# .999 = 1? **URL:** \ **Category:** Factual Questions\ **Created:** [July 31, 2000, 6:28am UTC](https://boards.straightdope.com/t/999-1/27517 "2000-07-31T06:28:03Z")\ **Posts on this page:** 20\ **Page:** 46
**Author:** ![Valmont314](https://avatars.discourse-cdn.com/v4/letter/v/3da27b/32.png) [@Valmont314](https://boards.straightdope.com/u/Valmont314)\ **Post date:** [September 29, 2012, 6:40am UTC](https://boards.straightdope.com/t/999-1/27517/901 "2012-09-29T06:40:34Z")
> [@Francis\_Vaughan](#): > > If that were true either 2 or 0.9999… is not a member of the reals. > > Since both are members, there must be such a value. How about 1.99999… > > You can say that either 2 is the integer 2, in which case we can define 2 x 0.999… as: > > 0.999… + 0.999… , and so on for x \* 0.999… for n a member of the integers. Then sum the terms starting from the left. > > or you can take the general case, of x \* 0.999… where x is a member of the reals. Then you just apply the multiply with carry rules starting from the left. (See an earlier post about starting from the left.) The exact value of 1 is known, the exact value of Pi is not known, the product of 1 x Pi is not known, therefore 1 is not a member of the reals??? ---
**Author:** ![Valmont314](https://avatars.discourse-cdn.com/v4/letter/v/3da27b/32.png) [@Valmont314](https://boards.straightdope.com/u/Valmont314)\ **Post date:** [September 29, 2012, 6:44am UTC](https://boards.straightdope.com/t/999-1/27517/902 "2012-09-29T06:44:36Z")
> [@Giles](#): > > Yes, “there is a value for 2 x 1” – it is 2. In other words, “2 x 1” and “2” are “different symbolic representations” of the same real number. > > Therefore, the fact that “1” and “.999…” are “different symbolic representations” does not imply that they are distinct real numbers. > > Please try to make your logic consistent! Yes I considered editing out the symbolic representation part and just simply asking the question without out…but while we are at it…2 x 1 is not a symbolic representation of a member of the set of reals, it involves an operation. ---
**Author:** ![Giles](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/giles/32/60_2.png) [@Giles](https://boards.straightdope.com/u/Giles)\ **Post date:** [September 29, 2012, 6:46am UTC](https://boards.straightdope.com/t/999-1/27517/903 "2012-09-29T06:46:48Z")
> [@Valmont314](#): > > Yes I considered editing out the symbolic representation part and just simply asking the question without out…but while we are at it…2 x 1 is not a symbolic representation of a member of the set of reals, it involves an operation. OK, how about 2 and 2.0? Are they different real numbers? ---
**Author:** ![Giles](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/giles/32/60_2.png) [@Giles](https://boards.straightdope.com/u/Giles)\ **Post date:** [September 29, 2012, 6:50am UTC](https://boards.straightdope.com/t/999-1/27517/904 "2012-09-29T06:50:02Z")
Incidentally, the simplest symbolic representation of a real number often includes an operation. For example, pi + e or pi x e. There’s no simpler way to represent those real numbers. ---
**Author:** ![Valmont314](https://avatars.discourse-cdn.com/v4/letter/v/3da27b/32.png) [@Valmont314](https://boards.straightdope.com/u/Valmont314)\ **Post date:** [September 29, 2012, 6:54am UTC](https://boards.straightdope.com/t/999-1/27517/905 "2012-09-29T06:54:54Z")
> [@Giles](#): > > OK, how about 2 and 2.0? Are they different real numbers? Neither of those symbols are technically correct representations of the mathematical object you are trying to represent. Any symbolic representation that conveys an infinite number of zeros is correct. Again I considered striking the symbolic part. We could pin this down I suspect but I really think it would be a pedantic exercise. ---
**Author:** ![Giles](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/giles/32/60_2.png) [@Giles](https://boards.straightdope.com/u/Giles)\ **Post date:** [September 29, 2012, 7:05am UTC](https://boards.straightdope.com/t/999-1/27517/906 "2012-09-29T07:05:37Z")
> [@Valmont314](#): > > Neither of those symbols are technically correct representations of the mathematical object you are trying to represent. Any symbolic representation that conveys an infinite number of zeros is correct. Again I considered striking the symbolic part. We could pin this down I suspect but I really think it would be a pedantic exercise. What nonsense! Here are some “technically correct” statements about these numbers: The real numbers are a field with the operations + and x as usually defined. The real number 0 is the additive identity of the real numbers as a field. The real number 1 is the multiplicative identity of the real numbers as a field. The real number 2 may be defined as 1 + 1. Real numbers do not have to be represented as infinite decimal fractions. ---
**Author:** ![Valmont314](https://avatars.discourse-cdn.com/v4/letter/v/3da27b/32.png) [@Valmont314](https://boards.straightdope.com/u/Valmont314)\ **Post date:** [September 29, 2012, 7:17am UTC](https://boards.straightdope.com/t/999-1/27517/907 "2012-09-29T07:17:47Z")
> [@Giles](#): > > What nonsense! Here are some “technically correct” statements about these numbers: > > The real numbers are a field with the operations + and x as usually defined. > The real number 0 is the additive identity of the real numbers as a field. > The real number 1 is the multiplicative identity of the real numbers as a field. > The real number 2 may be defined as 1 + 1. > > Real numbers do not have to be represented as infinite decimal fractions. \ Giles do you own a Pit bull by chance, a venemous snake, a pair of vice grips? This entire thread involves discussions of infinite precision. The zeros are there conceptually, we just don’t bother to write them down. The point is that some reals are known and symbolically represented as having infinite precision, others are not, 5 is known with infinite precision so we can put zeros there with an ellipsis and it is technically more precise that simply writing it as five…we cannot express pi as 3.14…, how much further do you want to debate this trivial point? ---
**Author:** ![Giles](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/giles/32/60_2.png) [@Giles](https://boards.straightdope.com/u/Giles)\ **Post date:** [September 29, 2012, 7:50am UTC](https://boards.straightdope.com/t/999-1/27517/908 "2012-09-29T07:50:21Z")
> [@Valmont314](#): > > This entire thread involves discussions of infinite precision. But 2 is “infinitely precise” as a real number. > [@Valmont314](#): > > The zeros are there conceptually, we just don’t bother to write them down. No, they’re not, because you don’t need decimal fractions to represent the real numbers. What’s more, any base will do if you want to define real numbers. If we’d all been born with 4 fingers on each hand, this thread would be about whether .777… is equal to 1. > [@Valmont314](#): > > The point is that some reals are known and symbolically represented as having infinite precision, others are not, 5 is known with infinite precision so we can put zeros there with an ellipsis and it is technically more precise that simply writing it as five…we cannot express pi as 3.14…, how much further do you want to debate this trivial point? Pi is a good example. We don’t define pi as equal to some decimal fraction 3.14159, etc., because we can’t write down that whole decimal fraction. So we define it geometrically, or if we want to stick to algebra, we define as the limit of some infinite series of rational numbers. However, that limit is infinitely precise, just as 1 divided by 3 or the square root of 2 is infinitely precise – and both of those are defined without referring to their representations as decimal fractions. ---
**Author:** ![Frylock](https://avatars.discourse-cdn.com/v4/letter/f/ce7236/32.png) [@Frylock](https://boards.straightdope.com/u/Frylock)\ **Post date:** [September 29, 2012, 9:35am UTC](https://boards.straightdope.com/t/999-1/27517/909 "2012-09-29T09:35:14Z")
> [@Valmont314](#): > > The point of my post, which clearly was not explained with sufficient rigor as to force binary responses from those responding was to make the following assertions: I understood the points you list in this post perfectly well from your longer post. I explained in my own response part of the reason why your argument for these points fails. What do you have to say in response to my latest post, where I point out that just because the number of zeroes that can appear in a decimal representation is unbounded, this does not mean there is a decimal representation with an unbounded number of zeroes? It appears to me to be a crucial point. > [@](#): > > 1. The equation .999… = 1 means that the Real elements .999… and 1 are co-located on the Real number line and for the purposes of measurement are the same point. Read carefully and literally, the above is true. > [@](#): > > 1. The fact that the elements are co-located on the real number line does not imply that they are the same mathematical object, That may or may not be, but what it does imply is that they are the same real number. > [@](#): > > that is they are still distinct members of the set of Real numbers, otherwise by definition they would not have different symbolic representations. There is no definition I am aware of which would force this conclusion. In fact it seems as though there are counterexamples that would be uncontroversial to you. For example, here are two distinct symbolic representations of a single real number: 0.5, and 1/2. [quote] > [@](#): > > This distinction can be observed by the fact there is a value for 2 x 1 whereas there is no known value for 2 x .999… Of course this is part of the point at contention between you and the general mathematical community. We don’t agree that there’s no known value. We maintain that the value is known, and that it is 2. Once again, with feeling–the two strings .999… and 1 are, by standard mathematical practice, simply two names for the same real number. It is pointless to argue with this because it is literally a matter of definition. Importantly, you _can_ do math in a way which treats 1 minus infinity as a distinct quantity from 1. And you could even work out a way to do it, I suspect, which treats .999… as representing the former and not the latter. But importantly, to do this is to apply a _new way of interpreting strings like .999…_ and hence is to offer a _new definition_ of those strings. No problem. Mathematicians have done this. But when they’ve done this, they haven’t taken themselves to have “proven” the old definitions “wrong.” They are, rather, strictly and only working on _new_ definitions. And so, when they do this, they aren’t discovering something new about the reals. Rather they’re defining a type of number distinct from the reals. On the standard interpretation, the two strings refer to the same real number. But on the new interpretation, 1 still refers to a real number but .999… refers to a member of the new set of numbers–numbers that are not real. Infinitessimals are one kind of number you could do this with. Hypperreals are another. Surreals are another. It’s all been done and well covered. You would do well to read about those three kinds of numbers–it should be very interesting to you, and as you study them you’ll eventually come to understand how pointless it is to argue over definitions, and instead how fruitful and interesting it can be to simply explore the systems various sets of definitions end up giving you. ---
**Author:** ![Francis\_Vaughan](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/francis_vaughan/32/3093_2.png) [@Francis\_Vaughan](https://boards.straightdope.com/u/Francis_Vaughan)\ **Post date:** [September 29, 2012, 9:49am UTC](https://boards.straightdope.com/t/999-1/27517/910 "2012-09-29T09:49:17Z")
> [@Valmont314](#): > > The exact value of 1 is known, the exact value of Pi is not known, > the product of 1 x Pi is not known, therefore 1 is not a member of the reals??? You missed the point. The reals are closed under multiplication. If you don’t believe this you are not talking about the reals, but some other number system, or subset of the reals (which appears to be the case.) If you have two real numbers the result is a real. It matters not if the decimal expansion of the number is known, terminating or transcendental. The value of Pi is of course known perfectly. It is Pi. Are you trying to say that Pi is not a real number? The decimal expansion is infinite - but we do know how to write it, it just takes a long time. The decimal representation is not the number. If any number that requires an infinite decimal representation is defined to not be real you are in big trouble. You claimed that 2 x 0.999… does not denote a real number. This if this is so, then either 2 or 0.999… can’t be real. Or you are wrong, and that 2 x 0.999… is a member of the reals. ---
**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:** [September 29, 2012, 5:03pm UTC](https://boards.straightdope.com/t/999-1/27517/911 "2012-09-29T17:03:35Z")
> [@Valmont314](#): > > Yes, you are right that > > “fill in the blank and answer this question” > > Could imply that you were to take one of the multiple choices and use it to fill in the blank…however, I thought it was implicit that the only words that could be filled in were “at infinity” much in the same way that there is only one word that can satisfy the statement “the value of the function y = 2x +3 at \_\_\_\_\_\_\_ is 9” (the only word is three). So I’m not sure where you are going with this but it would seem that you are looking for a degree of rigor in the way the question was formulated that is above and beyond that which is necessary to convey what you were supposed to do. This sums up the entire discussion between you and the others in this thread. When we all looked at the \_\_\_\_\_\_\_\_\_ in your statement, we were all baffled. That’s because the term “at infinity” does not even occur to any of us. Why should it? No such thing exists in mathematics. It is solely part of your intuition about how infinite numbers should behave, an intuition that none of us share. That should be sufficient to explain why intuitionist statements do not work in math. You also lack understanding of the most basic concepts. Pi, for example, is an exact point on the number line. (Take a perfect circle and roll it one diameter. The resulting point is one pi length away from 0. Exactly. It is a precise, defined, point just as 2 is and the square root of two is.) There are many infinite decimal expansions known for pi. They must converge exactly on pi, but can do so by overshooting and then undershooting alternatively forever. What is your last real in such a case? It lacks even imaginary existence. Numbers have many representations. 2 is written that way for convenience, but it should be thought of as 2/1 just as one and a half is 3/2. 1.9999~, 2, 2.0. 2.0000~, 2/1, 4/2, and 120099000/60049500 are equivalent in base 10. Not a single one of those representations exist in base 2 but their equivalent expressions can be given and they all refer to exactly the same unique point on the number line. This is not something that requires proof. It is definitionally true. You keep failing to give definitions of any of the terms or procedures or operations you do. They may be intuitive to you, but nobody else can see them. Your inability to understand the basic language of math trips you up every time. Nobody has to follow your words - even though many here are making valiant attempts to explain your mistakes to you - because it is immediately obvious to a math speaker than you are uttering gibberish. (As an analogy, if you gave this page to a Spanish speaker it would be immediately obvious that it is not Spanish even if some actual Spanish words popped up now and then.) Intuition must therefore be a wrong place to start. I have intuition and it disagrees with yours. How do you get around that? Rigorous definitions and procedures that everybody must agree on. Try them. You’ll like them in the end. ---
**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:** [September 29, 2012, 5:48pm UTC](https://boards.straightdope.com/t/999-1/27517/912 "2012-09-29T17:48:26Z")
> [@Valmont314](#): > > The exact value of 1 is known, the exact value of Pi is not known, What do you mean by “the exact value of Pi is not known”? This is only true if by “the exact value” you mean “the exact decimal representation.” From much of what you’ve been saying here, it sounds like you’re working with the premise that a number **is** its decimal representation, which may be at the heart of your disagreement with so many of us here. As **Exapno** notes, Pi is an exact point on the number line. We can specify, precisely and unambiguously, what/where it is. It’s true we can’t write down, and in a sense don’t know, the exact **decimal representation** of pi (although we do know how to find as many decimal places as desired). But we know exactly what pi **is** ; and to say I’m giving the exact value when I write down the symbol π should be no more problematic than to say I’m giving the exact value of the square root of 9, or the lowest odd prime, when I write down the symbol 3. ---
**Author:** ![Valmont314](https://avatars.discourse-cdn.com/v4/letter/v/3da27b/32.png) [@Valmont314](https://boards.straightdope.com/u/Valmont314)\ **Post date:** [September 29, 2012, 6:01pm UTC](https://boards.straightdope.com/t/999-1/27517/913 "2012-09-29T18:01:49Z")
> [@Frylock](#): > > It’s all been done and well covered. You would do well to read about those three kinds of numbers–it should be very interesting to you, and as you study them you’ll eventually come to understand how pointless it is to argue over definitions, and instead how fruitful and interesting it can be to simply explore the systems various sets of definitions end up giving you. I hear you loud and clear. Regarding the reals I have gone as far as I feel that I can to go to understand what is being discussed here. The final conclusion for me is this: The lack of a definition of a least element (which is also the fundamental representation of interval in the Reals) for the Reals results in the inability to define a relationship between a least element and the limits of precision for the Reals. The definition of this relationship is necessary to balance the limits of precision for the Reals against the magnitude of the least element . The balancing is necessary to preclude the creation of co-located elements. The inability to define such a relationship results in co-located elements. The creation of co-located elements results in the following situation and two possibilities: The two symbolic representations of the co-located elements represent distinct elements of the set. The two symbolic representation of the co-located elements represent the same element. At this point, assuming nobody can demonstrate an error in the logic above, I would say that any further discussion of this topic can be viewed as a discussion of whether or not a Necker cube can be viewed as a single object with two representations or two objects with a single representation. Either answer is correct. That is as far as I am going as far the discussion of this topic as it relates to mathematical rigor and the Reals. Needless to say I do not possess the rigor to go any further even if I wanted to. The original site that I posted, was not about a mathematically rigorous treatment of the question of why or why not .999… = 1. It was about how mathematical concepts can be related to other areas of understanding and how it might be possible to see a meaningful confluence. Thank you to all of you for responding to my posts and not clobbering me to hard when I overstepped (which was often) my limits of rigor. Over and out. ---
**Author:** ![Valmont314](https://avatars.discourse-cdn.com/v4/letter/v/3da27b/32.png) [@Valmont314](https://boards.straightdope.com/u/Valmont314)\ **Post date:** [September 29, 2012, 6:35pm UTC](https://boards.straightdope.com/t/999-1/27517/914 "2012-09-29T18:35:11Z")
I forgot to add the disclaimer for the above: Everything stated above applies to decimal representations of real numbers as that is what this entire thread is about in the first place. ---
**Author:** ![Valmont314](https://avatars.discourse-cdn.com/v4/letter/v/3da27b/32.png) [@Valmont314](https://boards.straightdope.com/u/Valmont314)\ **Post date:** [September 29, 2012, 7:35pm UTC](https://boards.straightdope.com/t/999-1/27517/915 "2012-09-29T19:35:00Z")
> [@Thudlow\_Boink](#): > > What do you mean by “the exact value of Pi is not known”? This is only true if by “the exact value” you mean “the exact decimal representation.” From much of what you’ve been saying here, it sounds like you’re working with the premise that a number **is** its decimal representation, which may be at the heart of your disagreement with so many of us here. I’m on the way out the door but I will do my best to clarify my clumsily worded statement about Pi. Is this thread not about the decimal representations of real numbers? What is not known about Pi is this: The nth digit of its decimal representation is not known without going through some sort of iterative process. The nth digit of .999… is known without going through an iterative process. In this sense every digit of the sequence .999… is known a priori, this pattern is what allows us to see intuitively that the limit of the sequence .999… is 1 What is not known about Pi is whether or not Pi’s decimal representation contains a pattern so it is impossible to say what the nth digit it in its decimal representation a priori. ---
**Author:** ![Derleth](https://avatars.discourse-cdn.com/v4/letter/d/b9e5f3/32.png) [@Derleth](https://boards.straightdope.com/u/Derleth)\ **Post date:** [September 29, 2012, 8:09pm UTC](https://boards.straightdope.com/t/999-1/27517/916 "2012-09-29T20:09:30Z")
> [@Valmont314](#): > > What is not known about Pi is this: > > The nth digit of its decimal representation is not known without going through some sort of iterative process. This, in fact, is incorrect. The [Bailey-Borwein-Plouffe formula](http://en.wikipedia.org/wiki/Bailey%E2%80%93Borwein%E2%80%93Plouffe_formula) allows us to do just that. ---
**Author:** ![Valmont314](https://avatars.discourse-cdn.com/v4/letter/v/3da27b/32.png) [@Valmont314](https://boards.straightdope.com/u/Valmont314)\ **Post date:** [September 29, 2012, 8:30pm UTC](https://boards.straightdope.com/t/999-1/27517/917 "2012-09-29T20:30:20Z")
> [@Derleth](#): > > This, in fact, is incorrect. The [Bailey-Borwein-Plouffe formula](http://en.wikipedia.org/wiki/Bailey%E2%80%93Borwein%E2%80%93Plouffe_formula) allows us to do just that. Interesting, but how do you arrive at an integer result without applying some sort of algorithm (which always involves an iterative process) for converting the expression ---
**Author:** ![Chipacabra](https://avatars.discourse-cdn.com/v4/letter/c/e19b73/32.png) [@Chipacabra](https://boards.straightdope.com/u/Chipacabra)\ **Post date:** [September 29, 2012, 8:44pm UTC](https://boards.straightdope.com/t/999-1/27517/918 "2012-09-29T20:44:04Z")
> [@Derleth](#): > > This, in fact, is incorrect. The [Bailey-Borwein-Plouffe formula](http://en.wikipedia.org/wiki/Bailey%E2%80%93Borwein%E2%80%93Plouffe_formula) allows us to do just that. The BBP extracts hexadecimal digits. I think it’s been proven there’s no BBP-like formula for decimal digits of pi. Apparently, there are known spigot algorithms that will extract the nth decimal digit, but they’re all slower than the algorithms that get you all the digits up to the nth 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:** [September 29, 2012, 10:10pm UTC](https://boards.straightdope.com/t/999-1/27517/919 "2012-09-29T22:10:54Z")
> [@Valmont314](#): > > The nth digit of .999… is known without going through an iterative process. This is also wrong. You absolutely have to go through an iterative process to get to the nth digit of any repeating decimal. It’s just that they are predictable after the first set. But 1/7 requires an infinite number of digits even though it is not a single repeated number in the decimal expansion. ---
**Author:** ![Valmont314](https://avatars.discourse-cdn.com/v4/letter/v/3da27b/32.png) [@Valmont314](https://boards.straightdope.com/u/Valmont314)\ **Post date:** [September 29, 2012, 11:46pm UTC](https://boards.straightdope.com/t/999-1/27517/920 "2012-09-29T23:46:00Z")
> [@Exapno\_Mapcase](#): > > This is also wrong. You absolutely have to go through an iterative process to get to the nth digit of any repeating decimal. It’s just that they are predictable after the first set. But 1/7 requires an infinite number of digits even though it is not a single repeated number in the decimal expansion. Dude, you just don’t give up do you? Let’s just say that in this case we don’t have to do anything regarding .999… because for practical purposes we know it has the same value as 1 shall we? I would like to keep playing Tic-Tac-Toe with you but staring at a Necker cube is starting to strain my eyes…adieu. [Previous page](https://boards.straightdope.com/t/999-1/27517.md?page=45) [Next page](https://boards.straightdope.com/t/999-1/27517.md?page=47)