com/v4/letter/j/ccd318/32.png) [@John\_Kentzel-Griffin](https://boards.straightdope.com/u/John_Kentzel-Griffin)\
**Post date:** [August 16, 2001, 4:57pm UTC](https://boards.straightdope.com/t/99999999-equal-to-1/76468/69 "2001-08-16T16:57:19Z")

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> [@](#):
>
> \*Originally posted by kinoons \*  
> Two numbers cannot occupy the same point on the number line. It’s that simple. I’ve understood that since elementary school as well.

Two _different_ numbers cannot occupy the same point on the number line. One and 0.999… are not two _different_ numbers. They are two _names_ for the exact same number. Would you say that zero does not equal 1 - 1, because they cannot occupy the same point?

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**Author:** ![WordMan](https://avatars.discourse-cdn.com/v4/letter/w/e8c25b/32.png) [@WordMan](https://boards.straightdope.com/u/WordMan)\
**Post date:** [August 16, 2001, 5:02pm UTC](https://boards.straightdope.com/t/99999999-equal-to-1/76468/70 "2001-08-16T17:02:22Z")

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> [@](#):
>
> \*Originally posted by kinoons \*  
> \*\*okay, call me ignorant, stupid, so on and so forth – No matter how you manipulate the math to make the proofs work, .0999… cannot equal to one. It can become infinitely close, but it cannot equal one. The only thing that can equal one is one. It is truly less a mathematical debate than a logical one. Two numbers cannot occupy the same point on the number line. It’s that simple. I’ve understood that since elementary school as well. \*\*

Ultimately, it’s your choice, kinoons - the arguments provided are pretty strong, so I don’t see a lot of value in pushing this further - I have to ask if you have taken the time to explore the Zeno’s paradox concept I described in this thread and that is laid out quite well in the other thread pointed at by DrMatrix, but that’s about it. The math types (and I am not one of them) are arguing that .999… vs. 1 is the same as 6 vs. a half dozen: same amount, different names. If you choose to disagree, no amount of arguing will change that…

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**Author:** ![lucwarm](https://avatars.discourse-cdn.com/v4/letter/l/e19adc/32.png) [@lucwarm](https://boards.straightdope.com/u/lucwarm)\
**Post date:** [August 16, 2001, 5:23pm UTC](https://boards.straightdope.com/t/99999999-equal-to-1/76468/71 "2001-08-16T17:23:47Z")

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**ultrafilter** is correct that conventional mathematics won’t let you manipulate divergent series as if they were numbers.

My point is that you should feel free to reject convergent series if you like.

People should not claim that (0.9999 . . . ) = 1 is some sort of cosmic truth. It’s merely simple, consistent, and elegant to equate the two expressions.

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**Author:** ![John\_Kentzel-Griffin](https://avatars.discourse-cdn.com/v4/letter/j/ccd318/32.png) [@John\_Kentzel-Griffin](https://boards.straightdope.com/u/John_Kentzel-Griffin)\
**Post date:** [August 16, 2001, 5:30pm UTC](https://boards.straightdope.com/t/99999999-equal-to-1/76468/72 "2001-08-16T17:30:24Z")

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> [@](#):
>
> \*Originally posted by \*\*\*lucwarm \*\*  
> My point is that you should feel free to reject convergent series if you like.

Umm. . . OK. Let’s say that I reject convergent series. Now, 1 does not equal 0.999… because there is no such thing as 0.999… I rejected it.

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**Author:** ![lucwarm](https://avatars.discourse-cdn.com/v4/letter/l/e19adc/32.png) [@lucwarm](https://boards.straightdope.com/u/lucwarm)\
**Post date:** [August 16, 2001, 5:55pm UTC](https://boards.straightdope.com/t/99999999-equal-to-1/76468/73 "2001-08-16T17:55:23Z")

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> [@](#):
>
> \*Originally posted by DrMatrix \*  
> \*\*Umm. . . OK. Let’s say that I reject convergent series. Now, 1 does not equal 0.999… because there is no such thing as 0.999… I rejected it. \*\*

If you want to leave (9/10 + 9/100 + 9/1000 . . .) undefined, that’s fine by me.

Or, you could simply say that infinite series may not be manipulated using the traditional rules of arithmetic.

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**Author:** ![kinoons](https://avatars.discourse-cdn.com/v4/letter/k/a698b9/32.png) [@kinoons](https://boards.straightdope.com/u/kinoons)\
**Post date:** [August 16, 2001, 6:12pm UTC](https://boards.straightdope.com/t/99999999-equal-to-1/76468/74 "2001-08-16T18:12:04Z")

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but 0.999… is not the same number as 1.0 – even simple place value shows that any number that begins with 0.9 must be less than any number that begins with 1.0

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**Author:** ![Scott\_Dickerson](https://avatars.discourse-cdn.com/v4/letter/s/f04885/32.png) [@Scott\_Dickerson](https://boards.straightdope.com/u/Scott_Dickerson)\
**Post date:** [August 16, 2001, 7:06pm UTC](https://boards.straightdope.com/t/99999999-equal-to-1/76468/75 "2001-08-16T19:06:22Z")

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There is no doubt whatsoever that the logic of standard arithmetic proves that .9rep = 1. Now, let’s move beyond the logic of standard arithmetic.

We will make three parallel columns. In the first, the numeral “1” and its decimal extension (ie, 1.00000…etc.); in the second, a representation of .9rep (ie,0.99999…etc.); in the third, the difference between columns 1 and 2. Each row of the first two columns, reading down, represents the next decimal place.

1 0 1  
0 .9 -.9  
0 .09 -.09  
0 .009 -.009  
0 .0009 -.0009  
0 .00009 -.00009  
(…etc…)

For each and every new entry in column one, there is a new entry in column 2. This continues without end, by definition. There is no entry anywhere in column 2 that “makes up the remaining difference” between column 1 (a known totality, as further zeroes add nothing) and column 2 (an accumulating sum which is never completed). If by definition something never happens, how does mathematical logic allow us to say just the reverse? In other words, why are we permitted to understand as a totality a string of numerals of which “totality” cannot be predicated?

Doesn’t something called Robinsonian Nonstandard Analysis allow a real difference between the two terms?

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**Author:** ![MrDeath](https://avatars.discourse-cdn.com/v4/letter/m/22d042/32.png) [@MrDeath](https://boards.straightdope.com/u/MrDeath)\
**Post date:** [August 16, 2001, 7:08pm UTC](https://boards.straightdope.com/t/99999999-equal-to-1/76468/76 "2001-08-16T19:08:19Z")

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Gee, I appear to have touched on a nerve here. Unfortunately, my copy of _Infinity and the Mind_ is at the office, so I’m going to have to wing this one.

Okay, many of you have probably heard of John Conway, the 20th century’s most eccentric mathematician. He invented the “game” of Life, discovered the Conway group, and almost in his spare time built up the framework of modern game theory. All these are amazing things, but I expect that the thing he will be most remembered for among mathematicians is the surreal numbers.

A _very_ quick course on the surreal numbers:  
[/list][li]A surreal number is a _pair_ of sets of surreal numbers, a left set and a right set, with every element of the left set less than every element of the right set; conversely, every such pair of sets is a surreal number. If the left set is A and the right set is B, we denote the number by (A|B).[/li]li \<= (C|D) if every element of A is less than (C|D) and (A|B) is less than every element of D. (A|B) = (C|D) if (A|B) \<= (C|D) and (C|D) \<= (A|B). (A|B) \< (C|D) if (A|B) \<= (C|D) and ((C|D) \<= (A|B) is false).[/li][li]Hi Opal![/li]li + (C|D) = (a + (C|D), c + (A|B) | (A|B) + d, (C|D) + b), where the lower case letters are numbers that are elements of the respective capital letters.[/li][/list]  
There is a similar rule for multiplication but it gets a little tricky. Since the empty set is (vacuously) a set of surreal numbers, the “simplest” surreal number is ( | ) (when writing out sets in surreal numbers, we leave the braces off), which we call 0. The next-simplest numbers are (0| ) and ( |0). You cannot have (0| ) \<= 0, since by our second rule that would mean 0 \< 0; but 0 \<= 0, so 0 \< 0 is false. (I know, the definitions are a little circular, but you _can_ make it work.) However, 0 \<= (0| ), so we have 0 \< (0| ); likewise ( |0) \< 0. We call (0| ) 1 and ( |0) -1. It turns out by the addition rule we have above, 0 is the additive identity, and by the multiplication rule that I didn’t write out, 1 is the multiplicative identity. By simple repetition of the rules, we can construct (among many other things) all the real numbers in a pretty straightforward way, so that the addition and multiplication on surreal numbers is the same as conventional addition and multiplication. I’m going to ask you to buy that for a moment.

It turns out that there is a very natural “binary expansion” for every surreal number. I’m not going to go into enormous detail, except to say that it matches up pretty closely with our usual concept of binary expansion; i.e., 1 = 1.0000…, 1/2 = 0.100000…, 1/3 = 0.0101010101… and so on. The tricky bit is, every binary expansion turns out to correspond to a _unique_ surreal number. Where does that leave us? Well, it leaves us with 0.11111… being a different number than 1.00000…, for starters, but that isn’t a lot of help. What number _is_ it?

Well, let’s go back to our other notation. (Sometimes, having two notations can help explain things that one notation alone can’t.) 0.10000… = (0|1) = 1/2, 0.110000… = (0,1/2|1) = 3/4, 0.1110000… = (0,1/2,3/4|1) = 7/8, … so intuitively we should have 0.111111… = (0,1/2,3/4,7/8,15/16,…|1), where the left set is infinite (but each of its elements is strictly less than 1). This surreal number is less than 1, but greater than every _real_ number between 0 and 1. The normal way to express it is 1 - 1/omega, where you can think of omega as being infinity.\*

How does this work with the usual proof that .999999… = 1?  
The proof as mathematicians usually write it out goes as follows. Call the value of 0.9999999… S. Then

10S = 9.9999999…  
S = 0.9999999…  
Subtracting on both sides, we get  
9S = 9

Therefore, S = 1. But clearly we’re missing something if we try this out with .999999… = 1 - 1/omega. The problem is, we can get at 9.99999… two different ways - we can multiply .999999… by 10, or we can add 9 to it instead. (This is the trick that makes the proof work.) In the world of surreal numbers, the two methods are _not_ compatible and give you different answers: 9 + (1 - 1/omega) = 10 - 1/omega, but 10(1 - 1/omega) = 10 - _10_/omega, which is a distinct surreal number, although the two are only infinitesimally different.

Bored yet?

- In fact, omega is the surreal number (0,1,2,3,…| ); it’s the first of the “transfinite” surreal numbers, but it’s neither the greatest nor the smallest. For instance, (|omega) \< omega \< (omega|), and so on. Reciprocals of infinite surreal numbers exist; in fact, every surreal except 0 has a unique inverse. If this sort of stuff is endlessly fascinating to you, see if you can track down a copy of Donald Knuth’s little book _Surreal Numbers_. If you’re still not satisfied, then try Conway’s book _On Numbers and Games_, preferrably the second edition. Or send me an e-mail; I find this stuff endlessly fascinating too.

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<div class="post-metadata">

**Author:** ![MrDeath](https://avatars.discourse-cdn.com/v4/letter/m/22d042/32.png) [@MrDeath](https://boards.straightdope.com/u/MrDeath)\
**Post date:** [August 16, 2001, 7:11pm UTC](https://boards.straightdope.com/t/99999999-equal-to-1/76468/77 "2001-08-16T19:11:16Z")

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> [@](#):
>
> _Originally posted by MrDeath \*  
> John Conway, the 20th century’s most eccentric mathematician_\*

Let’s hedge our bets here and call him that century’s third-most-eccentric. Paul Erdos was certainly more eccentric (but probably not quite as brilliant), and I’ll leave another slot open so that people can say, “But \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ was more eccentric too!”

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**Author:** ![ModernRonin2](https://avatars.discourse-cdn.com/v4/letter/m/c37758/32.png) [@ModernRonin2](https://boards.straightdope.com/u/ModernRonin2)\
**Post date:** [August 16, 2001, 7:22pm UTC](https://boards.straightdope.com/t/99999999-equal-to-1/76468/78 "2001-08-16T19:22:34Z")

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If it weren’t for the fact that we’re dealing with  
(disguised forms of) infinity here.

The “…” in “0.9…” is just a symbol for infinity, really. And strange things happen when you deal with infinity. As someone pointed out infinity - 1 = infinity. As a matter of face, infinity minus any constant still equals infinity. Now that’s bizzare. That doesn’t work with real numbers. But it works with infinity. Transfinite numbers are strange. Sometimes one kind of infinity doesn’t equal another one.

This happens, I guess, because infinity is not properly a number. It’s a concept, and different rules apply to manipulating concepts than numbers.

I’m not trying to talk you out of your belief that .9… != 1 here. More like, trying to explain how I managed to understand it myself. I can accept that .9… = 1 because I know there’s an infinity involved. And strange things start happening when you deal with infinities.  
-Ben

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<div class="post-metadata">

**Author:** ![John\_Kentzel-Griffin](https://avatars.discourse-cdn.com/v4/letter/j/ccd318/32.png) [@John\_Kentzel-Griffin](https://boards.straightdope.com/u/John_Kentzel-Griffin)\
**Post date:** [August 16, 2001, 7:58pm UTC](https://boards.straightdope.com/t/99999999-equal-to-1/76468/79 "2001-08-16T19:58:33Z")

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> [@](#):
>
> \*Originally posted by ModernRonin2 \*  
> This happens, I guess, because infinity is not properly a number. It’s a concept, and different rules apply to manipulating concepts than numbers.

What is a number? A number is an element of whatever set you decide to call the set of numbers. If you are using Natural numbers, then one is a number, but minus one is not. If you are using the integers one and minus one are numbers, but 1/2 and _i_ are not numbers. There are perfectly consistent definitions of numbers that allow infinity (or infinities) to be a number. Transfinite Ordinals and Transfinite Cardinals have numbers that are not finite. Conways Surreal Numbers, mentioned by \*_MrDeath_, contain all the real numbers plus all the transfinite numbers too.

**MrDeath**  
I loved _Surreal Numbers_. I read it when I was in college. The full title of the book is _Surreal Numbers: How two ex-students turned on to pure mathematics and found total happiness_. Great book – great title. I used to have a copy of _On Numbers and Games_. John Conway is brilliant. Such a rich system based upon such simple rules.

* * *

Virtually yours,

**DrMatrix** — If I’ve told you once, I’ve told you 0.999… times.

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**Author:** ![DMark](https://avatars.discourse-cdn.com/v4/letter/d/e68b1a/32.png) [@DMark](https://boards.straightdope.com/u/DMark)\
**Post date:** [August 16, 2001, 9:15pm UTC](https://boards.straightdope.com/t/99999999-equal-to-1/76468/80 "2001-08-16T21:15:53Z")

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so, I guess the Buck finally stops at a 99 Cent store.

By the way, Pi ends with a 6.  
1/2 of zero is C  
1+1+1 could be 3 or 111, depending on how you see it.

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