# An infinite question: Why doesn't .999~ = 1? **URL:** \ **Category:** Cecil's Columns/Staff Reports\ **Created:** [September 24, 2016, 7:29am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711 "2016-09-24T07:29:45Z")\ **Posts on this page:** 20\ **Page:** 20
**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:** [November 2, 2016, 4:41pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/381 "2016-11-02T16:41:23Z")
> [@watchwolf49](#): > > If you could please say what that reason is, if you don’t mind … are you saying that 16/32 doesn’t equal 1/2 for the same reasons that 7/16 doesn’t equal 8/16? … and you’re second paragraph is not fully complete … “getting closer to” 1 needs to be defined in relation to something else “getting closer to” something … “as you go” is too indistinct, as we go to the store, as we go to the moon, as we go where? I read the sentence as : It’s not like … that it is “getting closer to” 1 “as you go”. AHunter3’s negating a mistaken notion, not advocating it. But, netzweltler, while you’re here, please take the time to answer my questions. This kind of behavior absolutely fascinates me and people like you won’t explain it. Why are you spending so much time arguing this with us? What are you getting out of it? You say you’re not using it for math. So what are you doing? You can’t convince the entire world of mathematicians, professional and amateur, that you’re right and they’re wrong. 1 is \*defined \*as the answer. You can’t argue that a definition is wrong. You can substitute a different definition and get a different set of mathematics, but a definition is by definition always right. You’ve posted 79 times so far. You’ve made the same argument 79 times. You’re not convincing anyone who disagreed the first 78 times. So why are you doing this? Please help me understand. That would be the most wonderful thing you could do to lessen our ignorance. ---
**Author:** ![AHunter3](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/ahunter3/32/368_2.png) [@AHunter3](https://boards.straightdope.com/u/AHunter3)\ **Post date:** [November 2, 2016, 9:08pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/382 "2016-11-02T21:08:13Z")
I’m going to bow out of a discussion that I’m not qualified to participate in. ---
**Author:** ![John\_W.Kennedy](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/john_w.kennedy/32/1031_2.png) [@John\_W.Kennedy](https://boards.straightdope.com/u/John_W.Kennedy)\ **Post date:** [November 3, 2016, 12:02am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/383 "2016-11-03T00:02:05Z")
> [@watchwolf49](#): > > This is getting tensor and tensor … Eight, sir; seven, sir; Six, sir; five, sir; Four, sir; three, sir; Two, sir; one! Tenser, said the Tensor. Tenser, said the Tensor. Tension, apprehension, And dissension have begun. —_The Demolished Man_ ---
**Author:** ![Riemann](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/riemann/32/3133_2.png) [@Riemann](https://boards.straightdope.com/u/Riemann)\ **Post date:** [November 3, 2016, 12:19am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/384 "2016-11-03T00:19:15Z")
Then there was Sherping Tensor, that guy who climbed 99.9999~% of the way up Everest. ---
**Author:** ![DMC](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/dmc/32/18049_2.png) [@DMC](https://boards.straightdope.com/u/DMC)\ **Post date:** [November 3, 2016, 3:25am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/385 "2016-11-03T03:25:26Z")
> [@Snarky\_Kong](#): > > Has any person that wasn’t immediately convinced by the usual arguments been convinced eventually? I tried to get through this thread, but it was too painful due to all of the various sidetracks. In case this wasn’t one of the usual arguments, the one that worked best for me back in the day was “For any two different real numbers, there are an infinite amount of real numbers between them. Name any one real number between 0.999~ and 1.0” Of course, I’ve seen absolutely bizarre attempts to counter that. My favorite was 0.999~1. Um, okay. ---
**Author:** ![netzweltler](https://avatars.discourse-cdn.com/v4/letter/n/a3d4f5/32.png) [@netzweltler](https://boards.straightdope.com/u/netzweltler)\ **Post date:** [November 3, 2016, 6:55am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/386 "2016-11-03T06:55:54Z")
> [@Budget\_Player\_Cadet](#): > > Then what point have we reached? > > It can’t be 0.999 because we reached that at t=0.99. > It can’t be 0.9999 because we reached that at t=0.999. > It can’t be 0.99999 because we reached that at t=0.9999. > It can’t be 0.99999999999999999999999999999999 because we reached that at t=0.9999999999999999999999999999999. > > I can keep adding 9s arbitrarily. It doesn’t matter how small the difference between 0.9999999999999999999999999999999999999 and 1 is, I can find the step where we went smaller. The only conclusion from this is that there _is_ no distance between 0.99… and 1. If the “residual distance” between 0.99… and 1 is “not defined”, how can you even say it exists in mathematics? Depends on what “not exist” means in mathematics. True is that we haven’t reached 1 because none of the steps on the list shows how this happens. Reaching 1 would be the only condition to define a residual distance of 0. ---
**Author:** ![netzweltler](https://avatars.discourse-cdn.com/v4/letter/n/a3d4f5/32.png) [@netzweltler](https://boards.straightdope.com/u/netzweltler)\ **Post date:** [November 3, 2016, 7:10am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/387 "2016-11-03T07:10:27Z")
> [@naita](#): > > Good, there are infinitely many nines in [0, 1], we’re on the same page here. > > > [@netzweltler](#): > > > > What you don’t want to see is, that these infinitely many 9s (0.9, 0.99, 0.999, …) are in [0, 1) either. So, point 1 isn’t needed at all. Look up set theory. > > What do you mean “either”, do you mean also? Yes, sorry. Wrong wording. > [@naita](#): > > A number is in the interval if if it is less than 1 by some non-zero _ε_. What is the difference between 0.999… and 1? Actual mathematicians say that the difference is 0, and that 0.999… is 1. You need to either come up with a definition for what that difference is, or abandon your infatuation with 0.999… as a process as unmathematical. > > If you insist on keeping 0.999… in [0, 1\> it will have the absurd distinction of being the only number in the interval that doesn’t have a definable difference from 1. What I always insisted on is that 0.999… is not a point on the number line. So, it is not a point in the interval [0, 1) either (this time correct wording). [0, 1) is simply the union of the intervals [0, 0.9], [0.9, 0.99], [0.99, 0.999], … ---
**Author:** ![netzweltler](https://avatars.discourse-cdn.com/v4/letter/n/a3d4f5/32.png) [@netzweltler](https://boards.straightdope.com/u/netzweltler)\ **Post date:** [November 3, 2016, 7:32am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/388 "2016-11-03T07:32:44Z")
> [@watchwolf49](#): > > We have looked up set theory, and we find your conclusions wanting still … have you looked up limits yet? How does the limit 1 come into play? All we can see is that it doesn’t. And again: t = 0: I move my pen from point 0 to point 0.9 of the number line t = 0.9: I move my pen from point 0.9 to point 0.99 of the number line t = 0.99: I move my pen from point 0.99 to point 0.999 of the number line … There is no action on the list that includes the limit. > [@watchwolf49](#): > > I think we’re making progress here, now we need to clarify exactly what value this represents … Also, we need to find a value that exists between (0.999…) and 1, as to demonstrate that we’re complying with the law of math that says between any two real number there exists yet another real number … > > ========= As long as it is not clarified that (0.999…) is a point on the number line? Why should we find a point between point 1 and a non-point? > [@watchwolf49](#): > > I’m going to go out on a limb here, so I’m not saying I’m right, but something to consider: I think we can agree with **netzweltler** that this particular proof of (0.999…) = 1 isn’t as robust as maybe it should be … the mistake is that **netzweltler** is then saying this then acts as a counter-proof, but fails to acknowledge that as counter-proof it suffers the same flaw, it’s not as robust as it should be. > > There are other proofs that (0.999…) = 1 … some of which are quite rigid … your mission … **netzweltler** … if you choose to accept it … is to show that these other proofs also give you the same results. I would like to see a counter-proof to my claim, that we don’t reach 1 even after completing infinitely many steps by t = 1. ---
**Author:** ![netzweltler](https://avatars.discourse-cdn.com/v4/letter/n/a3d4f5/32.png) [@netzweltler](https://boards.straightdope.com/u/netzweltler)\ **Post date:** [November 3, 2016, 7:38am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/389 "2016-11-03T07:38:44Z")
> [@The\_Great\_Unwashed](#): > > Seems to me that the problem isn’t that > 1 is not an element of [0 , 1) (which is certainly true) > but that > 0.999… is not an element of {0.9, 0.99, 0.999, … } (being greater that every single element in that set) Greater than any element of { 0.9, 0.99, 0.999, … }, yes. But take into account, that there is no segment in 0.999… which you cannot find already in one of the elements of the set { 0.9, 0.99, 0.999, … }. > [@The\_Great\_Unwashed](#): > > There’s nothing wrong with any of the formulations proposed by the pro side. ---
**Author:** ![netzweltler](https://avatars.discourse-cdn.com/v4/letter/n/a3d4f5/32.png) [@netzweltler](https://boards.straightdope.com/u/netzweltler)\ **Post date:** [November 3, 2016, 7:54am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/390 "2016-11-03T07:54:39Z")
> [@Exapno\_Mapcase](#): > > I read the sentence as : It’s not like … that it is “getting closer to” 1 “as you go”. AHunter3’s negating a mistaken notion, not advocating it. > > But, netzweltler, while you’re here, please take the time to answer my questions. This kind of behavior absolutely fascinates me and people like you won’t explain it. > > Why are you spending so much time arguing this with us? What are you getting out of it? You say you’re not using it for math. So what are you doing? You can’t convince the entire world of mathematicians, professional and amateur, that you’re right and they’re wrong. 1 is \*defined \*as the answer. You can’t argue that a definition is wrong. You can substitute a different definition and get a different set of mathematics, but a definition is by definition always right. > > You’ve posted 79 times so far. You’ve made the same argument 79 times. You’re not convincing anyone who disagreed the first 78 times. So why are you doing this? Please help me understand. That would be the most wonderful thing you could do to lessen our ignorance. Well, I am convinced that my argument is correct. And I am interested in the counter-arguments coming up during this thread. As soon as I can see that we are stuck in a loop I might stop participating this thread. I am not convinced that we are stuck in a loop yet. ---
**Author:** ![naita](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/naita/32/5862_2.png) [@naita](https://boards.straightdope.com/u/naita)\ **Post date:** [November 3, 2016, 1:36pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/391 "2016-11-03T13:36:36Z")
> [@netzweltler](#): > > What I always insisted on is that 0.999… is not a point on the number line. So, it is not a point in the interval [0, 1) either (this time correct wording). [0, 1) is simply the union of the intervals [0, 0.9], [0.9, 0.99], [0.99, 0.999], … Okay, let me bring back another statement of yours then: > [@](#): > > There is no largest number of nines. We are passing infinitely many digits without reaching a last point. There is no last step after an infinite number of steps! Here you’re saying we are passing infinitely many digits. 0.999… is infinitely many digits. Are you saying we pass a point that doesn’t exist on the number line? You can’t have it both ways. If 0.999… is not on the number line there is a number with a finite number of 9’s that is the largest number in the interval [0, 1\>. Now it’s possible to make a number system that has such numbers, but then we’re no longer dealing with the reals, and you’ve made no attempt at introducing the consistent and meaningful definition of infinitesimals required for such an expansion beyond the reals. Or put another way: If 0.999… is a number it is on the number line. If it’s not a number, your entire argument becomes the boring old “there are no infinities!”. ---
**Author:** ![watchwolf49](https://avatars.discourse-cdn.com/v4/letter/w/e9c0ed/32.png) [@watchwolf49](https://boards.straightdope.com/u/watchwolf49)\ **Post date:** [November 3, 2016, 2:10pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/392 "2016-11-03T14:10:53Z")
For the 83rd time … > [@netzweltler](#): > > [snip] … What I always insisted on is that 0.999… is not a point on the number line … [snip] Correct me if I’m wrong, but originally you claimed you cannot find this point on the number line using you methodology … you can certainly insist all you want to … but that doesn’t make it true … > [@netzweltler](#): > > How does the limit 1 come into play? All we can see is that it doesn’t. And again: > > t = 0: I move my pen from point 0 to point 0.9 of the number line > t = 0.9: I move my pen from point 0.9 to point 0.99 of the number line > t = 0.99: I move my pen from point 0.99 to point 0.999 of the number line > … > > There is no action on the list that includes the limit. Sure it does … as t gets closer and closer to 1, the number gets closer and closer to 1 … do you agree? … now compare that to the definition of a limit … once again you’re treating the infinitesimal as an extremely small amount, but an infinitesimal is infinitely small … it has no “amount”, or to say an amount of 0 … > [@netzweltler](#): > > As long as it is not clarified that (0.999…) is a point on the number line? Why should we find a point between point 1 and a non-point? Again and again and again … just because this number isn’t clearly on the number line in your mind doesn’t mean it’s not on the number line … by this same reasoning you would be claiming π is not on the number line because you can’t explicitly find yourself … hopefully you can see how ridiculous such a claim is … of course π is a number, how can you justify saying it’s not? > [@netzweltler](#): > > I would like to see a counter-proof to my claim, that we don’t reach 1 even after completing infinitely many steps by t = 1. The limit as t approaches 1 of your summation is equal to 1 … this is when you move your pen an infinitely short distance, or a distance of zero … I’ll have to leave the proof of an infinitely short distance being equal to zero to your second year calculus instructor … > [@netzweltler](#): > > Well, I am convinced that my argument is correct. And I am interested in the counter-arguments coming up during this thread. As soon as I can see that we are stuck in a loop I might stop participating this thread. > I am not convinced that we are stuck in a loop yet. We are definitely stuck in a loop here … we say such-and-such is defined this way … and you say it’s not … it is … it’s not … it is … it’s not … it is … it’s not … etc etc etc … ```auto for ($i = 1; $i <= 10; $i = 1) { echo "it is ... it's not ...
"; } ``` ---
**Author:** ![Budget\_Player\_Cadet](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/budget_player_cadet/32/205_2.png) [@Budget\_Player\_Cadet](https://boards.straightdope.com/u/Budget_Player_Cadet)\ **Post date:** [November 3, 2016, 3:26pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/393 "2016-11-03T15:26:05Z")
> [@netzweltler](#): > > Depends on what “not exist” means in mathematics. True is that we haven’t reached 1 because none of the steps on the list shows how this happens. Reaching 1 would be the only condition to define a residual distance of 0. But we’re not at any of the steps. We’ve done them all. The steps are over. We’ve completed every single step. So no matter how arbitrarily small you set the distance between 1 and 0.99… I can point to a step that is smaller. The only reasonable conclusion, then, is that by completing infinitely many steps, the distance is zero. Because, again, if it weren’t zero, _we wouldn’t be done yet_, because at every step it comes closer to zero. > [@netzweltler](#): > > How does the limit 1 come into play? All we can see is that it doesn’t. And again: > > t = 0: I move my pen from point 0 to point 0.9 of the number line > t = 0.9: I move my pen from point 0.9 to point 0.99 of the number line > t = 0.99: I move my pen from point 0.99 to point 0.999 of the number line > … > > There is no action on the list that includes the limit. You’re right. Because the limit is _outside that list_. The list goes on forever; we’re looking at what happens after that point. In mathematics, we say that the limit of a series is defined as thus: lim from n to infinity of the sequence a(n) = L when for every real number ε \> 0, there exists a natural number N such that for all n \> N, we have | a(n) − L | \< ε. This only applies to 1. Resolve the above equation for your sequence, {0.9, 0.99, …}, and you’ll find that once again, we reach the same conclusion: ε becomes arbitrarily small. No matter how close to zero you take it, you can always find an N such that all ns larger than N lead to a(n) - 1 being smaller than ε. This doesn’t work with any other number. > [@](#): > > I would like to see a counter-proof to my claim, that we don’t reach 1 even after completing infinitely many steps by t = 1. But you’re still looking at finitely many steps. Again, what is the distance between 1 and the point we reach after infinitely many steps? Is it 0.00001? Nope, because we passed that step ages ago. Is it 0.000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001? No, again, we passed that step. It does not matter what distance you claim it to be, you cannot find a distance we have not already gone smaller than. Ergo, the distance must be zero. ---
**Author:** ![The\_Great\_Unwashed](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/the_great_unwashed/32/4791_2.png) [@The\_Great\_Unwashed](https://boards.straightdope.com/u/The_Great_Unwashed)\ **Post date:** [November 3, 2016, 4:01pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/394 "2016-11-03T16:01:32Z")
Here’s a thing: If a number p ∈ (0, 1) then p \< √p \< 1 So if 0.999… ∈ (0, 1) what is its square root? Or simpler still: what is the first digit of its square root? And the second? And the third? … ---
**Author:** ![Chingon](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chingon/32/17645_2.png) [@Chingon](https://boards.straightdope.com/u/Chingon)\ **Post date:** [November 3, 2016, 5:17pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/395 "2016-11-03T17:17:32Z")
Better yet maybe, square it. If p = 0.9999… is p^2 \< p? By how much? How about p^3, p^4, etc… ---
**Author:** ![septimus](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/septimus/32/410_2.png) [@septimus](https://boards.straightdope.com/u/septimus)\ **Post date:** [November 3, 2016, 6:03pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/396 "2016-11-03T18:03:46Z")
Let me try to change the subject by posing a challenging problem loosely related to this .9999… question. The set of infinite sequences of the digits 0-9 is closely related to the set of real numbers in the half-open interval [0,1), e.g. the sequence 3,3,3,3,3,… corresponds to 1/3 and so on. _ **Construct a bijection between the two sets.** _ At first glance, it seems like you’re done at once. 3,3,3,3,3,… ←→ the real number 1/3, and so on. But this doesn’t quite work. 2,0,0,0,0,… → 1/5 and 1,9,9,9,9,9,… → 1/5. This duplication means we do not have a perfect bijection. Can anyone fix this problem and find a perfect bijection? I do not have an answer yet. (I can do it with some difficulty for base 2 or base 6, but not yet base 10. :mad: ) ---
**Author:** ![septimus](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/septimus/32/410_2.png) [@septimus](https://boards.straightdope.com/u/septimus)\ **Post date:** [November 3, 2016, 7:57pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/397 "2016-11-03T19:57:19Z")
> [@septimus](#): > > The set of infinite sequences of the digits 0-9 is closely related to the set of real numbers in the half-open interval [0,1), e.g. the sequence 3,3,3,3,3,… corresponds to 1/3 and so on. > _ **Construct a bijection between the two sets.** _ I came up with a solution. (Indeed, a solution I’d found, but since forgotten :smack:, when I posed the same problem to myself years ago.) [SPOILER] We write Diff(x) if there are two decimal representations of the real number x, i.e. if x = A × 10[sup]-B[/sup] for integers A,B. We write Seq(x) to denote the string of decimal digits in a decimal rendition of x, e.g. Seq(1/7) = 1,4,2,8,5,7,1,4,2,8,5,7,… When Diff(x), Seq(x) denotes the representational sequence ending in 000…; Seq9(x) will denote the alternate sequence ending in 999… We now describe a f(x) which provides a bijection between the reals in the half-open interval [0,1) and the set of infinite sequences of decimal digits. If NOT Diff(x) then f(x) = Seq(x). If Diff(x) and x \< 1/2. then f(x) = Seq(2x) If Diff(x) and x \> 1/2. then f(x) = Seq9(2x-1) If x = 1/2 then f(x) = 9,9,9,9,9,9,…[/SPOILER] ---
**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:** [November 3, 2016, 9:09pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/398 "2016-11-03T21:09:23Z")
**netzweltler** , I must confess that I’ve lost track of what your argument is. So I’m going to try to restate it, and you can tell me if I have it right. OK? As I understand it, your argument is that, while any finite sequence of digits has meaning, and corresponds to a real number, no infinite string of digits can represent a number, and that there are therefore some numbers which simply cannot be represented by a string of digits. 0.9 is an infinite string of digits, and therefore is an example of a non-number, as is 0.3, which means that there is only one string of digits which represents 1, and no string of digits which represents 1/3. Is that an accurate statement of your claim? ---
**Author:** ![The\_Great\_Unwashed](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/the_great_unwashed/32/4791_2.png) [@The\_Great\_Unwashed](https://boards.straightdope.com/u/The_Great_Unwashed)\ **Post date:** [November 3, 2016, 9:54pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/399 "2016-11-03T21:54:18Z")
nm ---
**Author:** ![Lemur866](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/lemur866/32/434_2.png) [@Lemur866](https://boards.straightdope.com/u/Lemur866)\ **Post date:** [November 4, 2016, 3:16am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/400 "2016-11-04T03:16:18Z")
> [@netzweltler](#): > > As soon as I can see that we are stuck in a loop I might stop participating this thread. I am not convinced that we are stuck in a loop yet. Head asplodes. Seriously, I have a hard time understanding this. You agree that 1/3 is a number. You don’t agree that 0.33333… is a number. You agree that 0.1(base3) is a number. But 0.33333…(base 10) and 0.1(base3) are the same number! And both are the same number as 1/3. And 10/30. And π/3π. And 1.00000…/3.0000… All those are different ways of representing the same number. Doesn’t it seem strange to you that in base 10 mathematics we can represent some numbers that according to you couldn’t be represented in base 3, and that in base 3 we can represent some numbers that we can’t represent in base 10? [Previous page](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711.md?page=19) [Next page](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711.md?page=21)