# A method for counting real numbers? **URL:** \ **Category:** Factual Questions\ **Created:** [June 14, 2011, 11:38am UTC](https://boards.straightdope.com/t/a-method-for-counting-real-numbers/585412 "2011-06-14T11:38:23Z")\ **Posts on this page:** 20\ **Page:** 1
**Author:** ![AaronX](https://avatars.discourse-cdn.com/v4/letter/a/7bcc69/32.png) [@AaronX](https://boards.straightdope.com/u/AaronX)\ **Post date:** [June 14, 2011, 11:38am UTC](https://boards.straightdope.com/t/a-method-for-counting-real-numbers/585412/1 "2011-06-14T11:38:23Z")
Maths says that there are more real numbers than integers, even though there are infinitely many of both types (yes, there are different types of infinity). What if I could come up with a way of matching real numbers one to one with integers, would that prove there are equally many? How about this method of matching integers with real numbers between 0 and 1: Take an integer (123) Order the digits backwards (321) Put the resulting number to the right of the decimal point (0.321) This way, the counting would go: 0.1 0.2 0.3 … 0.9 0.01 0.11 0.21 … 0.99 0.001 0.101 … It’s not in any order, but doesn’t this match every single real number (0,1) to every integer? ---
**Author:** ![Lance\_Turbo](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/lance_turbo/32/6156_2.png) [@Lance\_Turbo](https://boards.straightdope.com/u/Lance_Turbo)\ **Post date:** [June 14, 2011, 11:45am UTC](https://boards.straightdope.com/t/a-method-for-counting-real-numbers/585412/2 "2011-06-14T11:45:06Z")
Which integer maps to one third? ---
**Author:** ![AaronX](https://avatars.discourse-cdn.com/v4/letter/a/7bcc69/32.png) [@AaronX](https://boards.straightdope.com/u/AaronX)\ **Post date:** [June 14, 2011, 11:46am UTC](https://boards.straightdope.com/t/a-method-for-counting-real-numbers/585412/3 "2011-06-14T11:46:45Z")
333… infinitely recurring. Ok come to think of it, if you include 0 and infinity, it maps to [0,1], including 0 and 1. ---
**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:** [June 14, 2011, 11:53am UTC](https://boards.straightdope.com/t/a-method-for-counting-real-numbers/585412/4 "2011-06-14T11:53:14Z")
Okay, so you’ve mapped [0, 1\> to every conceivable integer. What are you going to do with the infite other intervalls between integers? ---
**Author:** ![Xema](https://avatars.discourse-cdn.com/v4/letter/x/9de053/32.png) [@Xema](https://boards.straightdope.com/u/Xema)\ **Post date:** [June 14, 2011, 11:55am UTC](https://boards.straightdope.com/t/a-method-for-counting-real-numbers/585412/5 "2011-06-14T11:55:56Z")
> [@AaronX](#): > > …doesn’t this match every single real number (0,1) to every integer? No - no matter how extensive your enumeration of reals, [Cantor’s Diagonal](http://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument) scheme shows how to construct a real number you haven’t included. ---
**Author:** ![Lance\_Turbo](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/lance_turbo/32/6156_2.png) [@Lance\_Turbo](https://boards.straightdope.com/u/Lance_Turbo)\ **Post date:** [June 14, 2011, 11:58am UTC](https://boards.straightdope.com/t/a-method-for-counting-real-numbers/585412/6 "2011-06-14T11:58:33Z")
> [@AaronX](#): > > 333.. infinitely recurring. Ok come to think of it, if you include 0 and infinity, it maps to [0,1], including 0 and 1. That’s not an integer. Which integer maps to Cos(pi/4)? ---
**Author:** ![Lord\_Mondegreen](https://avatars.discourse-cdn.com/v4/letter/l/91b2a8/32.png) [@Lord\_Mondegreen](https://boards.straightdope.com/u/Lord_Mondegreen)\ **Post date:** [June 14, 2011, 12:08pm UTC](https://boards.straightdope.com/t/a-method-for-counting-real-numbers/585412/7 "2011-06-14T12:08:54Z")
> [@Lance\_Turbo](#): > > That’s not an integer. It is possible to show a way to count the integers - they are countably infinite. The reals however are uncountably infinite. (I’m not arguing against you - I know you are essentially saying the same thing.) ---
**Author:** ![Lance\_Turbo](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/lance_turbo/32/6156_2.png) [@Lance\_Turbo](https://boards.straightdope.com/u/Lance_Turbo)\ **Post date:** [June 14, 2011, 12:13pm UTC](https://boards.straightdope.com/t/a-method-for-counting-real-numbers/585412/8 "2011-06-14T12:13:28Z")
> [@Lord\_Mondegreen](#): > > It is possible to show a way to count the integers - they are countably infinite. The reals however are uncountably infinite. I know that. I’m trying to point out the flaws in the OP’s argument. He claims to have produced a map from the naturals to the reals in (0,1), but his map is actually from the naturals to the rationals that can be written as terminating decimals on (0,1). Asking him what integers map to numbers outside this set exposes this discrepancy. ---
**Author:** ![AaronX](https://avatars.discourse-cdn.com/v4/letter/a/7bcc69/32.png) [@AaronX](https://boards.straightdope.com/u/AaronX)\ **Post date:** [June 14, 2011, 12:14pm UTC](https://boards.straightdope.com/t/a-method-for-counting-real-numbers/585412/9 "2011-06-14T12:14:43Z")
> [@Lance\_Turbo](#): > > That’s not an integer. > > Which integer maps to Cos(pi/4)? Doesn’t the integers include infinite numbers? If so, then why isn’t 333.. an integer? Ah, but Cantor’s Diagonal requires you to arrange the numbers in ascending order. This method doesn’t. If you were to try the same, the integer that maps to it is the previous one + 111… The point is, there are many schemes to map between 2 sets of numbers. Some work, and some don’t. As long as there is one scheme that works, isn’t that enough to show they’re equal? ---
**Author:** ![AaronX](https://avatars.discourse-cdn.com/v4/letter/a/7bcc69/32.png) [@AaronX](https://boards.straightdope.com/u/AaronX)\ **Post date:** [June 14, 2011, 12:17pm UTC](https://boards.straightdope.com/t/a-method-for-counting-real-numbers/585412/10 "2011-06-14T12:17:58Z")
> [@Lance\_Turbo](#): > > I know that. I’m trying to point out the flaws in the OP’s argument. He claims to have produced a map from the naturals to the reals in (0,1), but his map is actually from the naturals to the rationals that can be written as terminating decimals on (0,1). > > Asking him what integers map to numbers outside this set exposes this discrepancy. To get an integer that maps to an irrational number in this scheme, you need to know the last digit of the irrational number, which sounds impossible. But.. is 14159264… (the digits of pi after 3.) an integer? ---
**Author:** ![Lance\_Turbo](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/lance_turbo/32/6156_2.png) [@Lance\_Turbo](https://boards.straightdope.com/u/Lance_Turbo)\ **Post date:** [June 14, 2011, 12:29pm UTC](https://boards.straightdope.com/t/a-method-for-counting-real-numbers/585412/11 "2011-06-14T12:29:29Z")
No, the integers do not include infinite numbers. Neither 333… nor 11415… are integers. ---
**Author:** ![AaronX](https://avatars.discourse-cdn.com/v4/letter/a/7bcc69/32.png) [@AaronX](https://boards.straightdope.com/u/AaronX)\ **Post date:** [June 14, 2011, 12:32pm UTC](https://boards.straightdope.com/t/a-method-for-counting-real-numbers/585412/12 "2011-06-14T12:32:55Z")
> [@Lance\_Turbo](#): > > No, the integers do not include infinite numbers. Neither 333… nor 11415… are integers. Oh, if they don’t include infinitely long numbers then this scheme doesn’t work then. I guess there’s a difference between infinitely many and infinitely long. ---
**Author:** ![Xema](https://avatars.discourse-cdn.com/v4/letter/x/9de053/32.png) [@Xema](https://boards.straightdope.com/u/Xema)\ **Post date:** [June 14, 2011, 12:33pm UTC](https://boards.straightdope.com/t/a-method-for-counting-real-numbers/585412/13 "2011-06-14T12:33:55Z")
> [@AaronX](#): > > Cantor’s Diagonal requires you to arrange the numbers in ascending order. How does it require that? It shows how to construct a real number that differs (in at least one digit) from every one in any enumeration. Order is unimportant. > [@](#): > > As long as there is one scheme that works, isn’t that enough to show they’re equal? Yes. Unfortunately, this one doesn’t work. And it seems to be well accepted that Cantor has shown no scheme can work. ---
**Author:** ![Lord\_Mondegreen](https://avatars.discourse-cdn.com/v4/letter/l/91b2a8/32.png) [@Lord\_Mondegreen](https://boards.straightdope.com/u/Lord_Mondegreen)\ **Post date:** [June 14, 2011, 12:40pm UTC](https://boards.straightdope.com/t/a-method-for-counting-real-numbers/585412/14 "2011-06-14T12:40:00Z")
> [@Lance\_Turbo](#): > > I know that. I’m trying to point out the flaws in the OP’s argument. He claims to have produced a map from the naturals to the reals in (0,1), but his map is actually from the naturals to the rationals that can be written as terminating decimals on (0,1). > > Asking him what integers map to numbers outside this set exposes this discrepancy. Sorry, I misunderstood. My apologies. ---
**Author:** ![AaronX](https://avatars.discourse-cdn.com/v4/letter/a/7bcc69/32.png) [@AaronX](https://boards.straightdope.com/u/AaronX)\ **Post date:** [June 14, 2011, 12:49pm UTC](https://boards.straightdope.com/t/a-method-for-counting-real-numbers/585412/15 "2011-06-14T12:49:16Z")
> [@Xema](#): > > How does it require that? > > It shows how to construct a real number that differs (in at least one digit) from every one in any enumeration. Order is unimportant. > > Yes. Unfortunately, this one doesn’t work. > > And it seems to be well accepted that Cantor has shown no scheme can work. I thought it had to be in ascending order to show it wasn’t in the list. On further thought, it doesn’t. Well, that’s the thing about theories, you can’t prove they work, you can only prove they don’t don’t work 😃 ---
**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:** [June 14, 2011, 1:04pm UTC](https://boards.straightdope.com/t/a-method-for-counting-real-numbers/585412/16 "2011-06-14T13:04:41Z")
> [@AaronX](#): > > Well, that’s the thing about theories, you can’t prove they work, you can only prove they don’t don’t work 😃 Unless you’re doing math, then you **can** prove they work. ---
**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:** [June 14, 2011, 3:10pm UTC](https://boards.straightdope.com/t/a-method-for-counting-real-numbers/585412/17 "2011-06-14T15:10:01Z")
> [@naita](#): > > Unless you’re doing math, then you **can** prove they work. Unless they are undecidable in your axiom system. 🙂 ---
**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:** [June 14, 2011, 4:23pm UTC](https://boards.straightdope.com/t/a-method-for-counting-real-numbers/585412/18 "2011-06-14T16:23:31Z")
It’s been well known for a long time that there are no from the integers bijections to the reals or any interval of the reals. The most famous proof is [Cantor’s diagonalization argument](http://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument). As others have pointed out what the OP maps to is numbers with terminatining decimals in (0,1) i.e. a subset of the rational numbers in (0,1). ---
**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:** [June 14, 2011, 4:45pm UTC](https://boards.straightdope.com/t/a-method-for-counting-real-numbers/585412/19 "2011-06-14T16:45:23Z")
> [@AaronX](#): > > How about this method of matching integers with real numbers between 0 and 1: > Take an integer (123) > Order the digits backwards (321) > Put the resulting number to the right of the decimal point (0.321) > This way, the counting would go: > 0.1 > 0.2 > 0.3 > … > 0.9 > 0.01 > 0.11 > 0.21 > … > 0.99 > 0.001 > 0.101 > … > It’s not in any order, but doesn’t this match every single real number (0,1) to every integer? Just to echo what others have said… Here’s the problem: Your list only consists of terminating decimals. That means it leaves out all of the irrational, and some of the rational, real numbers between 0 and 1. For example, as **Lance Turbo** points out, 1/3 (= 0.33333…) never appears on your list. Neither does pi/4. Since it has been proved (by Cantor) that there can be no way of listing (or denumerating, or matching integers to) the set of all real numbers between 0 and 1, any such scheme is doomed to failure. You’re like a person who claims to have a way to trisect the angle with only straightedge and compass. I mean none of this as a dis on you. Thinking about such things, and asking “What’s wrong with this argument,” is a perfectly good thing to be doing. ---
**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:** [June 14, 2011, 4:49pm UTC](https://boards.straightdope.com/t/a-method-for-counting-real-numbers/585412/20 "2011-06-14T16:49:38Z")
By the way, for anyone reading this thread who’s wondering what we’re talking about (different sizes of infinity? Cantor’s diagonalization argument?), one really good online explanation can be found here (at the Platonic Realms website): [Infinity: You Can’t Get There From Here](http://www.mathacademy.com/pr/minitext/infinity/) [Next page](https://boards.straightdope.com/t/a-method-for-counting-real-numbers/585412.md?page=2)