Timeline for "Gaps" or "holes" in rational number system
Current License: CC BY-SA 4.0
Post Revisions
23 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Nov 10, 2022 at 14:25 | history | unprotected | Eric Wofsey | ||
| Nov 10, 2022 at 13:52 | history | protected | Ѕᴀᴀᴅ | ||
| Jul 6, 2020 at 8:37 | answer | added | tomasz | timeline score: 4 | |
| May 12, 2020 at 0:13 | audit | First posts | |||
| May 12, 2020 at 0:13 | |||||
| May 8, 2020 at 6:36 | audit | First posts | |||
| May 8, 2020 at 6:36 | |||||
| May 5, 2020 at 18:29 | comment | added | student | By a "gap" he says that although rationals are arbitrarily close to $\sqrt{2}$ both smaller and larger than it, no rational is equal to $\sqrt{2}$ . So the mere fact that $\sqrt{2}$ is irrational does not in itself illustrate the assertion that the rational number system has "gaps". For example, $-1$ is never the square of a real number. But this fact alone does not say that the real number system has "gaps". It does not, the reals form a complete ordered field. | |
| May 1, 2020 at 3:09 | audit | First posts | |||
| May 1, 2020 at 3:09 | |||||
| Apr 30, 2020 at 15:39 | comment | added | Rex Butler | This answer may be helpful math.stackexchange.com/a/95366/6455. There are physical and intuitive explanations for the use of the words "hole" and or "gap". | |
| Apr 30, 2020 at 14:46 | answer | added | user13618 | timeline score: 13 | |
| Apr 30, 2020 at 11:53 | audit | First posts | |||
| Apr 30, 2020 at 11:53 | |||||
| Apr 30, 2020 at 3:15 | answer | added | Paramanand Singh♦ | timeline score: 27 | |
| Apr 30, 2020 at 1:48 | comment | added | fleablood | But the hole comes in noting we can get close to it. We can hone in and find infinitely many $p_i$ and $q_i$ where $q_i-p_i$ can get as small as we like but $p_i^2 < 2 < q_i^2$. So the road between $p_i$ to $q_i$ ought to be "smooth" (they can be as close as we like) but the have a "jump" somehow we jump fro $p_i^2 < 2$ to $q_i^2> 2$ without passing a $r^2 = 2$ in between. | |
| Apr 30, 2020 at 1:48 | comment | added | fleablood | 1) It's subtly but there is not reason to think that rationals not have a number $q$ so that $q^2=2$ is a "hole"? After all there is no rational number $q$ where $q$ has sharp teeth and eats rabbits, is a hole. And there is not rational number $q$ where $q^2=-1$ is a hole. $x^2=2$ could simply be... something that doesn't exist. .... | |
| Apr 30, 2020 at 1:36 | answer | added | David Pement | timeline score: 21 | |
| Apr 30, 2020 at 1:10 | comment | added | PM 2Ring | FWIW, this {A, B} partition is called a Dedekind cut | |
| Apr 29, 2020 at 23:49 | history | became hot network question | |||
| Apr 29, 2020 at 17:01 | vote | accept | Larry | ||
| Apr 29, 2020 at 16:19 | review | Close votes | |||
| May 4, 2020 at 3:04 | |||||
| Apr 29, 2020 at 16:13 | history | edited | Xander Henderson♦ | CC BY-SA 4.0 |
added 11 characters in body; edited title
|
| Apr 29, 2020 at 16:09 | answer | added | Xander Henderson♦ | timeline score: 55 | |
| Apr 29, 2020 at 16:09 | answer | added | joriki | timeline score: 112 | |
| Apr 29, 2020 at 15:51 | review | First posts | |||
| Apr 29, 2020 at 15:59 | |||||
| Apr 29, 2020 at 15:48 | history | asked | Larry | CC BY-SA 4.0 |