# 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:** 8
**Author:** ![moes\_lotion](https://avatars.discourse-cdn.com/v4/letter/m/7ba0ec/32.png) [@moes\_lotion](https://boards.straightdope.com/u/moes_lotion)\ **Post date:** [October 12, 2016, 4:03am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/141 "2016-10-12T04:03:16Z")
> [@engjs](#): > > Watchwolf49: What is the decimal expansion of 1/3? > > It has no actual decimal expansion. It’s normal practice to use the expression 0.3~ to represent it, because 1/3 is the limit of the series (0.3, 0.33, 0.333, …). But the two things are not equal, one is just used as a label for the other. In the same way that the pi symbol is used to represent the number. > > \ > > Watchwolf49: I understand the limit of a function, I don’t understand the limit of (0.3, 0.33, 0.333 …). > > So you didn’t do first year analysis? Okay. As a function it is (excuse the notation) f(n) = the sum from i=1 to n of [3 times (10 to the power of -i)], with the limit being taken as n tends to infinity. When you say that it is normal practice to use the expression 0.3 to represent 1/3, what exactly do you mean by represent? To everyone else responding to your original post, the “expression” 0.3 means exactly the result of dividing 1 by 3, and it “represents” an infinite string of 3’s after the decimal point. If you are proposing an alternate definition of the notation, please share it with us. In the same post quoted above you also define a function, f(n) = [sum i = 1 to n](3 x 10^(-i)) and speak of the limit of this as n tends to infinity. Please share with us what you think this limit is equal to. > [@engjs](#): > > Chronos: Not true. 1/3 is a finite number, and yet it has only one decimal representation. > > 🙂 Ya got me. I should have said: Every real number that has a finite decimal representation also has an infinite one, except 0. > > \ > 0.3~ does not mean 0. followed by all the threes, because that is nonsense. It does not mean 0. followed by a number of threes whose count is a number called infinity, because that is also nonsense. It mean 0. followed by a non-terminating sequence of threes. Because the sequence of numbers formed by stopping the process after each three forms a strictly monotonically increasing series, and because the sequence has no final three, the sequence can never be equal to 1/3. It’s impossible for the same reason that it is impossible for the two function to ever touch. That’s why in order to use 0.3~ to represent 1/3 you have to define it as being equal to 1/3. Sorry, 0.3 does mean 0. followed by an infinite number of zeros to everyone but you, and it is not nonsense, it is 1/3. ---
**Author:** ![netzweltler](https://avatars.discourse-cdn.com/v4/letter/n/a3d4f5/32.png) [@netzweltler](https://boards.straightdope.com/u/netzweltler)\ **Post date:** [October 12, 2016, 4:51am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/142 "2016-10-12T04:51:41Z")
> [@moes\_lotion](#): > > Sorry, 0.3 does mean 0. followed by an infinite number of zeros to everyone but you, and it is not nonsense, it is 1/3. How can 0.333… be any defined number on the number line? It (the sum 0.3 + 0.03 + 0.003 + …) doesn’t stop at any defined point on the number line, does it? ---
**Author:** ![Trinopus](https://avatars.discourse-cdn.com/v4/letter/t/2bfe46/32.png) [@Trinopus](https://boards.straightdope.com/u/Trinopus)\ **Post date:** [October 12, 2016, 4:57am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/143 "2016-10-12T04:57:45Z")
> [@netzweltler](#): > > How can 0.333… be any defined number on the number line? > > It (the sum 0.3 + 0.03 + 0.003 + …) doesn’t stop at any defined point on the number line, does it? It’s an exactly specified number (except to some people who don’t agree…) It doesn’t have to “stop.” It’s an infinite number of 3s. This thread has dealt with the problem of thinking of it as an algorithm. 10 add another 3 20 goto 10 But that isn’t how it works – and, worse, even if that was how it works, it goes to 1/3 anyway, because it can’t go anywhere else! It gets arbitrarily close to 1/3. Another challenge: how would you express 1/3 in decimal notation? If 0.333~ isn’t it, what is? Are you really ready to declare that the number 1/3 does not have a decimal expansion? (Repeating decimals are fun. As a specific case, easily generalizable, abcdef/999999 = .abcdefabcdefabcdef… This leads to a rather charming – and somewhat sexist – number puzzle. Each letter stands for a decimal digit. Solve… EVE/DID = .TALKTALKTALKTALK…) (From Martin Gardner.) ---
**Author:** ![netzweltler](https://avatars.discourse-cdn.com/v4/letter/n/a3d4f5/32.png) [@netzweltler](https://boards.straightdope.com/u/netzweltler)\ **Post date:** [October 12, 2016, 5:52am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/144 "2016-10-12T05:52:24Z")
> [@Trinopus](#): > > It gets arbitrarily close to 1/3. So, if it (the sum) doesn’t stop at point 1/3 on the number line, the meaning of the equal sign (0.333… **=** 1/3) is being close enough to in this case? ---
**Author:** ![netzweltler](https://avatars.discourse-cdn.com/v4/letter/n/a3d4f5/32.png) [@netzweltler](https://boards.straightdope.com/u/netzweltler)\ **Post date:** [October 12, 2016, 6:58am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/145 "2016-10-12T06:58:00Z")
> [@Trinopus](#): > > (Repeating decimals are fun. As a specific case, easily generalizable, abcdef/999999 = .abcdefabcdefabcdef… This leads to a rather charming – and somewhat sexist – number puzzle. Each letter stands for a decimal digit. Solve… > > EVE/DID = .TALKTALKTALKTALK…) > > (From Martin Gardner.) abcdef = 876543 ---
**Author:** ![netzweltler](https://avatars.discourse-cdn.com/v4/letter/n/a3d4f5/32.png) [@netzweltler](https://boards.straightdope.com/u/netzweltler)\ **Post date:** [October 12, 2016, 7:04am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/146 "2016-10-12T07:04:22Z")
> [@netzweltler](#): > > abcdef = 876543 And many others 🙂 ---
**Author:** ![watchwolf49](https://avatars.discourse-cdn.com/v4/letter/w/e9c0ed/32.png) [@watchwolf49](https://boards.straightdope.com/u/watchwolf49)\ **Post date:** [October 12, 2016, 12:31pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/147 "2016-10-12T12:31:52Z")
“Even when the _Far Dareis Mai_ stop talking they can’t stop talking” ---
**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:** [October 12, 2016, 1:08pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/148 "2016-10-12T13:08:36Z")
> [@](#): > > So, if it (the sum) doesn’t stop at point 1/3 on the number line, the meaning of the equal sign (0.333… = 1/3) is being close enough to in this case? It means “being close enough, for _every possible definition of close enough_”. ---
**Author:** ![DSYoungEsq](https://avatars.discourse-cdn.com/v4/letter/d/c6cbf5/32.png) [@DSYoungEsq](https://boards.straightdope.com/u/DSYoungEsq)\ **Post date:** [October 12, 2016, 1:20pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/149 "2016-10-12T13:20:57Z")
> [@Chronos](#): > > It means “being close enough, for _every possible definition of close enough_”. Well, apparently except for whatever definition **engjs** is using… :smack: ---
**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:** [October 12, 2016, 3:25pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/150 "2016-10-12T15:25:37Z")
> [@](#): > > Or to put it another way, 0.3333… isn’t a sequence. I think there is an easier way to look at all this. 0.333333… _is_ a sequence of digits; it is a sequence that extends infinitely in one direction. The real numbers can be _defined_ as the set of such sequences. Without going into details (sign and decimal point), there will be, with one class of exceptions, a one-to-one correspondence between the set of reals and the set of such infinite sequences. The exceptions are sequences that end in 000… or 999… There are _two_ sequences which map to reals of that form. There are many such numbers, e.g. 1/2 = .500… = .499999… but 1/3 = .33333… is _not_ one of the exceptions. (This leads to a simple proof that 1 = .999…: 3/3 = 3 \* .33333… = .99999…) ---
**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:** [October 12, 2016, 3:29pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/151 "2016-10-12T15:29:55Z")
> [@septimus](#): > > I think there is an easier way to look at all this. 0.333333… _is_ a sequence of digits; it is a sequence that extends infinitely in one direction. > > The real numbers can be _defined_ as the set of such sequences. Without going into details (sign and decimal point), there will be, with one class of exceptions, a one-to-one correspondence between the set of reals and the set of such infinite sequences… Yup, that’s a much, much easier way. ---
**Author:** ![netzweltler](https://avatars.discourse-cdn.com/v4/letter/n/a3d4f5/32.png) [@netzweltler](https://boards.straightdope.com/u/netzweltler)\ **Post date:** [October 12, 2016, 3:46pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/152 "2016-10-12T15:46:08Z")
> [@DSYoungEsq](#): > > Well, apparently except for whatever definition **engjs** is using… :smack: He might have some problem in understanding what “being close enough, for every possible definition of close enough” means exactly. Actually, I do. :o ---
**Author:** ![DSYoungEsq](https://avatars.discourse-cdn.com/v4/letter/d/c6cbf5/32.png) [@DSYoungEsq](https://boards.straightdope.com/u/DSYoungEsq)\ **Post date:** [October 12, 2016, 4:04pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/153 "2016-10-12T16:04:04Z")
I tried to point out that the issue is “=”, but no one pays attention. As I see it, **engjs** is essentially saying that, if you cannot carry out the decimal representation of 0.333… to the point that it exactly has the same value as 1/3, the two are not equal, and since you can never write enough threes to get there, there’s always some infinitesimally small difference, so the two are not equal. Everyone else is saying that, if you cannot identify the difference between the two, then they are equal. Since you are using an infinitely long string of 3s, there’s no point where you can identify the difference between 0.333… and 1/3. Or, if you will, that 1/3 - 0.333… is not equal to some value distinct from zero. Thus, they are equal. So long as the two “sides” to the argument cannot agree upon the definition of “=”, there’s really no point to the discussion, is there? ---
**Author:** ![DSYoungEsq](https://avatars.discourse-cdn.com/v4/letter/d/c6cbf5/32.png) [@DSYoungEsq](https://boards.straightdope.com/u/DSYoungEsq)\ **Post date:** [October 12, 2016, 4:10pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/154 "2016-10-12T16:10:41Z")
> [@netzweltler](#): > > He might have some problem in understanding what “being close enough, for every possible definition of close enough” means exactly. Actually, I do. :o The more difficult aspect in my mind was always accepting that “close enough” was the same as “equals.” But that was back in my youth, when I didn’t stop to wonder about the fact that negative integers didn’t really “exist”, but were merely constructs of the human brain. Liberating mathematics from “reality” tends to make dealing with math much easier. ---
**Author:** ![netzweltler](https://avatars.discourse-cdn.com/v4/letter/n/a3d4f5/32.png) [@netzweltler](https://boards.straightdope.com/u/netzweltler)\ **Post date:** [October 12, 2016, 4:20pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/155 "2016-10-12T16:20:18Z")
> [@DSYoungEsq](#): > > As I see it, **engjs** is essentially saying that, if you cannot carry out the decimal representation of 0.333… to the point that it exactly has the same value as 1/3, the two are not equal, and since you can never write enough threes to get there, there’s always some infinitesimally small difference, so the two are not equal. As long as you don’t write enough 3s (only a finite number of them) the difference is not even infinitesimally small (whatever infinitesimal means in this context). But we can write enough 3s: t = 0: write 0.3 t = 0.5: append another 3 (0.33) t = 0.75: append another 3 (0.333) … at t = 1s we have written 0.333… (taking into account that we have liberated mathematics from “reality”) ---
**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:** [October 12, 2016, 4:55pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/156 "2016-10-12T16:55:59Z")
> [@DSYoungEsq](#): > > negative integers didn’t really “exist”, but were merely constructs of the human brain. Tell that to my bank manager. ---
**Author:** ![DSYoungEsq](https://avatars.discourse-cdn.com/v4/letter/d/c6cbf5/32.png) [@DSYoungEsq](https://boards.straightdope.com/u/DSYoungEsq)\ **Post date:** [October 12, 2016, 5:01pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/157 "2016-10-12T17:01:54Z")
> [@netzweltler](#): > > As long as you don’t write enough 3s (only a finite number of them) the difference is not even infinitesimally small (whatever infinitesimal means in this context). _I_ understand this point. But I think what **engjs** is trying to say (I’ll let him actually SAY it, if he’ll ever respond to my questions) is that, to quantify that difference requires that you identify exactly where the two are not matched up. Can’t do that. But conceptually, you know it exists, because you can’t match them up. Hence there is a difference, even if it cannot be quantified (thus, is infinitesimally small). I think that asserting that a point exists on the number line, which resides between 0.333… and 1/3, without being able to identify it is unhelpful at best. I conceive that would be the inherent result of there being an infinitesimally small difference between the two. Unless it’s intended to mean something smaller in dimension than even a single number… ---
**Author:** ![DSYoungEsq](https://avatars.discourse-cdn.com/v4/letter/d/c6cbf5/32.png) [@DSYoungEsq](https://boards.straightdope.com/u/DSYoungEsq)\ **Post date:** [October 12, 2016, 5:03pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/158 "2016-10-12T17:03:16Z")
> [@The\_Great\_Unwashed](#): > > Tell that to my bank manager. Show me the negative money you owe him. 😉 Or, as I used to challenge my students: hand me negative three pencils. ---
**Author:** ![watchwolf49](https://avatars.discourse-cdn.com/v4/letter/w/e9c0ed/32.png) [@watchwolf49](https://boards.straightdope.com/u/watchwolf49)\ **Post date:** [October 12, 2016, 5:07pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/159 "2016-10-12T17:07:18Z")
> [@The\_Great\_Unwashed](#): > > Tell that to my bank manager. Your bank manager would define you being overdrawn as a positive value … what you think as - $35 is to him + $85 … or whatever … ---
**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:** [October 12, 2016, 5:43pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/160 "2016-10-12T17:43:44Z")
> [@DSYoungEsq](#): > > Show me the negative money you owe him. 😉 > > Or, as I used to challenge my students: hand me negative three pencils. I’ve tried that. It’s why I have no pencils. [Previous page](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711.md?page=7) [Next page](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711.md?page=9)