# 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:** 6
**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 9, 2016, 11:29am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/101 "2016-10-09T11:29:23Z")
OK, if you want to include 0.3 in your set as well, then we can do that. But in that case, this statement: > [@](#): > > Consequently, given that each element of this set is a different real number, there is no largest number in the set. is false: If we’re including 0.3 in the set, then it is clearly and unambiguously the largest element of that set, because there is no other element which is larger than it. ---
**Author:** ![DSYoungEsq](https://avatars.discourse-cdn.com/v4/letter/d/c6cbf5/32.png) [@DSYoungEsq](https://boards.straightdope.com/u/DSYoungEsq)\ **Post date:** [October 9, 2016, 3:23pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/102 "2016-10-09T15:23:40Z")
> [@The\_Great\_Unwashed](#): > > _ **Thank god you shouted that immediately after I had merely said it** _ I figured he needed it emphasized. 😛 ---
**Author:** ![watchwolf49](https://avatars.discourse-cdn.com/v4/letter/w/e9c0ed/32.png) [@watchwolf49](https://boards.straightdope.com/u/watchwolf49)\ **Post date:** [October 9, 2016, 5:21pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/103 "2016-10-09T17:21:00Z")
> [@DSYoungEsq](#): > > I figured he needed it emphasized. 😛 Thank God he didn’t yell it in red. ---
**Author:** ![engjs](https://avatars.discourse-cdn.com/v4/letter/e/6f9a4e/32.png) [@engjs](https://boards.straightdope.com/u/engjs)\ **Post date:** [October 9, 2016, 11:57pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/104 "2016-10-09T23:57:23Z")
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: Infinity … is not a number Yes I know that. Don’t assume that I know less than you do. 🙂 The Great Unwashed says he has an honours degree in maths and that he teaches the subject. Don’t assume that I know less than he does either. 🙂 Watchwolf49: So in your first post, “some number” cannot be include infinity. So how do you define 0.3~? As an infinite number of threes? 🙂 When I say “some number” I mean that everything after the decimal point is a three. Did you really not understand that? Because it looks to me like you are deliberately trying to misunderstand me. 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. ---
**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 10, 2016, 12:27am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/105 "2016-10-10T00:27:15Z")
So, 1/3 is the limit, but it is not equal to the limit? What definition of “equal” are you using? ---
**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:** [October 10, 2016, 1:00am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/106 "2016-10-10T01:00:56Z")
> [@engjs](#): > > 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. You lost me. How can a symbol that represents a number not be exactly equal to that number? ---
**Author:** ![watchwolf49](https://avatars.discourse-cdn.com/v4/letter/w/e9c0ed/32.png) [@watchwolf49](https://boards.straightdope.com/u/watchwolf49)\ **Post date:** [October 10, 2016, 2:34am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/107 "2016-10-10T02:34:50Z")
> [@engjs](#): > > Watchwolf49: What is the decimal expansion of 1/3? > > It has no actual decimal expansion … [snip] Both the set of fractions and the set of decimal expansions form vector spaces. If you’ll remember from your linAlgebra class, between the elements of two vector spaces, there exists a one to one correspondence. For every fraction, there exists one and only one decimal representation. That might be the _only_ thing I remember from that class, so I’ll let my betters explain why this is so. So, 1/3 = 0.333…; otherwise we’re in violation of the most basic theorem of abstract mathematics. If the basis of your claim is that the proof of this theorem is in error, please enlighten us. You’ve reminded us twice now that you’re knowledgeable in math, but no one here has questioned that. Now we need you to explain why the fraction 1/3 has no decimal representation, in spite the rules of math saying it must. ---
**Author:** ![engjs](https://avatars.discourse-cdn.com/v4/letter/e/6f9a4e/32.png) [@engjs](https://boards.straightdope.com/u/engjs)\ **Post date:** [October 10, 2016, 12:07pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/108 "2016-10-10T12:07:17Z")
Watchwolf49: Now we need you to explain why the fraction 1/3 has no decimal representation, in spite the rules of math saying it must. 🙂 It does have a decimal representation: 0.3~. My point is that the fraction and the decimal notation are not the same number, they vary by an infinitesimal amount. 0.3~ doesn’t equal 1/3 because it is the same number, it equals 1/3 because it is defined to do so. You could define 1 + 1 to equal 7 if you wanted, but it wouldn’t be useful and would in fact produce massive contradictions. Defining 1/3 as 0.3~ is useful. Watchwolf49: Both the set of fractions and the set of decimal expansions form vector spaces. … between the elements of two vector spaces, there exists a one to one correspondence. For every fraction, there exists one and only one decimal representation. That might be the only thing I remember from that class, so I’ll let my betters explain why this is so. 🙂 The set of fractions (the rationals) in enumerable, meaning it can be bijected (the fancy word for what you said) onto the natural numbers. The set of reals is not. A guy called Cantor demonstrated this using his diagonalisation method, which showed that for any enumeration of the real numbers there must be one extra decimal expansion of a real number that can’t be found in the enumeration. So the set of decimal expansions is bigger than the set of fractions. (I don’t actually agree with this, but it’s me versus the world on that one. 😃 ) That said, you can biject (0,1) onto (7,8). It doesn’t mean that any number in the first range is equal to any number in the second. Even if you could demonstrate a bijection between decimal expansions and fractions, it doesn’t mean that every decimal expansion is equal to a fraction, or that every fraction is equal to a decimal expansion. ---
**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 10, 2016, 1:06pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/109 "2016-10-10T13:06:11Z")
You seem to think that there’s some book of definitions somewhere that says that 0.3 == 1/3, and 0.6 == 2/3, and 0.142857 == 1/7, and so on, and that we could if we chose just strike any line from that book. But of course there can’t be such a book: It would be infinitely long, and no mathematician has ever had time to write it. What we have instead is a definition for how the entire system of decimal representations of numbers works. And sure, you can do without that entire system if you choose, too… Mathematics is all about choosing some set of rules and seeing what follows. But if you do that, then you lose not only 0.3 = 1/3 and the like; you also lose 0.1 = 1/10 and 0.25 = 1/4 and all the rest. In short, if 0.3 does not equal 1/3, then no decimal representation is equal to anything. ---
**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:** [October 10, 2016, 2:37pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/110 "2016-10-10T14:37:43Z")
> [@engjs](#): > > 🙂 It does have a decimal representation: 0.3~. My point is that the fraction and the decimal notation are not the same number, they vary by an infinitesimal amount. 0.3~ doesn’t equal 1/3 because it is the same number, it equals 1/3 because it is defined to do so. You’re contradicting yourself here. If it equals 1/3 by definition, then it is the same number. You can’t have it both ways. > [@](#): > > (I don’t actually agree with this, but it’s me versus the world on that one. 😃 ) It’s you versus the parts of the world that matters when it comes to 0.333… being equal to 1/3 as well. There are places in math where infinitesimals are useful, interesting and mathematical. You defining there is one between 0.333… and 1/3 isn’t one such place. ---
**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:** [October 10, 2016, 2:59pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/111 "2016-10-10T14:59:18Z")
> [@engjs](#): > > 🙂 It does have a decimal representation: 0.3~. My point is that the fraction and the decimal notation are not the same number, they vary by an infinitesimal amount. It would help me to follow your argument if you explained this. You seem to be saying that 0.333~ + 0.333~ + 0.333~ /= 1 because each is missing an infinitesimal. So what does it equal? Do they add up to 0.999~? Or do the three infinitesimals add up to something larger than one infinitesimal? Are all infinitesimals the same size? Can you add your infinitesimals? What if you add an infinite number of infinitesimals? You keep using the word infinitesimal without any definition. Definitions really are important, at least to me. Please provide one, something I grasp onto. ---
**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 10, 2016, 3:13pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/112 "2016-10-10T15:13:54Z")
> [@watchwolf49](#): > > For every fraction, there exists one and only one decimal representation. Let’s keep our fingers crossed that this isn’t true (0.9 = 1, remember!) ---
**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:** [October 10, 2016, 3:20pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/113 "2016-10-10T15:20:53Z")
> [@The\_Great\_Unwashed](#): > > Let’s keep our fingers crossed that this isn’t true (0.9 = 1, remember!) Again for my own clarity, how is this different from 1/2 = 2/4 = 3/6 = … ? ---
**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 10, 2016, 3:27pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/114 "2016-10-10T15:27:45Z")
> [@Exapno\_Mapcase](#): > > Again for my own clarity, how is this different from 1/2 = 2/4 = 3/6 = … ? Exactly! (I presume you’re addressing **watchwolf**.) ---
**Author:** ![watchwolf49](https://avatars.discourse-cdn.com/v4/letter/w/e9c0ed/32.png) [@watchwolf49](https://boards.straightdope.com/u/watchwolf49)\ **Post date:** [October 10, 2016, 4:14pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/115 "2016-10-10T16:14:26Z")
> [@engjs](#): > > Watchwolf49: Now we need you to explain why the fraction 1/3 has no decimal representation, in spite the rules of math saying it must. > > 🙂 It does have a decimal representation: 0.3~. My point is that the fraction and the decimal notation are not the same number, they vary by an infinitesimal amount. 0.3~ doesn’t equal 1/3 because it is the same number, it equals 1/3 because it is defined to do so. You could define 1 + 1 to equal 7 if you wanted, but it wouldn’t be useful and would in fact produce massive contradictions. Defining 1/3 as 0.3~ is useful. We seem to be running around in circles … first you claim 1/3 has _no_ decimal representation … now you say it does. You seem to think that 1/3 = 0.333… by definition is somehow relevant. It’s not, in the context of vector spaces, (scalar) multiplication is very strictly defined. You will have to post a different definition of multiplication if you want 1/3 ≠ 0.333…; and I’m not saying this isn’t valid, just that it’s not valid over a vector space. Can we please stay within this context or state what context you are speaking about. > [@engjs](#): > > Watchwolf49: Both the set of fractions and the set of decimal expansions form vector spaces. … between the elements of two vector spaces, there exists a one to one correspondence. For every fraction, there exists one and only one decimal representation. That might be the only thing I remember from that class, so I’ll let my betters explain why this is so. > > 🙂 The set of fractions (the rationals) in enumerable, meaning it can be bijected (the fancy word for what you said) onto the natural numbers. The set of reals is not. A guy called Cantor demonstrated this using his diagonalisation method, which showed that for any enumeration of the real numbers there must be one extra decimal expansion of a real number that can’t be found in the enumeration. So the set of decimal expansions is bigger than the set of fractions. (I don’t actually agree with this, but it’s me versus the world on that one. 😃 ) Whoa whoa whoa … how did rational numbers and natural numbers come about here. We’ve already discussed in this thread that fractions include irrational numbers. For example π/2 is a common occurrence, the number of radians in a right angle, and this number is profoundly irrational. Additionally, neither the rationals nor naturals form a vector space. I understand that, technically, the real numbers also do not form a vector space; but that distinction is trivial in this discussion. The reals aren’t closed when multiplied by a complex number. I’ve scanned through some of the pages on Wikipedia about set theory, that’s all fine and dandy in that it establishes why we can define a specific set. However, once we’ve defined the specific set we’re discussing then we need to stay within the axioms and definitions of that specific set. Here we have the complete set of complex decimal representations and the complete set of complex fractions having the same number of elements. I’m guessing this is the “absolute” infinite condition. The reals behave the same way if we limit scalar multiplication to multiplication by reals only. > [@engjs](#): > > That said, you can biject (0,1) onto (7,8). It doesn’t mean that any number in the first range is equal to any number in the second. Even if you could demonstrate a bijection between decimal expansions and fractions, it doesn’t mean that every decimal expansion is equal to a fraction, or that every fraction is equal to a decimal expansion. (7,8) is not a vector space, it lacks a unique additive identity and it is not closed under addition. Whatever principles of “bijection” you’re using here is outside the context of the basic vector space. I’m not saying this is invalid, but you do have to clearly state what context you’re working in. If you say hyperring or multilinear algebra, you’re _way_ overthinking this. You’ve stated that 1/3 = 0.333… by definition. If we ignore the definition of mathematics; then yes, we can say that 1/3 = (an apple) or that a tensor is where grad students keep their beer cold. We can also use a claw hammer to cut a 2x4 to length … Wikipedia is using SDMB as their reference … so sad … just so sad … ---
**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 10, 2016, 4:33pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/116 "2016-10-10T16:33:01Z")
**watchwolf** , with all due respect, perhaps you might want to sit this one out? The fact that real numbers are a vector space is in no way relevant to this topic, and the properties that you’re claiming that vector spaces have are false. ---
**Author:** ![watchwolf49](https://avatars.discourse-cdn.com/v4/letter/w/e9c0ed/32.png) [@watchwolf49](https://boards.straightdope.com/u/watchwolf49)\ **Post date:** [October 10, 2016, 5:10pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/117 "2016-10-10T17:10:47Z")
> [@Chronos](#): > > **watchwolf** , with all due respect, perhaps you might want to sit this one out? The fact that real numbers are a vector space is in no way relevant to this topic, and the properties that you’re claiming that vector spaces have are false. Understood … although I’m curious as to which properties I’m claiming are false. Thank you for waving me off this discussion, but you can put the rolled up newspaper down now … please … ---
**Author:** ![Shodan](https://avatars.discourse-cdn.com/v4/letter/s/9f8e36/32.png) [@Shodan](https://boards.straightdope.com/u/Shodan)\ **Post date:** [October 10, 2016, 5:48pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/118 "2016-10-10T17:48:11Z")
Correct me if I am wrong\*, but is at least part of the issue that representing 1/3 as .333333… does not mean “if you repeat the division long enough you eventually reach 0” - just that .333… is how mathematicians represent 1/3? Regards, Shodan \*If ever there were an unnecessary statement on the SDMB… ---
**Author:** ![Trinopus](https://avatars.discourse-cdn.com/v4/letter/t/2bfe46/32.png) [@Trinopus](https://boards.straightdope.com/u/Trinopus)\ **Post date:** [October 10, 2016, 8:27pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/119 "2016-10-10T20:27:02Z")
> [@Shodan](#): > > Correct me if I am wrong\*, but is at least part of the issue that representing 1/3 as .333333… does not mean “if you repeat the division long enough you eventually reach 0” - just that .333… is how mathematicians represent 1/3? . . . Essentially, yeah. It isn’t an algorithm, just a notation. > [@](#): > > \*If ever there were an unnecessary statement on the SDMB… Grin! Anyway, I think **Cartoonacy** won the game for us, back in post #87. [QUOTE=Cartoonacy] 1 - 0.9~ = 0.0~, no? So if 1 \> 0.9~ then that means 0.0~ \> 0 Sorry, I don’t see that. [/QUOTE] ---
**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:** [October 10, 2016, 8:53pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/120 "2016-10-10T20:53:54Z")
> [@The\_Great\_Unwashed](#): > > Exactly! (I presume you’re addressing **watchwolf**.) I don’t think you’re helping me. I can’t tell what you’re agreeing with. You seem to have a problem with “one and only one decimal notation.” Is that correct? If so, could you explain that? If it’s true, could you then explain what your argument is? [Previous page](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711.md?page=5) [Next page](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711.md?page=7)