# An infinite question: Why doesn't .999~ = 1? **URL:** \ **Category:** Cecil's Columns/Staff Reports\ **Created:** [July 27, 2003, 6:37am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/191230 "2003-07-27T06:37:04Z")\ **Posts on this page:** 14\ **Page:** 1
**Author:** ![Wassup](https://avatars.discourse-cdn.com/v4/letter/w/f0a364/32.png) [@Wassup](https://boards.straightdope.com/u/Wassup)\ **Post date:** [July 27, 2003, 6:37am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/191230/1 "2003-07-27T06:37:04Z")
I read your answer with interest, Cecil, and you are correct. However, in addition to your explanation that it is a fluke of the limitations of our decimal system, there is also a physical limitation. I believe that Max Planck conluded that time and space cannot be subdivided forever. The fundamental quantum of length is Planck length, and is the distance that light can travel in Planck time (10^-43 seconds). It is 10^-35 of a meter. In other words, one millimeter contains 100,000,000,000,000,000,000,000,000,000,000 Planck lengths. Thus, a measurement cannot be cut into thirds infinitely. Thus, at some point in repeating 0.999~ you will reach Planck length, and the answer becomes zero when all added up. In regards to Zeno’s paradox, it was solved by Sir Isaac Newton when he developed calculus, originally used to calculate the moon’s orbit accurately. If the moon was stationary at the beginning of every second, adding up all of its tiny “falls” does not give the right answer. ---
**Author:** ![Wassup](https://avatars.discourse-cdn.com/v4/letter/w/f0a364/32.png) [@Wassup](https://boards.straightdope.com/u/Wassup)\ **Post date:** [July 27, 2003, 6:40am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/191230/2 "2003-07-27T06:40:15Z")
“and the answer becomes zero when all added up.” er, sorry. It should read “and the answer becomes one when all added up.” I apologize for my poor proofreading. ---
**Author:** ![Wassup](https://avatars.discourse-cdn.com/v4/letter/w/f0a364/32.png) [@Wassup](https://boards.straightdope.com/u/Wassup)\ **Post date:** [July 27, 2003, 6:43am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/191230/3 "2003-07-27T06:43:06Z")
[http://www.straightdope.com/columns/030711.html](http://www.straightdope.com/columns/030711.html) theres the link. sorry for being a stupid newb. I’m used to the edit button working. ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [July 27, 2003, 6:33pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/191230/4 "2003-07-27T18:33:47Z")
The whole misunderstanding of the Planck scales is a minor pet peeve of mine. We don’t know that you can’t divide up space finer than the Planck length. In fact, we know precious little at all about what happens at the Planck length. That’s just what the Planck scale is: It’s a back-of-the-envelope estimate of the scale at which our current theories break down. The problem, basically, is this. We have General Relativity, which involves the constants G (Newton’s gravitational constant) and c (the speed of light), and which works very well. We also have quantum mechanics, which involves the constants hbar (Planck’s quantum of action) and c, and which also works very well. So long as we’re in a situation where only one of those theories applies (which is everything we have any experimental experience with at all), we’re fine: We apply one theory, ignore the other, and get an answer. But if we squish all three of those constants together in the right way, we get a unit of length (squish them slightly differently, and you can get a time, or mass, or energy, or most any other sort of unit). If we’re working with something with about that size, then both GR and QM will be significant simultaneously. Except that GR and QM don’t really seem to play well together. Surely, there’s some way of explaining what happens in such situations, but we have no idea what it is. ---
**Author:** ![Wassup](https://avatars.discourse-cdn.com/v4/letter/w/f0a364/32.png) [@Wassup](https://boards.straightdope.com/u/Wassup)\ **Post date:** [July 28, 2003, 4:58am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/191230/5 "2003-07-28T04:58:48Z")
Still, the fact that spacetime cannot be subdivided is still relevant, right? Or is it simply the fact that our theories break down on the Planck scale that makes it seem so? ---
**Author:** ![Achernar](https://avatars.discourse-cdn.com/v4/letter/a/e274bd/32.png) [@Achernar](https://boards.straightdope.com/u/Achernar)\ **Post date:** [July 28, 2003, 6:30am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/191230/6 "2003-07-28T06:30:07Z")
Hmmm, where do you get the idea that spacetime can’t be subdivided? ---
**Author:** ![RM\_Mentock](https://avatars.discourse-cdn.com/v4/letter/r/e274bd/32.png) [@RM\_Mentock](https://boards.straightdope.com/u/RM_Mentock)\ **Post date:** [July 28, 2003, 8:15am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/191230/7 "2003-07-28T08:15:52Z")
Yeah, why would we not be able to subdivide the Planck mass, but we can the Planck length? ---
**Author:** ![CurtC](https://avatars.discourse-cdn.com/v4/letter/c/ce73a5/32.png) [@CurtC](https://boards.straightdope.com/u/CurtC)\ **Post date:** [July 28, 2003, 5:24pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/191230/8 "2003-07-28T17:24:34Z")
'Cause the Planck mass is big, that’s why! ---
**Author:** ![Achernar](https://avatars.discourse-cdn.com/v4/letter/a/e274bd/32.png) [@Achernar](https://boards.straightdope.com/u/Achernar)\ **Post date:** [July 28, 2003, 5:29pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/191230/9 "2003-07-28T17:29:49Z")
What are you talking about? It’s exactly the same size as the Planck length! …in Planck units. 😉 ---
**Author:** ![Exapno\_Mapcase](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/exapno_mapcase/32/1051_2.png) [@Exapno\_Mapcase](https://boards.straightdope.com/u/Exapno_Mapcase)\ **Post date:** [July 28, 2003, 6:30pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/191230/10 "2003-07-28T18:30:37Z")
> [@](#): > > \*Originally posted by Wassup \* > \*\*Still, the fact that spacetime cannot be subdivided is still relevant, right? Or is it simply the fact that our theories break down on the Planck scale that makes it seem so? \*\* No, it is entirely irrelevant. Zeno’s paradox merely uses a physical metaphor to talk about infinite mathematical series. What does or doesn’t happen in the real world is of no matter, so to speak. And if spacetime cannot be subdivided indefinitely, it it even less relevant, because then distance is merely the addition of a finite number of increments, so the whole problem of converging or diverging infinite sums doesn’t come into it. It is almost always a mistake to put a physical structure onto purely mathematical problems. People who try to trisect an angle with straightedge and compass often make this same mistake. Infinity is not a physical concept. Don’t think about distance here. Think about pure numbers. ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [July 28, 2003, 9:17pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/191230/11 "2003-07-28T21:17:14Z")
> [@](#): > > 'Cause the Planck mass is big, that’s why! This illustrates part of the difficulty of the Planck units. Definitely, the Planck Speed (the Planck Length divided by the Planck Time; just a fancy name for the Speed of Light) is very big. And the Planck Action is assuredly very small, since it’s impossible to get any action (or angular momentum, which has the same units) which is any smaller. And everyone will agree that the Planck Length and Time are very small. But what about the Planck Mass? Is that big or small? Well, that depends on whom you ask. The Planck Mass is about a microgram, which is inconceivably small from the perspective of someone doing GR, but it’s inconceivably big to a particle physicist. And to someone working with Newtonian systems, it’s a perfectly reasonable mass (about the mass of a single bacterium). You can have things with half that mass, or twice that mass, and they behave exactly as you’d expect. And if you look at the Planck Momentum (mass times the speed of light), you’ll find that it’s about the same as the momentum of a running housecat. You don’t even need a microscope to study systems like that. ---
**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:** [July 30, 2003, 6:24pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/191230/12 "2003-07-30T18:24:59Z")
Ney, non, niet, the Planck length is not a quantized unit of space, infact observations by astronomers have proved that if space is quantized it must be on a much smaller scale than the Planck length. The Planck length is the length over which yet undiscoverd quantum gravitational effects should take over from classical gravitational effects. ---
**Author:** ![Darrell](https://avatars.discourse-cdn.com/v4/letter/d/54ee81/32.png) [@Darrell](https://boards.straightdope.com/u/Darrell)\ **Post date:** [July 31, 2003, 4:31pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/191230/13 "2003-07-31T16:31:18Z")
> [@](#): > > \*Originally posted by Wassup \* > \*\*I read your answer with interest, Cecil, and you are correct. > > However, in addition to your explanation that it is a fluke of the limitations of our decimal system, there is also a physical limitation.\*\* \<…\> I am no physicist. I wouldn’t even call myself a mathematician, but even **I** know there is no “physical limitation” in a purely mathematical statement such as 1=.99… ---
**Author:** ![MeanRoy](https://avatars.discourse-cdn.com/v4/letter/m/dec6dc/32.png) [@MeanRoy](https://boards.straightdope.com/u/MeanRoy)\ **Post date:** [August 3, 2003, 7:34am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/191230/14 "2003-08-03T07:34:41Z")
Bottom line is By Definition 1=.9999… where the dots indicate completion to infinity. But I’ve had three beers.