# Why do people say irrational numbers must contain any sequence of numbers? **URL:** \ **Category:** Factual Questions\ **Created:** [December 22, 2011, 10:18am UTC](https://boards.straightdope.com/t/why-do-people-say-irrational-numbers-must-contain-any-sequence-of-numbers/607064 "2011-12-22T10:18:30Z")\ **Posts on this page:** 20\ **Page:** 2
**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 20, 2013, 5:08pm UTC](https://boards.straightdope.com/t/why-do-people-say-irrational-numbers-must-contain-any-sequence-of-numbers/607064/21 "2013-06-20T17:08:40Z")
> [@Senegoid](#): > > Given an artificially constructed abnormal number – in particular, the one you gave here – How do you demonstrate that it’s algebraic? Actually tbh, I think I’ve erred: the binary expression “0.1010010001…” is algebraic, but the deciml expression “0.1010010001…” is almost certainly transcendental. Proving such a number is algebraic/irrational revolves around [transcendence theory](http://en.wikipedia.org/wiki/Transcendence_theory) ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [June 20, 2013, 5:26pm UTC](https://boards.straightdope.com/t/why-do-people-say-irrational-numbers-must-contain-any-sequence-of-numbers/607064/22 "2013-06-20T17:26:25Z")
> [@ultrafilter](#): > > That idea can be used to show that the set of numbers in [0, 1] which are not normal in a particular base has measure zero (although you’ll need to do some work to make it rigorous), and you should be able to extend it to the entire line with a little more work, but I think you need a fundamentally different approach to prove anything about numbers that are normal in every base. A countable union of measure zero sets has measure zero. So once you’ve proven that, for each b, almost all values in [0, 1] are normal in base b, you’ve automatically established that almost all values in [0, 1] are absolutely normal. ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [June 20, 2013, 5:55pm UTC](https://boards.straightdope.com/t/why-do-people-say-irrational-numbers-must-contain-any-sequence-of-numbers/607064/23 "2013-06-20T17:55:04Z")
> [@Phnord\_Prephect](#): > > Probably unrelated, but since this is a minor zombie thread anyway, and sorta-kinda on the topic, I’ve got a question. I’m not a math wiz by any means, so this may have already been answered and I just didn’t understand it. > > So, pi, 3.141… etc, continues infinitely without repeating… When people say that pi’s decimal expansion continues infinitely without repeating, they don’t mean that “There’s never a point at which some initial part of the sequence appears again.” After all, there are plenty of 3s in the decimal expansion of pi. Instead, what they mean is that “There’s never a point after which the sequence just cycles through the same values over and over forever”. As opposed to, say, 46.2815412412412412412412412…[repeating 412 forever]… And this sort of thing, indeed, provably doesn’t happen for pi. (The easy part of the proof is showing that this sort of thing happens just for numbers which are ratios of integers. The hard part of the proof is showing that pi cannot be a ratio of integers.) ---
**Author:** ![robert\_columbia](https://avatars.discourse-cdn.com/v4/letter/r/e79b87/32.png) [@robert\_columbia](https://boards.straightdope.com/u/robert_columbia)\ **Post date:** [June 20, 2013, 7:59pm UTC](https://boards.straightdope.com/t/why-do-people-say-irrational-numbers-must-contain-any-sequence-of-numbers/607064/24 "2013-06-20T19:59:54Z")
> [@ultrafilter](#): > > The most interesting thing about normality of numbers is that almost every real number is normal, and yet it’s very difficult to show that any particular one is. So in some sense, all the numbers that we’re familiar with are very unusual. The number 4 must be very odd indeed. But the only number that is both even and odd is infinity, therefore 4=infinity. ---
**Author:** ![pulykamell](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/pulykamell/32/3166_2.png) [@pulykamell](https://boards.straightdope.com/u/pulykamell)\ **Post date:** [June 20, 2013, 9:07pm UTC](https://boards.straightdope.com/t/why-do-people-say-irrational-numbers-must-contain-any-sequence-of-numbers/607064/25 "2013-06-20T21:07:35Z")
Infinity is not a number. ---
**Author:** ![Peter\_Morris](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/peter_morris/32/359_2.png) [@Peter\_Morris](https://boards.straightdope.com/u/Peter_Morris)\ **Post date:** [June 20, 2013, 9:38pm UTC](https://boards.straightdope.com/t/why-do-people-say-irrational-numbers-must-contain-any-sequence-of-numbers/607064/26 "2013-06-20T21:38:59Z")
> [@ultrafilter](#): > > So in some sense, all the numbers that we’re familiar with are very unusual. [All numbers are interesting,](http://en.wikipedia.org/wiki/Interesting_number_paradox) especially 1729. ---
**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 20, 2013, 9:49pm UTC](https://boards.straightdope.com/t/why-do-people-say-irrational-numbers-must-contain-any-sequence-of-numbers/607064/27 "2013-06-20T21:49:59Z")
> [@pulykamell](#): > > Infinity is not a number. Except for large values of ‘number’ ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [June 20, 2013, 10:38pm UTC](https://boards.straightdope.com/t/why-do-people-say-irrational-numbers-must-contain-any-sequence-of-numbers/607064/28 "2013-06-20T22:38:07Z")
My bugaboo: I generally don’t find it useful to say (or think!) things like “X is not a number”. Often, it’s just a way to shout down what would otherwise be a fruitful mathematical investigation, out of reflexive inertia. There’s not just one number system, and not just one sense of the term “number”. > [@pulykamell](#): > > Infinity is not a number. It depends on what you take “number” to encompass. It seems perfectly ordinary to me to take it to encompass various senses of the term “infinity”, given that people already so standardly take it to encompass fractions, negations, imaginaries, and the like. Why draw the line _here_ (and then chastise others for not happening to draw it in the same place)? > [@Ludwig Wittgenstein](#): > > And for instance the kinds of number form a family in the same way. Why do we call something a ‘number’? Well, perhaps because it has a—direct—relationship with several things that have hitherto been called number; and this can be said to give it an indirect relationship to other things we call the same name. And we extend our concept of number as in spinning a thread we twist fibre on fibre. And the strength of the thread does not reside in the fact that some one fibre runs through its whole length, but in the overlapping of many fibres. ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [June 20, 2013, 10:44pm UTC](https://boards.straightdope.com/t/why-do-people-say-irrational-numbers-must-contain-any-sequence-of-numbers/607064/29 "2013-06-20T22:44:59Z")
(I will concede that it’s not often useful to take 4 to equal infinity…) ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [June 20, 2013, 11:36pm UTC](https://boards.straightdope.com/t/why-do-people-say-irrational-numbers-must-contain-any-sequence-of-numbers/607064/30 "2013-06-20T23:36:26Z")
That’s cause you’re not a statistician. ---
**Author:** ![Senegoid](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/senegoid/32/6606_2.png) [@Senegoid](https://boards.straightdope.com/u/Senegoid)\ **Post date:** [June 21, 2013, 12:03am UTC](https://boards.straightdope.com/t/why-do-people-say-irrational-numbers-must-contain-any-sequence-of-numbers/607064/31 "2013-06-21T00:03:00Z")
> [@Indistinguishable](#): > > (I will concede that it’s not often useful to take 4 to equal infinity…) Especially since we’ve already just decided that [∞ = -1/12](http://boards.straightdope.com/sdmb/showthread.php?t=420045) 🙂 > [@Indistinguishable](#): > > My bugaboo: I generally don’t find it useful to say (or think!) things like “X is not a number”. Often, it’s just a way to shout down what would otherwise be a fruitful mathematical investigation, out of reflexive inertia. There’s not just one number system, and not just one sense of the term “number”. This. Sort of. When I was a wee infinitesimal, just learning algebra in Junior High, it was standard pedagogy to beat into our heads that infinity is NOT a number!!! This usually came up when some student asserted something like “Infinity is the largest number.” Problem with that, as I saw it: Nobody teacher ever really discussed was infinity IS, if it’s not a number. (Unless, maybe, some students connected enough dots to understand infinity as a size that a set can be.) I got the sense that a lot of students got the sense that infinity is an invalid concept that we aren’t ever supposed to think about at all, because it’s just total baloney or something. ---
**Author:** ![Omphaloskeptic](https://avatars.discourse-cdn.com/v4/letter/o/bcef8e/32.png) [@Omphaloskeptic](https://boards.straightdope.com/u/Omphaloskeptic)\ **Post date:** [June 21, 2013, 12:50am UTC](https://boards.straightdope.com/t/why-do-people-say-irrational-numbers-must-contain-any-sequence-of-numbers/607064/32 "2013-06-21T00:50:11Z")
> [@Asympotically\_fat](#): > > Actually tbh, I think I’ve erred: the binary expression “0.1010010001…” is algebraic[…]. I’m interpreting this expression as the binary number with 1s at the triangular-number positions (positions 1, 3, 6, 10, 15, …) — do you have a cite or explanation for the statement that this is algebraic? My understanding (which is not very deep) is that there are no known non-normal irrational algebraic numbers (e.g., [Wikipedia](http://en.wikipedia.org/wiki/Normal_number) gives this as a conjecture). This expression also can be written as a Jacobi theta function, and I think these are supposed to be transcendental at algebraic arguments. ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [June 21, 2013, 10:16am UTC](https://boards.straightdope.com/t/why-do-people-say-irrational-numbers-must-contain-any-sequence-of-numbers/607064/33 "2013-06-21T10:16:59Z")
> [@ultrafilter](#): > > That’s cause you’re not a statistician. I thought statisticians took infinity to be 30… ---
**Author:** ![robert\_columbia](https://avatars.discourse-cdn.com/v4/letter/r/e79b87/32.png) [@robert\_columbia](https://boards.straightdope.com/u/robert_columbia)\ **Post date:** [June 21, 2013, 2:38pm UTC](https://boards.straightdope.com/t/why-do-people-say-irrational-numbers-must-contain-any-sequence-of-numbers/607064/34 "2013-06-21T14:38:16Z")
> [@pulykamell](#): > > Infinity is not a number. This statement is irrational. ---
**Author:** ![BigT](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/bigt/32/12044_2.png) [@BigT](https://boards.straightdope.com/u/BigT)\ **Post date:** [June 21, 2013, 11:07pm UTC](https://boards.straightdope.com/t/why-do-people-say-irrational-numbers-must-contain-any-sequence-of-numbers/607064/35 "2013-06-21T23:07:10Z")
> [@pulykamell](#): > > Infinity is not a number. What definition of number are you using here? I know someone else has already said that it depends on the definition. I’m just wondering what definition you use. I’m wanting a definition that is consistent with everything else we consider a number, but does not include infinity (or, I assume, negative infinity). ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [June 23, 2013, 4:20am UTC](https://boards.straightdope.com/t/why-do-people-say-irrational-numbers-must-contain-any-sequence-of-numbers/607064/36 "2013-06-23T04:20:05Z")
The real numbers. ---
**Author:** ![Learjeff](https://avatars.discourse-cdn.com/v4/letter/l/94ad74/32.png) [@Learjeff](https://boards.straightdope.com/u/Learjeff)\ **Post date:** [June 23, 2013, 1:33pm UTC](https://boards.straightdope.com/t/why-do-people-say-irrational-numbers-must-contain-any-sequence-of-numbers/607064/37 "2013-06-23T13:33:31Z")
Ah, so it’s not a real number. It must be a fake number, then. 😉 ---
**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:** [June 23, 2013, 5:47pm UTC](https://boards.straightdope.com/t/why-do-people-say-irrational-numbers-must-contain-any-sequence-of-numbers/607064/38 "2013-06-23T17:47:22Z")
> [@](#): > > Quoth **Indistinguishable** : > > (I will concede that it’s not often useful to take 4 to equal infinity…) There is actually at least one calculation in physics where 3 is taken as being approximately equal to infinity. I can’t remember the details, but there’s a result that holds only in the limit of a large number of particle generations, and it’s used as an approximation to our world of three particle generations. Mind you, physicists don’t particularly _like_ using this approximation, but it’s the only way anyone has yet found of even approximating that problem, and so it’s (reluctantly) accepted. ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [June 23, 2013, 10:20pm UTC](https://boards.straightdope.com/t/why-do-people-say-irrational-numbers-must-contain-any-sequence-of-numbers/607064/39 "2013-06-23T22:20:23Z")
After that post, I tried to contrive some examples in which it is useful to identify 4 with infinity, in some sense, for some purposes. Only to realize, hell, there are many non-contrived contexts in which even 1 and infinity ought be identified [e.g., when one doesn’t care exactly how many elements a set has, but just whether it is inhabited or not, one may carry out the ordinary cardinality calculation in an arithmetic where 1 = 2 = 3 = … = infinity]. Perhaps all I should have conceded was that it’s not often useful to identify “odd (as in peculiar)” with “odd (as in 1, 3, 5, and perhaps others)”… ---
**Author:** ![Cheshire\_Human](https://avatars.discourse-cdn.com/v4/letter/c/5f8ce5/32.png) [@Cheshire\_Human](https://boards.straightdope.com/u/Cheshire_Human)\ **Post date:** [July 7, 2013, 12:07pm UTC](https://boards.straightdope.com/t/why-do-people-say-irrational-numbers-must-contain-any-sequence-of-numbers/607064/40 "2013-07-07T12:07:44Z")
> [@Chronos](#): > > … you can’t say “I found the sequence 314159, and that counts, because six digits is good enough”, because if six digits were good enough, you’d be far from reaching the place where you found that. 314159 shows up 21 times in the first 33 million digits. My phone number shows up, too. [Previous page](https://boards.straightdope.com/t/why-do-people-say-irrational-numbers-must-contain-any-sequence-of-numbers/607064.md?page=1) [Next page](https://boards.straightdope.com/t/why-do-people-say-irrational-numbers-must-contain-any-sequence-of-numbers/607064.md?page=3)