# Enola's Rule of Small Numbers **URL:** \ **Category:** Factual Questions\ **Created:** [June 7, 2018, 2:21am UTC](https://boards.straightdope.com/t/enolas-rule-of-small-numbers/815585 "2018-06-07T02:21:37Z")\ **Posts on this page:** 20\ **Page:** 1
**Author:** ![Enola\_Straight](https://avatars.discourse-cdn.com/v4/letter/e/dec6dc/32.png) [@Enola\_Straight](https://boards.straightdope.com/u/Enola_Straight)\ **Post date:** [June 7, 2018, 2:21am UTC](https://boards.straightdope.com/t/enolas-rule-of-small-numbers/815585/1 "2018-06-07T02:21:37Z")
Hydrogen…an atom of only one proton and one electron (_maybe_ one or rarely two neutrons) can arguably be placed on more than one position on the Periodic Table: > **[Why is hydrogen in the periodic table not placed in the halogen family, why...](https://www.quora.com/Why-is-hydrogen-in-the-periodic-table-not-placed-in-the-halogen-family-why-is-it-in-the-alkali-metal-family)** > > Answer: It belongs to no groups > > Hydrogen is the confused element. Its properties are a strange combination of metals and nonmetals. For example, it has a simple molecular structure like nitrogen but forms stable +1 ions like potassium. Thus some... Except for two and three, all prime numbers are of the form 6x + or -1 [https://solvemymaths.files.wordpress.com/2015/01/pat.png](https://solvemymaths.files.wordpress.com/2015/01/pat.png) Before you cite one of these: > **[Search results](https://en.wikipedia.org/w/index.php?fulltext=1&profile=default&search=Law%2Bof%2Bsmall%2Bnumbers&searchToken=9ot1n6sl5mckqk2bdopeasjv3&title=Special:Search&ns0=1)** > > There is a page named "Law of small numbers" on Wikipedia no…I’m talking about something different. If a number, object, physical process, etc. is small enough…that is, it’s lack of internal structure makes it so basic/trivial/degenerate, it will lack the distinguishing characteristics of larger numbers/complex processes. Do my fellow Dopers know if this “rule” is in fact formulated in academia? Do you know other examples? ---
**Author:** ![DavidwithanR](https://avatars.discourse-cdn.com/v4/letter/d/3bc359/32.png) [@DavidwithanR](https://boards.straightdope.com/u/DavidwithanR)\ **Post date:** [June 7, 2018, 3:42am UTC](https://boards.straightdope.com/t/enolas-rule-of-small-numbers/815585/2 "2018-06-07T03:42:33Z")
I do know that small numbers are treated unfairly, for example, when people say “There’s strength in numbers” and you think 0.186 could do the trick and then they say “No that’s not what I meant”, or when you’re told that a number of unsuccessful attempts have already been made to do what you’re trying to do, and you think probably the number is 2, and again with the “No, that’s not what I meant” trick. ---
**Author:** ![watchwolf49](https://avatars.discourse-cdn.com/v4/letter/w/e9c0ed/32.png) [@watchwolf49](https://boards.straightdope.com/u/watchwolf49)\ **Post date:** [June 7, 2018, 4:12am UTC](https://boards.straightdope.com/t/enolas-rule-of-small-numbers/815585/3 "2018-06-07T04:12:07Z")
AFAIK with real numbers the answer is no … there’s no real number too small to add or multiply, every one behaves exactly like every other one … if fact, there’s no real number that’s too small to have a real number even smaller, which is something of the basis for calculus … ---
**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 7, 2018, 10:53am UTC](https://boards.straightdope.com/t/enolas-rule-of-small-numbers/815585/4 "2018-06-07T10:53:26Z")
You can phrase that as real numbers having no scale to them. Though if you’re considering both addition and multiplication, they sort of do. ---
**Author:** ![Saint\_Cad](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/saint_cad/32/18907_2.png) [@Saint\_Cad](https://boards.straightdope.com/u/Saint_Cad)\ **Post date:** [June 7, 2018, 11:27am UTC](https://boards.straightdope.com/t/enolas-rule-of-small-numbers/815585/5 "2018-06-07T11:27:52Z")
> [@Chronos](#): > > You can phrase that as real numbers having no scale to them. Though if you’re considering both addition and multiplication, they sort of do. Isn’t that the idea behind infintesimals? Non-zero numbers too small to be a real number? ---
**Author:** ![DPRK](https://avatars.discourse-cdn.com/v4/letter/d/4491bb/32.png) [@DPRK](https://boards.straightdope.com/u/DPRK)\ **Post date:** [June 7, 2018, 11:30am UTC](https://boards.straightdope.com/t/enolas-rule-of-small-numbers/815585/6 "2018-06-07T11:30:57Z")
One rule that comes to mind is that a “system”, for instance a thermodynamic system, a random variable, or some such, will roughly speaking have an entropy that increases with the number of states available, or, even more basically, small systems simply have fewer possible states or configurations. There are not too many graphs with 3 vertices, etc, so there is more room for combinatorial coincidences at low numbers. ---
**Author:** ![DPRK](https://avatars.discourse-cdn.com/v4/letter/d/4491bb/32.png) [@DPRK](https://boards.straightdope.com/u/DPRK)\ **Post date:** [June 7, 2018, 11:37am UTC](https://boards.straightdope.com/t/enolas-rule-of-small-numbers/815585/7 "2018-06-07T11:37:31Z")
> [@Saint\_Cad](#): > > Isn’t that the idea behind infintesimals? Non-zero numbers too small to be a real number? No need to invoke infinitesimal numbers to consider real numbers with respect to scaling. If you take real numbers with respect to, say, addition, and a is a scaling factor, you have a(x + y) = ax + ay. One thing you can say is that if you have a locally compact topological group, then up to a multiplicative constant there is a unique invariant measure on the group. You can still scale it, though. ---
**Author:** ![Isilder](https://avatars.discourse-cdn.com/v4/letter/i/cc9497/32.png) [@Isilder](https://boards.straightdope.com/u/Isilder)\ **Post date:** [June 7, 2018, 11:59am UTC](https://boards.straightdope.com/t/enolas-rule-of-small-numbers/815585/8 "2018-06-07T11:59:23Z")
> [@Enola\_Straight](#): > > Do my fellow Dopers know if this “rule” is in fact formulated in academia? Its a boundary condition, an edge condition. What is element 1, is 1 odd ? is 1 prime ? Its common that the edge or boundary or start is not indicative of the behaviour of the system. Although its not clear that Hydrogen has to be at the edge for every possible periodic table… it could be in the middle ? Also, its not the only metal that is also participating in covalent bonds. There’s actually degrees inbetween ionic and covalent, and somewhere down the middle, the oxides of the metals are nearly covalent. But your misunderstanding of hydrogen is not important. Your first terms may also be axioms… since thats the starting point. If so , its connected with Godel’s incompletement theorems, which imply that you cannot prove the axiom is true… you’ve already assumed it to be true, you can only show that the set of axioms are consistent. Read through this … > **[Gödel's incompleteness theorems | First incompleteness theorem](https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems#First_incompleteness_theorem)** > > Gödel's first incompleteness theorem first appeared as "Theorem VI" in Gödel's 1931 paper "On Formally Undecidable Propositions of Principia Mathematica and Related Systems I". The hypotheses of the theorem were improved shortly thereafter by J. Barkley Rosser (1936) using Rosser's trick. The resulting theorem (incorporating Rosser's improvement) may be paraphrased in English as follows, where "formal system" includes the assumption that the system is effectively generated. ---
**Author:** ![Leo\_Bloom](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/leo_bloom/32/10377_2.png) [@Leo\_Bloom](https://boards.straightdope.com/u/Leo_Bloom)\ **Post date:** [June 7, 2018, 3:03pm UTC](https://boards.straightdope.com/t/enolas-rule-of-small-numbers/815585/9 "2018-06-07T15:03:11Z")
> [@DPRK](#): > > … > > One thing you can say is that if you have a locally compact topological group, then up to a multiplicative constant there is a unique invariant measure on the group. You can still scale it, though. So there we have it: Enola’s Rule of Small Numbers. Congratulations! ---
**Author:** ![DPRK](https://avatars.discourse-cdn.com/v4/letter/d/4491bb/32.png) [@DPRK](https://boards.straightdope.com/u/DPRK)\ **Post date:** [June 7, 2018, 5:22pm UTC](https://boards.straightdope.com/t/enolas-rule-of-small-numbers/815585/10 "2018-06-07T17:22:16Z")
> [@Leo\_Bloom](#): > > So there we have it: Enola’s Rule of Small Numbers. > > Congratulations! Er, first of all, that is [Haar’s theorem](https://en.wikipedia.org/wiki/Haar_measure), proved by Alfréd Haar. More importantly, that has to do with **Chronos** ’s remark about the real numbers having no scale, and dimensional analysis as in physics, not to the original question about small numbers (to which I think the simple answer is that when you have more to work with you have more complexity, and vice versa). ---
**Author:** ![Leo\_Bloom](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/leo_bloom/32/10377_2.png) [@Leo\_Bloom](https://boards.straightdope.com/u/Leo_Bloom)\ **Post date:** [June 7, 2018, 6:14pm UTC](https://boards.straightdope.com/t/enolas-rule-of-small-numbers/815585/11 "2018-06-07T18:14:12Z")
**Enola** , I rescind my congratulations, and in fact now consider you an outright plagiarist. And mistaken in the application of your plagiarized theorem. ---
**Author:** ![Colibri](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/colibri/32/1841_2.png) [@Colibri](https://boards.straightdope.com/u/Colibri)\ **Post date:** [June 7, 2018, 8:14pm UTC](https://boards.straightdope.com/t/enolas-rule-of-small-numbers/815585/12 "2018-06-07T20:14:54Z")
> [@Leo\_Bloom](#): > > **Enola** , I rescind my congratulations, and in fact now consider you an outright plagiarist. And mistaken in the application of your plagiarized theorem. **Moderator Note** **Leo** , I have no idea what you think you are on about, but personal attacks on other posters are not permitted in General Questions. No warning issued. However, I think it would be better that you not post further in this thread in order to avoid further misunderstandings. Colibri General Questions Moderator ---
**Author:** ![DPRK](https://avatars.discourse-cdn.com/v4/letter/d/4491bb/32.png) [@DPRK](https://boards.straightdope.com/u/DPRK)\ **Post date:** [June 7, 2018, 8:43pm UTC](https://boards.straightdope.com/t/enolas-rule-of-small-numbers/815585/13 "2018-06-07T20:43:42Z")
I found the following reference: [http://www.maa.org/sites/default/files/pdf/upload\_library/22/Ford/Guy697-712.pdf](http://www.maa.org/sites/default/files/pdf/upload_library/22/Ford/Guy697-712.pdf) “The Strong Law of Small Numbers”, by Richard K. Guy. (1988) > [@](#): > > There aren’t enough small numbers to meet the many demands made of them. > > Superficial similarities spawn spurious statements. > > Capricious coincidences cause careless conjectures. > > Early exceptions eclipse eventual essentials. > > Initial irregularities inhibit incisive intuition. > > When two numbers look equal, it ain’t necessarily so. and so on. He gives several examples, for instance, the numbers 31, 331, 3331, 33331, 333331, 3333331, 33333331 are all prime. The name of this “law”, the “Strong law of small numbers”, is obviously a play on the (unrelated) law of large numbers in probability theory, ---
**Author:** ![smithsb](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/smithsb/32/12107_2.png) [@smithsb](https://boards.straightdope.com/u/smithsb)\ **Post date:** [June 7, 2018, 9:26pm UTC](https://boards.straightdope.com/t/enolas-rule-of-small-numbers/815585/14 "2018-06-07T21:26:01Z")
I thought Leo was being humorous. I look forward to his reply. ---
**Author:** ![DavidwithanR](https://avatars.discourse-cdn.com/v4/letter/d/3bc359/32.png) [@DavidwithanR](https://boards.straightdope.com/u/DavidwithanR)\ **Post date:** [June 7, 2018, 9:44pm UTC](https://boards.straightdope.com/t/enolas-rule-of-small-numbers/815585/15 "2018-06-07T21:44:10Z")
> [@smithsb](#): > > I thought Leo was being humorous. I look forward to his reply. I had the same thought - it seemed to me that the whole “law” idea was at least to some extent tongue-in-cheek, and that he joined in enthusiastically with that theme of poking fun at over-serious laws. ---
**Author:** ![Hari\_Seldon](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/hari_seldon/32/5173_2.png) [@Hari\_Seldon](https://boards.straightdope.com/u/Hari_Seldon)\ **Post date:** [June 7, 2018, 10:21pm UTC](https://boards.straightdope.com/t/enolas-rule-of-small-numbers/815585/16 "2018-06-07T22:21:20Z")
There are all sorts of examples where things are different for one or a few small numbers. For example, all primes are odd–with one minor exception. More esoteric: the only symmetric group with an outer automorphism is the group of permutations of 6 things. ---
**Author:** ![Colibri](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/colibri/32/1841_2.png) [@Colibri](https://boards.straightdope.com/u/Colibri)\ **Post date:** [June 7, 2018, 10:39pm UTC](https://boards.straightdope.com/t/enolas-rule-of-small-numbers/815585/17 "2018-06-07T22:39:46Z")
> [@smithsb](#): > > I thought Leo was being humorous. I look forward to his reply. > [@DavidwithanR](#): > > I had the same thought - it seemed to me that the whole “law” idea was at least to some extent tongue-in-cheek, and that he joined in enthusiastically with that theme of poking fun at over-serious laws. If that’s the case, I’ll apologize (with the observation that if a remark might be taken as an insult, it is always wise to add a smilie to indicate the intent). ---
**Author:** ![DPRK](https://avatars.discourse-cdn.com/v4/letter/d/4491bb/32.png) [@DPRK](https://boards.straightdope.com/u/DPRK)\ **Post date:** [June 7, 2018, 10:53pm UTC](https://boards.straightdope.com/t/enolas-rule-of-small-numbers/815585/18 "2018-06-07T22:53:37Z")
On “laws” we have [Stigler’s law of eponymy](https://en.wikipedia.org/wiki/Stigler%27s_law_of_eponymy): that no scientific discovery is named after its original discoverer. ---
**Author:** ![RTFirefly](https://avatars.discourse-cdn.com/v4/letter/r/c77e96/32.png) [@RTFirefly](https://boards.straightdope.com/u/RTFirefly)\ **Post date:** [June 8, 2018, 2:14pm UTC](https://boards.straightdope.com/t/enolas-rule-of-small-numbers/815585/19 "2018-06-08T14:14:53Z")
> [@Hari\_Seldon](#): > > There are all sorts of examples where things are different for one or a few small numbers. For example, all primes are odd–with one minor exception. The injustice of it all! An infinite number of odd primes, yet only one lousy even prime! (Tokenism, no doubt. ;)) Here’s mathematician Richard K. Guy’s MAA humorous paper on “[The Strong Law of Small Numbers](https://www.maa.org/sites/default/files/pdf/upload_library/22/Ford/Guy697-712.pdf)”. The “law” in question states: “There aren’t enough small numbers to meet the many demands made of them.” Which is going to result in your seeing patterns among small integers that aren’t really there, because there are so many different things happening there. (E.g. even a sequence that gets really big really fast - the factorials - has five terms in the sequence that are under 25.) ---
**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 8, 2018, 3:39pm UTC](https://boards.straightdope.com/t/enolas-rule-of-small-numbers/815585/20 "2018-06-08T15:39:28Z")
> [@RTFirefly](#): > > The injustice of it all! An infinite number of odd primes, yet only one lousy even prime! Yeah, there’s only one lousy prime that’s divisible by 2. Not only that, but there’s only one prime divisible by 3, only one prime divisible by 5, only one prime divisible by 7… (And no primes at all divisible by 4, or 6.) [Next page](https://boards.straightdope.com/t/enolas-rule-of-small-numbers/815585.md?page=2)