Timeline for Is there an algorithm for determining whether an expression involving nested radicals is rational?
Current License: CC BY-SA 4.0
22 events
| when toggle format | what | by | license | comment | |
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| Oct 26, 2019 at 17:52 | comment | added | Oleg Lobachev | People talk here about adjunction and are fully correct. I, however, have a lingering suspicion you could do something with continued fractions. Rational numbers have finite continued fraction expansion, irrational numbers have only an infinite representation as a continued fraction, as proved by Euler. Nice 16th century maths! | |
| Oct 25, 2019 at 21:09 | answer | added | ahulpke | timeline score: 5 | |
| Oct 25, 2019 at 16:40 | comment | added | Eric Towers | Does "Post it on MSE with the question 'Is this rational?'." meet your definition of "algorithm"? | |
| Oct 24, 2019 at 21:03 | comment | added | François Brunault | Here is an ambiguous example with complex numbers: $i+\sqrt{2i}$ | |
| Oct 24, 2019 at 20:27 | answer | added | Dima Pasechnik | timeline score: 7 | |
| Oct 24, 2019 at 20:01 | answer | added | Will Sawin | timeline score: 9 | |
| Oct 24, 2019 at 19:26 | answer | added | Esteban Crespi | timeline score: 12 | |
| Oct 24, 2019 at 16:16 | comment | added | LSpice | @YCor, perhaps a reasonable interpretation is: can the radicals be interpreted in some fashion that makes the resulting expression rational? Of course, only @Jim can indicate if this is what was meant …. | |
| Oct 24, 2019 at 16:13 | history | became hot network question | |||
| Oct 24, 2019 at 15:38 | comment | added | Emil Jeřábek | @YCor The Wikipedia article doesn’t restrict the radicals to positive numbers. The only occurrence of the word positive in the definition refers to the degree of the root. There is, however, an obvious typo in that it speaks of integer additions, subtractions, etc. I’ll fix that. | |
| Oct 24, 2019 at 13:30 | comment | added | Francesco Polizzi | Yes, apparently Wikipedia article is confusing "definible by radicals" and "definible by real radicals", right? | |
| Oct 24, 2019 at 13:21 | comment | added | YCor | By the way, Wikipedia is quite confusing too (see here): they define number definable by radicals saying one uses radicals of positive numbers, and then says "there are algebraic numbers that cannot be obtained in this manner. These numbers are roots of polynomials of degree 5 or higher". This is false as they appear as soon as degree 3 then. | |
| Oct 24, 2019 at 13:17 | comment | added | YCor | Yes I know, the point of my comment is to request clarification, since the OP was sloppy about this point. My example was here to illustrate the difficulties occurring if radicals are not properly defined (as they are not only used for positive numbers), even if the example with square root of a positive number is a bit caricatural. | |
| Oct 24, 2019 at 13:08 | comment | added | Francesco Polizzi | @YCor: yes, but over $\mathbb{C}$ the square root is not a well-defined function because there are monodromy issues. Over $\mathbb{R}$ there is a well-defined positive branch. I am pretty sure that writing $\sqrt{4}$ one usually intends $2$. Anyway, this is just a matter of notation. | |
| Oct 24, 2019 at 13:02 | comment | added | YCor | @FrancescoPolizzi I wrote explicitly "formally speaking", and also explicitly mentioned that one can interpret radicals using positivity. As far as I know, the Cardan formulas for roots of cubic polynomials use radical signs for complex numbers. | |
| Oct 24, 2019 at 13:00 | comment | added | Francesco Polizzi | @YCor: Are you sure? As far as I know, for a positive real number $x$ the expression $\sqrt{x}$ denotes the positive square root, the negative one being denoted by $-\sqrt{x}$. | |
| Oct 24, 2019 at 11:11 | comment | added | Ville Salo | I know nothing but doc.sagemath.org/html/en/reference/number_fields/sage/rings/… says "Converting from either AA or QQbar to ZZ or QQ succeeds only if the number actually is an integer or rational."; though I don't know if that means it is guaranteed to succeed if it IS rational. | |
| Oct 24, 2019 at 11:11 | answer | added | François Brunault | timeline score: 21 | |
| Oct 24, 2019 at 10:02 | history | edited | YCor |
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| Oct 24, 2019 at 10:01 | comment | added | YCor | Beware that formally speaking, the question is ambiguous for such an expression as $\sqrt{7+\sqrt{4}}$. Indeed $\sqrt{4}$ might mean both $2$ and $-2$. Possibly you only allow only positive radicals, which removes the ambiguity, but restricts the scope since in this way you for instance miss roots of totally real cubic (irreducible rational) polynomials. | |
| Oct 24, 2019 at 8:05 | review | First posts | |||
| Oct 24, 2019 at 8:45 | |||||
| Oct 24, 2019 at 8:02 | history | asked | Jim | CC BY-SA 4.0 |