# 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:** 5
**Author:** ![RTFirefly](https://avatars.discourse-cdn.com/v4/letter/r/c77e96/32.png) [@RTFirefly](https://boards.straightdope.com/u/RTFirefly)\ **Post date:** [October 3, 2016, 1:32pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/81 "2016-10-03T13:32:36Z")
> [@Riemann](#): > > Perhaps you should re-read **The Great Unwashed** more carefully, and acknowledge that he’s playing devil’s advocate here? Perhaps you realize that, but your confrontational tone suggests that you don’t. I re-read him, and he still comes across to me as deliberately obtuse. If he’s playing devil’s advocate, it’s up to HIM to make that clear. ---
**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 3, 2016, 1:45pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/82 "2016-10-03T13:45:27Z")
> [@RTFirefly](#): > > I re-read him, and he still comes across to me as deliberately obtuse. If he’s playing devil’s advocate, it’s up to HIM to make that clear. How is this not clear? > [@The\_Great\_Unwashed](#): > > And (I suppose you know) I have complete faith in delta-epsilon proofs, I just don’t believe that they are as intuitive as \*\*Trinopus \*\*wants them to be. > > Non-mathematicians already bent on the idea that 0.999… is different to 1 will fairly quickly home in on the construct 0.000…1 – that “looks” like a very reasonable thing to propose – in fact it’s the apparent compliment of the 0.999… in question. It’s the bit that the deniers suppose is missing. > > Getting all lawlerly and demanding they play the game by “our” rules isn’t as persuasive as one might hope. ‘demanding they play the game by “our” rules’. Seems rather obvious to me. ---
**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 3, 2016, 1:46pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/83 "2016-10-03T13:46:11Z")
> [@The\_Great\_Unwashed](#): > > And (I suppose you know) I have complete faith in delta-epsilon proofs, I just don’t believe that they are as intuitive as \*\*Trinopus \*\*wants them to be. > > Non-mathematicians already bent on the idea that 0.999… is different to 1 will fairly quickly home in on the construct 0.000…1 – that “looks” like a very reasonable thing to propose – in fact it’s the apparent compliment of the 0.999… in question. It’s the bit that the deniers suppose is missing. > > Getting all lawlerly and demanding they play the game by “our” rules isn’t as persuasive as one might hope. > [@The\_Great\_Unwashed](#): > > Well the game ain’t over to the gong sounds. > > If you’re saying it’s impossible to have an infinite number of ZEROS followed by a ONE, how can you expect “me” to believe that we can have a ZERO-POINT followed by an infinite number of NINES. > > Besides, I think I can have an infinite number of zeros followed by a one, it’s the limit of this suite (0.1, 0.01, 0.001, …). It’s already been stipulated that suites can have such weird limits (i.e. (0.9, 0.99, 0.999, …)) > > You should note you are arguing with the already persuaded. The obstinate and yet-to-be-persauded others are not going to be impressed with the mere assertion that 0.000…1 is a “bullshit number”. > [@RTFirefly](#): > > I re-read him, and he still comes across to me as deliberately obtuse. If he’s playing devil’s advocate, it’s up to HIM to make that clear. Sometimes I am a stupid twat – I had supposed that a note directed at you would have sufficed. Still, we live and learn… ---
**Author:** ![watchwolf49](https://avatars.discourse-cdn.com/v4/letter/w/e9c0ed/32.png) [@watchwolf49](https://boards.straightdope.com/u/watchwolf49)\ **Post date:** [October 3, 2016, 2:19pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/84 "2016-10-03T14:19:39Z")
The highest form of mathematical thought is accounting; for truly, money is the only thing worth counting. ---
**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 3, 2016, 2:43pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/85 "2016-10-03T14:43:45Z")
Epsilon-delta proofs themselves are not intuitive: They take too long to explain, and even if each step in the process is straightforward, there’s an upper bound to the number of steps that the human brain can hold in working memory at once. But the concept of a “limit” can be made intuitive: It’s just “I can get as close as I like to that number, even if I can’t necessarily exactly reach it”. Epsilon-delta proofs are just to reassure you that your intuition is reliable, that your sense of “limit” has a sound mathematical backing. But when you speak of the limit of (0.1, 0.01, 0.001, 0.0001…) as being something other than 0, that’s showing that you don’t have the concept of “limit” at all. ---
**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 3, 2016, 2:51pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/86 "2016-10-03T14:51:53Z")
> [@Chronos](#): > > But when you speak of the limit of (0.1, 0.01, 0.001, 0.0001…) as being something other than 0, that’s showing that you don’t have the concept of “limit” at all. Are you talking to me? ---
**Author:** ![Cartoonacy](https://avatars.discourse-cdn.com/v4/letter/c/848f3c/32.png) [@Cartoonacy](https://boards.straightdope.com/u/Cartoonacy)\ **Post date:** [October 3, 2016, 8:54pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/87 "2016-10-03T20:54:59Z")
1 - 0.9~ = 0.0~, no? So if 1 \> 0.9~ then that means 0.0~ \> 0 Sorry, I don’t see that. ---
**Author:** ![engjs](https://avatars.discourse-cdn.com/v4/letter/e/6f9a4e/32.png) [@engjs](https://boards.straightdope.com/u/engjs)\ **Post date:** [October 8, 2016, 10:18am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/88 "2016-10-08T10:18:16Z")
I’m amazed at how many people here are so ready to say I’m not a mathematician. Or perhaps not. You never know who or what the person at the other end of the keyboard is. It is clear that a lot of people here do not talk like mathematicians, and that may well be an indication that they are not mathematicians. So let’s talk like mathematicians. 🙂 Form the set of all decimal expansions of the form three point followed by some number of threes. It’s obvious that the possible number of threes is infinite, so this set is an infinite set. An infinite set does not have a size; there are always more elements. Consequently, given that each element of this set is a different real number, there is no largest number in the set. For each element in the set there are an infinite number of elements that are bigger than it. 0.3~ matches the definition of an element in the set – it starts with “0.” and the rest is composed of threes – so it is a member of the set. This is turn means that there are an infinite number of elements in the set that are bigger than it. Because it is an infinitely long element, all such elements must also be infinitely long. It’s also true to say that there are an infinite number of infinitely long elements that are smaller than it, but that’s by the by. In order to take a limit you must have a series. So order the set so that each element is followed by an element that is larger. This is possible because all elements in the set are different. Taking a limit does not mean figuring out what the last element in the series is. There is no last element. Taking a limit involves bounding the series using the methods of analysis, usually contradiction and mathematical induction. The process in this case means finding the smallest number that is larger than any term in the series. The mathematics for doing this are well known and the result of the process is 1/3. No term in the series has the value 1/3. If any did, the next term would necessarily have to be bigger. 0.3~ is not equal to 1/3. In fact, there are an infinite number of terms that are bigger than 0.3~ and smaller than 1/3. It is convenient to be able to write 1/3 in a decimal form, and 0.3~ is a useful way of doing so, but it is very much a case of using a label that means one thing and applying it to something different. Hence 0.3~ is not equal to 1/3; it is simply a label that is defined to mean 1/3. But the two terms actually differ by an infinitesimal amount. RTFirefly: ‘being there’ means ‘given any epsilon \> 0, there’s an N such that every term after the Nth term is within epsilon of ‘there’.’ So you are saying that ‘being there’ means not actually being there, but rather being very close. If a term is not ‘there’ but is within epsilon of ‘there’ then it is not actually ‘there’ unless epsilon is zero. You are just specifying the infinitesimal difference as a number greater than zero but less than epsilon. 🙂 ---
**Author:** ![Xema](https://avatars.discourse-cdn.com/v4/letter/x/9de053/32.png) [@Xema](https://boards.straightdope.com/u/Xema)\ **Post date:** [October 8, 2016, 12:01pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/89 "2016-10-08T12:01:35Z")
> [@engjs](#): > > Form the set of all decimal expansions of the form three point followed by some number of threes … > 0.3~ matches the definition of an element in the set – it starts with “0.” and the rest is composed of threes – so it is a member of the set. It also satisfies all requirements to be larger than any member of your set (interestingly, by an amount that is exactly equal to the difference between that member and 1/3). Your misunderstanding may be tied up with the idea that infinity is “some number”. ---
**Author:** ![Thudlow\_Boink](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/thudlow_boink/32/320_2.png) [@Thudlow\_Boink](https://boards.straightdope.com/u/Thudlow_Boink)\ **Post date:** [October 8, 2016, 2:21pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/90 "2016-10-08T14:21:16Z")
> [@engjs](#): > > It’s obvious that the possible number of threes is infinite, so this set is an infinite set. An infinite set does not have a size; there are always more elements. Consequently, given that each element of this set is a different real number, there is no largest number in the set. For each element in the set there are an infinite number of elements that are bigger than it. I can’t tell for sure, but it may be that you’re basing your argument on the fallacious assumption that an infinite set cannot have a largest element. The closed interval [0, 1] is a counterexample: it contains infinitely many elements, but 1 is the largest. ---
**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 8, 2016, 2:35pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/91 "2016-10-08T14:35:39Z")
> [@Thudlow\_Boink](#): > > I can’t tell for sure, but it may be that you’re basing your argument on the fallacious assumption that an infinite set cannot have a largest element. The closed interval [0, 1] is a counterexample: it contains infinitely many elements, but 1 is the largest. Like you, I am unsure as to what is being argued, but I _think_ what **engjs** is imagining would be something more like the open-interval (0, 1) (which doesn’t have a maximum value). ---
**Author:** ![watchwolf49](https://avatars.discourse-cdn.com/v4/letter/w/e9c0ed/32.png) [@watchwolf49](https://boards.straightdope.com/u/watchwolf49)\ **Post date:** [October 8, 2016, 5:51pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/92 "2016-10-08T17:51:50Z")
> [@engjs](#): > > [snip] … Form the set of all decimal expansions of the form three point followed by **some number** of threes … [snip] … 0.3~ matches the definition of an element in the set … [snip] Emphases mine These two statement seem mutually exclusive, 0.333… is _not_ "followed by some number’ of threes, therefore it is not an element of the set you are describing. Your arguments require the condition that at some point we stop adding threes to the decimal expression and your conclusion is then that such a number doesn’t equal 1/3. The problem is your condition is invalid, we never stop adding threes. In terms of mathemagical doublespeak: The complete set of all complex numbers forms a vector space, therefore for each fractional representation of a complex number, the exists one and only one decimal representation. Your mission, if you choose to accept it, is to state the unique decimal representation of the fraction (1/3 + 0_i_). ---
**Author:** ![engjs](https://avatars.discourse-cdn.com/v4/letter/e/6f9a4e/32.png) [@engjs](https://boards.straightdope.com/u/engjs)\ **Post date:** [October 8, 2016, 9:07pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/93 "2016-10-08T21:07:24Z")
watchwolf49: These two statement seem mutually exclusive, 0.333… is not "followed by some number’ of threes, therefore it is not an element of the set you are describing. Your arguments require the condition that at some point we stop adding threes to the decimal expression and your conclusion is then that such a number doesn’t equal 1/3. The problem is your condition is invalid, we never stop adding threes. So which digit after the decimal point is not a three? The definition I’ve given does not say that there has to be a finite number of threes, only that all following digits must be threes. You seem to be arguing that an infinite set cannot contain an infinite element, or perhaps that it cannot contain more than one. The set of the real numbers contains pi, e, and root two, all of which are infinite. The power set of the natural numbers contains the natural numbers, the odd natural numbers, the even natural numbers, the squares, the cubes, and so on. All of these are distinct, all are infinite, and despite the fact that it’s possible to show that some are distinct subsets of others, all are the same size. I have not argued that at some point you must stop adding threes to the decimal expansion; if you did the result wouldn’t be infinite; I just question your claim that an infinite set can’t contain an infinite element. Perhaps you are arguing that all infinite elements of the set must have infinity threes, and hence the same number of threes, and therefore they must be the same element. The even numbers form an infinite set. So do the powers of two. They are both elements of the power set of the natural numbers. They are both infinite. They do have the same number of elements. They are not the same thing. But I think that what is really happening is that you are working backwards from your assumption that 0.3~ is equal to 1/3, and saying that because it is equal to the limit it can’t be part of the set, and from that trying to argue that the elements of the set must all be finite. That’s called circular logic. 🙂 Edit: I said “squares” rather than “powers of two.” Too early in the morning. 🙂 ---
**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 8, 2016, 9:15pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/94 "2016-10-08T21:15:39Z")
There is no digit which isn’t a 3. But it isn’t “some number of 3s”: If it is, then tell us what number. ---
**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 8, 2016, 10:12pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/95 "2016-10-08T22:12:21Z")
> [@engjs](#): > > But I think that what is really happening is that you are working backwards from your assumption that 0.3~ is equal to 1/3, and saying that because it is equal to the limit it can’t be part of the set, and from that trying to argue that the elements of the set must all be finite. That’s called circular logic. 🙂 Actually \*\*watchwolf \*\*has said more than once (and in more than one way) that you could help us out by telling us what you think the decimal expansion of 1/3 is. This is the second time I have pointed this out (that’s several opportunities you have had to read and respond to this question). If you will not now address the question: **WHAT IS THE DECIMAL EXPANSION OF 1/3?** (even if it’s only to say that you cannot say (or _one cannot say_) for some reason or other) then I will consider the “conversation” irredeemably one-sided. ---
**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, 1:29am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/96 "2016-10-09T01:29:59Z")
> [@engjs](#): > > [snip] … Form the set of all decimal expansions of the form three point followed by some number of threes … > [@engjs](#): > > [snip] … The definition I’ve given does not say that there has to be a finite number of threes … [snip] Let’s talk like mathemagicians, the word “number” has a very specific meaning in the mathemagical context. First we state the context, this is a vector space. This means every number has to follow a number of rules. Infinity fails these rules, therefore it is not a number in a vector space. So in your first post, “some number” cannot be include infinity. I’m not a mathemagician, so perhaps it would help me understand what you’re trying to explain if you could post the algebraic function you’re using. I understand the limit of a function, I don’t understand the limit of (0.3, 0.33, 0.333 …). ETA: What is the decimal expansion of 1/3? ---
**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, 3:13am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/97 "2016-10-09T03:13:51Z")
This thread would become much simpler if everyone would stop talking across the simple question asked early on, and yet to be answered by the OP: _ **WHAT IS THE DECIMAL EXPANSION OF THE FRACTION 1 DIVIDED BY 3?** _ Lots of interesting mathematical chatter going on, but as is often the case in these threads, the cross-talk obfuscates the main point. ☹ ---
**Author:** ![netzweltler](https://avatars.discourse-cdn.com/v4/letter/n/a3d4f5/32.png) [@netzweltler](https://boards.straightdope.com/u/netzweltler)\ **Post date:** [October 9, 2016, 7:22am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/98 "2016-10-09T07:22:03Z")
> [@engjs](#): > > Re: An infinite question: Why doesn’t .999~ = 1? > > .333~ is not 1/3. The two differ by an infinitesimal. An infinitesimal is a number so small that it may as well be 0. For practical purposes you can treat .333~ as the limit of the sequence (.3, .33, .333, …) which is equal to 1/3. Consequently you can say that .333~ is equal to 1/3 because it is defined as equal to it. There are situations where infinitesimals matter, such as probability theory and integration, but those situations aren’t common and aren’t something most people would ever care about. If 0.999… stands for infinitely many commands Add 0.9 + 0.09 Add 0.99 + 0.009 Add 0.999 + 0.0009 … then following all of these infinitely many commands won’t get you to point 1. If you reached point 1 you have disobeyed those commands, because every single of those infinitely many commands tells you to get closer to 1 but NOT reach 1. Therefore, if you want to define the position of a “point” 0.999… on the number line, it cannot be at position 1 – and for the same reason (“disobeying those commands”) it cannot be short of 1 nor can it be past 1. So, if you want to measure the distance |1 – 0.999…| you know where to start the measurement (at point 1) but you don’t know where to stop the measurement, because the position of a “point” 0.999… is not defined on the number line. ---
**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 9, 2016, 8:00am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/99 "2016-10-09T08:00:26Z")
> [@dsyoungesq](#): > > this thread would become much simpler if everyone would stop talking across the simple question asked early on, and yet to be answered by the op: > > \*\*\*what is the decimal expansion of the fraction 1 divided by 3? \*\*\* > > lots of interesting mathematical chatter going on, but as is often the case in these threads, the cross-talk obfuscates the main point. ☹ _ **Thank god you shouted that immediately after I had merely said it** _ ---
**Author:** ![Asympotically\_fat](https://avatars.discourse-cdn.com/v4/letter/a/e47c2d/32.png) [@Asympotically\_fat](https://boards.straightdope.com/u/Asympotically_fat)\ **Post date:** [October 9, 2016, 9:56am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/100 "2016-10-09T09:56:52Z")
Another way of looking at it is that the sequence 0.9, 0.99, 0.999… represents the smallest real number that is greater than (or equal to) all members of that sequence. That number is 1. [Previous page](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711.md?page=4) [Next page](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711.md?page=6)