# 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:** 11
**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 3, 2003, 9:37pm UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/201 "2003-06-03T21:37:52Z")
> [@](#): > > \*Originally posted by Phage \* > \*\*My credentials are not of any worth; I am not a mathematician. > > 1. The number 0.999… has an infinite number of 9s on the end. > 2. Infinity can never be reached. > 3. As more 9s in 0.999… are considered, the closer the number comes to 1. > 4. The only way for 0.999… to be equal to 1, is if an infinite number of 9s is placed on the end. This can be considered in theory, but as infinity cannot actually be reached in reality 0.999… will never equal 1. \*\* You’ve almost got it. The problem you’re hung up on is the infinite series of 9s. No one is _placing_ the 9s there. They are just _there_. Whether it’s physically possible to actually write them all our or not is immaterial, sice that’s not what we’re discussing. there _is_ an infinite number of 9s after the decimal point. Accept that. ---
**Author:** ![Terminus\_Est](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/terminus_est/32/3087_2.png) [@Terminus\_Est](https://boards.straightdope.com/u/Terminus_Est)\ **Post date:** [June 3, 2003, 9:38pm UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/202 "2003-06-03T21:38:58Z")
Such a small mind that cannot encompass the notion of infinity. ---
**Author:** ![Cervaise](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/cervaise/32/16693_2.png) [@Cervaise](https://boards.straightdope.com/u/Cervaise)\ **Post date:** [June 3, 2003, 9:50pm UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/203 "2003-06-03T21:50:00Z")
> [@](#): > > \*Originally posted by Skogcat \* > \*\ ***Phage** , you seem very knowledgeable and sure of yourself, are you a mathematician? \*\* No, he doesn’t have any formal training, but soon he will be introducing a revolutionary machine that produces more power than is put into it and that proves Einstein was wrong! ---
**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:** [June 3, 2003, 9:52pm UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/204 "2003-06-03T21:52:19Z")
**Phage** Phage, like many mathematicians and the mathematically inclined here, we’ll have to agree to disagree. A limit, properly applied, can provide an exact answer. You can disagree, if you like, but in doing so, you either define your own mathematics, or are arguing with the basis of established mathematics. The former is perfectly reasonable. The latter requires extraordinary proof, and counterexamples that have _already_ been given to you have to be shown to be flawed. You have yet to do so. You can define what 0.999… means to you, but it doesn’t mean that to mathematicians, and they have a better reason for doing so than you do – its consistent with the axioms of mathematics and everything deriving from them. As an example, if limits did not produce exact values, then it would not be appropriate to say that: d( x^3 ) ----------- = 3x^2 dx (That the derivative (slope at a point) of x cubed is three times x squared) for all values of x, not just for some values of x, or with some error component. The reverse, that S 3x^2 dx = x^3 + c (I use S as a cheap integral symbol, can’t figure out how to do it) is true for all values of x, not just some x, or with an error component. None of these would be provable without using some concept of limits, and allowing those limits to achieve exact values. They are not approximations. Because of the way reals are defined, and limits are defined, the result that 0.999… = 1 is a consequence, and an unavoidable one based on the axioms involved. If you accept the axioms, and still don’t accept the result, your logic is flawed, intuition aside. If you don’t accept the axioms, you aren’t using the same mathematics as we are. Since it is an iron-clad property that between two reals a and b there is at least one other real c such that c=(a+b)/2, there can never be a ‘next real’ or a ‘previous real’ in the real number line. Ever. This has already been discussed. There is no value difference between 0.999… and 1. There can’t be on the real number line. It may be non-intuitive, but sometimes that’s the way it goes. But as you say, we’re just going over old ground, and I’m ill-equipped to prove right from first principles, and the true mathematicians here won’t bother as the result is already well established in countless existing mathematics tomes, courses, and web-pages. ---
**Author:** ![LordVor](https://avatars.discourse-cdn.com/v4/letter/l/90ced4/32.png) [@LordVor](https://boards.straightdope.com/u/LordVor)\ **Post date:** [June 3, 2003, 9:53pm UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/205 "2003-06-03T21:53:57Z")
> [@](#): > > \*Originally posted by Phage \* > Take a look at this address, already posted at the start of the thread: [http://mathforum.org/library/drmath/view/55748.html](http://mathforum.org/library/drmath/view/55748.html) > > 1. The number 0.999… has an infinite number of 9s on the end. > 2. Infinity can never be reached. > 3. As more 9s in 0.999… are considered, the closer the number comes to 1. > 4. The only way for 0.999… to be equal to 1, is if an infinite number of 9s is placed on the end. This can be considered in theory, but as infinity cannot actually be reached in reality 0.999… will never equal 1. \*\* First, the cite listed is an answer appropriate for an 8th grade math class. The page cited BY that page for more information matches exactly at least two of the arguments presented against you. The problem with your thinking is at step 3. The elipse is defined as the representation of an infinite # of 9s. Since the elipse is designed to represent an infinite # of 9s, .999… is equal to 1, by your own fourth point. -lv ---
**Author:** ![Ogre](https://avatars.discourse-cdn.com/v4/letter/o/ecb155/32.png) [@Ogre](https://boards.straightdope.com/u/Ogre)\ **Post date:** [June 3, 2003, 10:28pm UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/206 "2003-06-03T22:28:46Z")
> [@](#): > > Such a small mind that cannot encompass the notion of infinity. Actually, the concept of infinity is a very complex and subtle idea, so I’m willing to give someone the benefit of the doubt who doesn’t grasp it. You guys have fought a good fight. You’ve explained it as many ways as there are to explain it. Six pages worth. It may be time to let the naysayers simply ruminate on the principles involved. I knew there’d be a problem when I read somebody say that 0.333… does not equal 1/3. 1/3 is **precisely** equal to 0.333… Therefore 0.999… is **precisely** equal to 3/3, which is **precisely** equal to 1. ---
**Author:** ![MC\_Master\_of\_Ceremonies](https://avatars.discourse-cdn.com/v4/letter/m/bb73d2/32.png) [@MC\_Master\_of\_Ceremonies](https://boards.straightdope.com/u/MC_Master_of_Ceremonies)\ **Post date:** [June 3, 2003, 10:31pm UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/207 "2003-06-03T22:31:18Z")
**Achenar** , of course 1/3 = 0.3333… is defined as having infinite decimal places otherwise if it has finite decimal places it is an inequality not an equality. ---
**Author:** ![wolfman](https://avatars.discourse-cdn.com/v4/letter/w/a8b319/32.png) [@wolfman](https://boards.straightdope.com/u/wolfman)\ **Post date:** [June 3, 2003, 10:34pm UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/208 "2003-06-03T22:34:34Z")
> [@](#): > > 1. The only way for 0.999… to be equal to 1, is if an infinite number of 9s is placed on the end. This can be considered in theory, but as infinity cannot actually be reached in reality 0.999… will never equal 1. I am the 18th person pointing this out, but what you or I or the entirety of humanity can do is irrelevant. We cannot create the long hand number of .9 repeating infinately. But we can create the shorthand representation of.999~. Now we have infinite 9s to work with. For example we have 2 circles one with a radius of 2 units and one with a radius of 4 units. now lets compare areas in base 10. circle one: pi _r[sup]2[/sup] = 4_pi square units circle two: pi _r[sup]2[/sup] = 16_pi square units. Circle 2 has exactly 4 times the area circle 1. Do you disagree with this calculation? It is only possible because we accept the shorthand ‘pi’ for the value it represents. In base ten the number cannot be written out because it would be infinite. Does this make the shorhandt calculation invalid? ---
**Author:** ![StephanieG](https://avatars.discourse-cdn.com/v4/letter/s/c77e96/32.png) [@StephanieG](https://boards.straightdope.com/u/StephanieG)\ **Post date:** [June 3, 2003, 10:34pm UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/209 "2003-06-03T22:34:52Z")
> [@](#): > > \*Originally posted by lucwarm \* > \*\*Let X = 1 + 2 + 4 + 8 + 16 . . . . > > then X - 1 = 2 + 4 + 8 + 16 . . . . > > so (X-1) / 2 = 1 + 2 + 4 + 8 + 16 . . . > > so (X-1) / 2 = X > > and X - 1 = 2X > > and X = -1 > > so 1 + 2 + 4 + 8 + 16 . . . = -1 > > 😃 \*\* ## I love proofs like this 🙂 Ok, here’s what I wrote to a friend: Let X = 1 + 2 + 4 + 8 + 16 . . . . + 2^n then X - 1 = 2 + 4 + 8 + 16 . . . . + 2^n so (X-1) / 2 = 1 + 2 + 4 + 8 + 16 . . . +2^(n-1) so ((X-1) / 2) + (2^n) = X and (X – 1) + 2^(n+1)= 2X and X = (2^(n+1)) - 1 so 1 + 2 + 4 + 8 + 16 . . . + 2^n = (2^(n+1)) - 1 * * * Just because it’s an infinite series, doesn’t mean to ignore certain terms. An infinite series is just a very very very very very very long finite series that doesn’t like to end. 🙂 X(0)= 1 =2-1=1 x(1)= 1+2 =4-1=3 x(2)= 1+2+4 =8-1=7 x(3)= 1+2+4+8=16-1=15 ---
**Author:** ![MC\_Master\_of\_Ceremonies](https://avatars.discourse-cdn.com/v4/letter/m/bb73d2/32.png) [@MC\_Master\_of\_Ceremonies](https://boards.straightdope.com/u/MC_Master_of_Ceremonies)\ **Post date:** [June 3, 2003, 10:37pm UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/210 "2003-06-03T22:37:05Z")
Also, I pointed out earlier that an arrow peforms this operation (9/10+9/100+9/1000…) to infinity in terms of l, it’s length, every time it is fired into a target (or indeed any moving object moving across an arbitary length, l) ---
**Author:** ![Pleonast](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/pleonast/32/1183_2.png) [@Pleonast](https://boards.straightdope.com/u/Pleonast)\ **Post date:** [June 3, 2003, 10:41pm UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/211 "2003-06-03T22:41:05Z")
**Phage** said: > [@](#): > > 1. The only way for 0.999… to be equal to 1, is if an infinite number of 9s is placed on the end. This can be considered _in theory_, but as infinity cannot actually be reached _in reality_ 0.999… will never equal 1. (Emphases added.) Here’s the crux of the disagreement. When mathematicians talk about 0.999…, they are talking mathematics. When **Phage** talks about 0.999…, he’s talking about “reality”. It doesn’t matter what reality _is_, it has no effect on mathematics. Mathematics is a formal system, deriving theorems from axioms, using a logic rule-set. In his point above, **Phage** basically admits that the mathematical proofs of 0.999…==1 are correct, but denies they are applicable to the real world. He can of course claim this, but should be aware that when most people write 0.999…==1 (or 0.333…==1/3) they are implicitly talking mathematics, not “reality”. Is there still a GQ here? I humbly suggest a moderator either close this thread or move it to GD. ---
**Author:** ![Orbifold](https://avatars.discourse-cdn.com/v4/letter/o/779978/32.png) [@Orbifold](https://boards.straightdope.com/u/Orbifold)\ **Post date:** [June 3, 2003, 10:42pm UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/212 "2003-06-03T22:42:24Z")
> [@](#): > > \*Originally posted by Phage \* > \*\*Take a look at this address, already posted at the start of the thread: [http://mathforum.org/library/drmath/view/55748.html](http://mathforum.org/library/drmath/view/55748.html) > \*\* Do you even read this stuff? Here are some quotes from that exact page: > [@](#): > > Thus, if you are going to assign a value to .9999… (going on > forever), the only sensible value is 1. > […] > 1/3 + 2/3 = .999999… = 1. And of course… > [@](#): > > I assume you’ve seen the Dr. Math FAQ: > > 0.9999… = 1 > [http://mathforum.org/dr.math/faq/faq.0.9999.html](http://mathforum.org/dr.math/faq/faq.0.9999.html) > > If not, take a look. I mean, really. Why do you quote this stuff when it doesn’t support your position? > [@](#): > > _Originally posted by Phage_ > **I think that anyone who would actually learn anything from this thread has already read it and come to a conclusion. Everyone else is set in their ways, and will not change their minds even when presented with a reasonable proof.** The evidence certainly seems to agree with this statement. ---
**Author:** ![ElJeffe](https://avatars.discourse-cdn.com/v4/letter/e/958977/32.png) [@ElJeffe](https://boards.straightdope.com/u/ElJeffe)\ **Post date:** [June 3, 2003, 10:44pm UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/213 "2003-06-03T22:44:38Z")
After perusing 4+ long pages of (interesting yet time-consuming) mathematics, I think the problem is that the supporters of Phage-math and the supporters of actual math are talking through each other. Phage, et al, keep bringing up “properties” of 0.9~ that no Real numbers have - ie, it’s the “closest number to 1”, and such. Basically, Phage looks at 0.9~ and imagines something that isn’t a Real number (note here that I’m referring to “Real numbers”, the rigidly defined class of numbers, as opposed to “real numbers”, which are just some random “number” that can possess any property you throw at it). And that’s fine, I suppose. Phage, go ahead and create some new class of numbers. It can include your version of 0.9~ (the closest number to 1, as you approach from the left). It can include 1.00…1 (the closest number to 1 as you approach from the right). It can include Bob, defined as 2\ **Author:** ![Azael](https://avatars.discourse-cdn.com/v4/letter/a/a8b319/32.png) [@Azael](https://boards.straightdope.com/u/Azael)\ **Post date:** [June 3, 2003, 10:48pm UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/214 "2003-06-03T22:48:55Z") > [@](#): > > 1. The number 0.999… has an infinite number of 9s on the end. This is exactly what the “…” means, we can write the “…” in this case because all of the values are known. > [@](#): > > 1. Infinity can never be reached. Reached no, expressed symbolically yes. > [@](#): > > 1. As more 9s in 0.999… are considered, the closer the number comes to 1. Not “as more 9s in 0.999… are considered,” _all_ of them are being considered here. That is what the “…” allows us to do. You yourself are saying that “as more are considered the closer the number comes to one,” the trick here is to realize that “more” is meaningless here because there are an infinte number of 9s being considered. > [@](#): > > 1. The only way for 0.999… to be equal to 1, is if an infinite number of 9s is placed on the end. Which is exactly what the “…” means and exactly what everyone here has been trying to show you. > [@](#): > > This can be considered in theory, but as infinity cannot actually be reached in reality 0.999… will never equal 1. Ah but in reality, .999… = 1. In order for them to be two different numbers you _have to_ show that there is a difference between them. What is that difference? You are confusing yourself when you say “infinity cannot be reached.” When you say that you mean that an infinite _amount_ can never be reached. However, when we are talking about an infinite number of nines after the decimal point we are talking about an infinite degree of precision. That means that the number we throw out is defined to be exactly as it is represented. Just like the number …0002.000… or …00027.34333… or even …000.999… If that doesn’t help then think back to the earlier example involving a triangle with area equal to 1 being divided into three parts. Each part will have an area equal to 0.333… and the sum of those three parts can be written both as 0.999… and as 1. If 0.999… != 1 then it seems you have a **real world** problem where a given **real world** area can be changed simply by performing math upon it. ---
**Author:** ![Jabba](https://avatars.discourse-cdn.com/v4/letter/j/b5e925/32.png) [@Jabba](https://boards.straightdope.com/u/Jabba)\ **Post date:** [June 3, 2003, 11:08pm UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/215 "2003-06-03T23:08:04Z")
**Pleonast** said: > [@](#): > > Is there still a GQ here? I humbly suggest a moderator either close this thread or move it to GD. While I understand your frustration, I think many of us would be worried by an official statement that mathematical quandaries were matters of debate rather than factual questions. ---
**Author:** ![JRootabega](https://avatars.discourse-cdn.com/v4/letter/j/8e7dd6/32.png) [@JRootabega](https://boards.straightdope.com/u/JRootabega)\ **Post date:** [June 3, 2003, 11:14pm UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/216 "2003-06-03T23:14:04Z")
Well, how about an "Asked and answered …and asked, and asked, and asked, and asked, and asked…" This is going to take a while. Hey, that brings up a good point! Phage’s pattern of posts are meant to mimic the infinite sequence of 9’s! He’s a GENIUS! ---
**Author:** ![Wumpus](https://avatars.discourse-cdn.com/v4/letter/w/8c91f0/32.png) [@Wumpus](https://boards.straightdope.com/u/Wumpus)\ **Post date:** [June 3, 2003, 11:16pm UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/217 "2003-06-03T23:16:47Z")
\ This is the thread that never ends, It just goes on and on my friends, Some people started writing it not knowing what it was And they’ll continue writing it forever just because This is the thread that never ends… ---
**Author:** ![Jabba](https://avatars.discourse-cdn.com/v4/letter/j/b5e925/32.png) [@Jabba](https://boards.straightdope.com/u/Jabba)\ **Post date:** [June 3, 2003, 11:23pm UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/218 "2003-06-03T23:23:19Z")
**JRootabega** said: > [@](#): > > Well, how about an "Asked and answered > > …and asked, and asked, and asked, and asked, and asked…" On that one you would get no argument from me. ---
**Author:** ![Achernar](https://avatars.discourse-cdn.com/v4/letter/a/e274bd/32.png) [@Achernar](https://boards.straightdope.com/u/Achernar)\ **Post date:** [June 3, 2003, 11:37pm UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/219 "2003-06-03T23:37:42Z")
> [@](#): > > \*Originally posted by MC Master of Cermonies \* > **Achenar** , of course 1/3 = 0.3333… is defined as having infinite decimal places otherwise if it has finite decimal places it is an inequality not an equality. I humbly request a cite, preferably from a text on Analysis or some other math of a similar level of advancement. I am not saying that the notation 0.333… represents any finite number of 3’s. I am saying that the notation 0.333… represents the limit of a sequence whose Nth term is 0.333…333, where the number of 3’s is N. In your mind, this may be the same thing as an infinite number of 3’s, but in my mind, it’s not. Specifically, exactly what “an infinite number of 3’s” means mathematically is not well defined. ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [June 3, 2003, 11:41pm UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/220 "2003-06-03T23:41:31Z")
> [@](#): > > \*Originally posted by StephanieG \* > **Just because it’s an infinite series, doesn’t mean to ignore certain terms. An infinite series is just a very very very very very very long finite series that doesn’t like to end. 🙂** This is incorrect. An infinite series is an infinite series. It has more terms than any finite series. The “proof” that you quoted is incorrect because laws of arithmetic only work for convergent series, but there are contexts in which it is a valid argument: add all the powers of 2, and you get -1. [Previous page](https://boards.straightdope.com/t/why-doesnt-9999-1/179019.md?page=10) [Next page](https://boards.straightdope.com/t/why-doesnt-9999-1/179019.md?page=12)