# Decimal Representation of 1/7 **URL:** \ **Category:** Factual Questions\ **Created:** [November 8, 2013, 2:49am UTC](https://boards.straightdope.com/t/decimal-representation-of-1-7/673379 "2013-11-08T02:49:49Z")\ **Posts on this page:** 6\ **Page:** 3
**Author:** ![KarlGauss](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/karlgauss/32/3713_2.png) [@KarlGauss](https://boards.straightdope.com/u/KarlGauss)\ **Post date:** [November 10, 2013, 2:53am UTC](https://boards.straightdope.com/t/decimal-representation-of-1-7/673379/41 "2013-11-10T02:53:39Z")
Unless I am misunderstanding, [so is 1/17](https://en.wikipedia.org/wiki/Heptadecagon). ETA: Oh, the irony! ---
**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:** [November 10, 2013, 3:05am UTC](https://boards.straightdope.com/t/decimal-representation-of-1-7/673379/42 "2013-11-10T03:05:58Z")
Definitively Answered…Flawless Victory. ---
**Author:** ![j\_sum1](https://avatars.discourse-cdn.com/v4/letter/j/8baadc/32.png) [@j\_sum1](https://boards.straightdope.com/u/j_sum1)\ **Post date:** [November 10, 2013, 3:56am UTC](https://boards.straightdope.com/t/decimal-representation-of-1-7/673379/43 "2013-11-10T03:56:31Z")
> [@ultrafilter](#): > > 1/7 is constructible. Maybe I misunderstood something here. 1/7 is indeed constructible. Any rational division of a straight line is. A 7-gon certainly is not constructible. And that is what I thought was being asked. ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [November 10, 2013, 8:19am UTC](https://boards.straightdope.com/t/decimal-representation-of-1-7/673379/44 "2013-11-10T08:19:57Z")
> [@Enola\_Straight](#): > > A better question is: > > Are cyclical numbers always non-constructable-by-compass-and-straightedge polygons? > [Constructible polygon - Wikipedia](http://en.wikipedia.org/wiki/Constructible_polygon) No. The 17-gon is constructible, but 1/17 is cyclical. Same for 257 and 65537. In fact, whenever the n-gon is constructible for a prime n \> 5, we have that 1/n is cyclical. ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [November 10, 2013, 8:31am UTC](https://boards.straightdope.com/t/decimal-representation-of-1-7/673379/45 "2013-11-10T08:31:40Z")
(By “the n-gon”, I mean “the regular n-gon” throughout, of course). Also, a proof sketch (though perhaps there’s something much simpler?): Using the existence of primitive roots modulo any prime and the observation that every odd multiple of a generator of a cyclic group of even order also generates the group, we find that x is a primitive root modulo a Fermat prime just in case x is not a quadratic residue modulo that Fermat prime. Sufficiently large Fermat primes must be of the form 16^n + 1, and thus have last decimal digit 7, and thus, by quadratic reciprocity, 10 will not be a quadratic residue of those primes. Thus, Fermat primes have cyclical reciprocals. Finally, invoke Gauss for the connection between Fermat primes and constructible regular polygons. [This depends on the fact that 10 = ten, mind you. This is not base-independent] ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [November 10, 2013, 5:11pm UTC](https://boards.straightdope.com/t/decimal-representation-of-1-7/673379/46 "2013-11-10T17:11:04Z")
> [@Indistinguishable](#): > > of a cyclic group of even order Sorry, I meant “of a finite cyclic group of order a power of two”. [Previous page](https://boards.straightdope.com/t/decimal-representation-of-1-7/673379.md?page=2)