# Talk:Descartes' rule of signs

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pay attention to the last example: it's rong... --Arirossa 23:19, 7 February 2007 (UTC)

fixed, thanks for spotting. DavidMcKenzie 12:37, 8 February 2007 (UTC)
Actually, it was correct before "fixing". The factorization and roots discussed in the final lines refer to the original polynomial, not the sign-flipped version. --mglg(talk) 16:42, 13 February 2007 (UTC)

What does «Multiple roots of the same value are counted separately» mean, exactly? Mariano (talk) 00:31, 8 February 2008 (UTC)

Take ${\displaystyle (x+1)^{2}(x-1)}$ this polynomial has two roots with the value x=-1, and one with value x=1. So counting roots we have 2 negative roots and one positive root. --Salix alba (talk) 00:46, 8 February 2008 (UTC)
Wouldn't the very much more standard ``Roots are counted taking into account their multiplicities be better? Mariano (talk) 18:33, 15 February 2008 (UTC)

## Proof

The example is all well and good, but it's not a proof.

I happen to know that the proof is rather long, so perhaps it would be apt to provide an external link to it, if anyone can find it.

Thanks, Glooper 20:24, 3 November 2007 (UTC)

There seem to be a few neat expositions of various proofs floating around, I added the one from Cut the Knot since they cater for most audiences. Chenxlee (talk) 14:25, 20 February 2008 (UTC)

## Moved back to "Descartes' rule of signs"

I have reversed the recent move to "Descartes's rule of signs". The usual possessive form of Descartes is Descartes' - this is the standard followed on other sites such as MathWorld and the Stanford Encyclopedia of Philosophy, and in the titles of books such as Descartes' Error and Descartes' Metaphysical Physics. Gandalf61 (talk) 16:53, 5 February 2008 (UTC)

## Second Example

"This polynomial has two sign changes, meaning the original polynomial has 2 or 0 negative roots and this second polynomial has 2 or 0 positive roots."

Aren't the positive roots counted from the original polynomial and the negative roots from the second? --Patrick McLaren (talk) 22:56, 15 October 2010 (UTC)

## Comment

The page lacks a proof; please put it in. 76.181.64.113 (talk) 20:02, 21 July 2011 (UTC)

Wikipedia is an encyclopedic reference, not a textbook. To see a proof you can follow the link under External links or consult an algebra textbook. Gandalf61 (talk) 09:34, 22 July 2011 (UTC)

## Complex Roots

This result is a bit off; it needs to take into account the fact that 0 could be a root. — Preceding unsigned comment added by 71.174.58.74 (talk) 17:45, 22 February 2012 (UTC)

True. I have updated that section of the article to add the condition that polynomials must not not have a root at 0. Gandalf61 (talk) 07:45, 3 October 2013 (UTC)

How about being patent nonsense? It relies solely on "(X + i)(X - i) = X^2 + b" to work in all forms. It only detects 'purely imaginary' (not complex) numbers in that manner. (X + i)(X + i) instantly defeats it, I believe... with 1 positive sign change and 1 negative sign change. http://mathworld.wolfram.com/DescartesSignRule.html uses a general polynomial [1 1 -1 -1 -1 1 -1], which has roots of

``` -1.66489 + 0.00000i
-0.62797 + 0.90355i
-0.62797 - 0.90355i
1.16202 + 0.00000i
0.37941 + 0.53195i
0.37941 - 0.53195i
```

again, "defeated".173.168.30.119 (talk) 06:08, 3 October 2013 (UTC)

You can't apply Dsecartes' Rule to
${\displaystyle (x+i)(x+i)=x^{2}+2ix-1}$
because it only applies to polynomials with real coefficients. As for your other example,
${\displaystyle f(x)=x^{6}+x^{5}-x^{4}-x^{3}-x^{2}+x-1}$
this has 3 sign changes, so f(x) has a maximum of 3 positive real roots. And we have
${\displaystyle f(-x)=x^{6}-x^{5}-x^{4}+x^{3}-x^{2}-x-1}$
which also has 3 sign changes so f(x) has a maximum of 3 negative real roots. The "complex roots" version of Descartes' Rule then says that f(x) has a minimum of 6-3-3 = 0 complex roots, which is entirely consistent with it having 6 complex roots. Gandalf61 (talk) 07:38, 3 October 2013 (UTC)
It only detects "purely imaginary pole pairs", now you COULD say If random(5) <= 5, a minimum of 0 "purely imaginary" roots exist, but that is a silly assertion... (But just as valid as this claim)
(x + i)(x - i)(x - 1) is another fun thing, which it fails to detect the imaginary pole pair.
This IS patent nonsense, it works under a select number of cases and does not work consistently.
And what is more... you don't even list the conditions under which it works! 173.168.30.119 (talk) 20:19, 3 October 2013 (UTC)
I don't know why you think that the complex roots version of the rule only applies to purely imaginary roots. This is not correct. Perhaps you were misled by the simple example in that section. I have replaced that example with a more general one where the complex roots are not purely imaginary. The complex roots version of the rule is actually a very simple extension of the real roots version - if you know there are a maximum of p+q real roots then there must be a minimum of n - (p+q) complex roots. If you accept that the real roots version of the rule is correct, then the complex roots version is quite obviously true also. Gandalf61 (talk) 10:28, 4 October 2013 (UTC)

## Original source + unchanging signs

Here is a source for Descartes Geometrie: http://www.gutenberg.org/ebooks/26400 The rule is on page 42. Maybe this is interesting: Descartes also makes the statement that you can determine the number of negative roots (called "false" by Descartes) by counting unchanging signs between consecutive coefficients. Descartes does not say what to do with coefficients that are zero, but you have to insert an arbitrary sign for any zero to make the rules work. Another point: Descartes is talking only about polynomials with all real roots. — Preceding unsigned comment added by 93.220.47.168 (talk) 17:56, 11 May 2014 (UTC)

## Requested move 31 December 2017

The following is a closed discussion of a requested move. Please do not modify it. Subsequent comments should be made in a new section on the talk page. Editors desiring to contest the closing decision should consider a move review. No further edits should be made to this section.

The result of the move request was: Not moved (non-admin closure) Ⓩⓟⓟⓘⓧ Talk 19:37, 6 January 2018 (UTC)

Descartes' rule of signsDescartes's rule of signs – The rule is to use "s's" per WP:MOS as Cherkash seems to insist. The move was reverted by David Eppstein twice. GeoffreyT2000 (talk) 06:04, 31 December 2017 (UTC)

Per WP:COMMONNAME, "Descartes' rule of signs" has about 4060 results in Google scholar; "Chasles's theorem" has 224. Also, to me, "s's" implies that both s's should be pronounced separately, which is the incorrect pronounciation in this case. —David Eppstein (talk) 06:43, 31 December 2017 (UTC)
I do not see that WP:MOS requires the use of "s's"; it specifically mentions that there are two systems and that one should be consistent within an article. Under these circumstances, I find David's appeal to WP:COMMONNAME to be very convincing. While you can certainly find both forms in the literature, you will also find many clearly incorrect forms, but the majority, in my experience, have used the apostrophe only form. --Bill Cherowitzo (talk) 19:48, 31 December 2017 (UTC)
I agree - WP:COMMONNAME trumps WP:MOS in my opinion. Wolfram, Cut The Knot and Purplemath all use the <s'> form. I oppose the proposed move. Gandalf61 (talk) 12:32, 1 January 2018 (UTC)
• Oppose per David Eppstein, Bill Cherowitzo and Gandalf61. —Roman Spinner (talk)(contribs) 19:10, 1 January 2018 (UTC)
• Support. Two main reasons: 1) Singular nouns, even the ones that end in s/z, take the " 's " as opposed to a single apostrophe in their possessive form (see here). 2) Singular nouns that terminate with a silent "s/z/x" (as in some French names, an example of which is the name Descartes discussed here), almost uniformly take the " 's " in their possessive form (see here), especially since this " 's " is actually pronounced: "de-CART" for the name itself, "de-CARTz" for its possessive (David Eppstein, note that the spelling doesn't imply the "de-CART-siz" pronunciation as you incorrectly suggested e.g. in your edit summary here). cherkash (talk) 01:23, 4 January 2018 (UTC)
Comment. I do not see how you arrive at these conclusions from what is written in the Apostrophe article. The statements there are clearly not definitive and the most that can be concluded from the passages is that there is a difference of opinion on the matter. Your statement about Descartes (above) I would place in the category of fake news as a simple google scholar search clearly shows its falsity.--Bill Cherowitzo (talk) 04:34, 4 January 2018 (UTC)
Not even sure how to interpret your last sentence. Which specific "statement" do you believe false? cherkash (talk) 02:21, 5 January 2018 (UTC)
This one. "Singular nouns that terminate with a silent "s/z/x" (as in some French names, an example of which is the name Descartes discussed here), almost uniformly take the " 's " in their possessive form".--Bill Cherowitzo (talk) 04:57, 5 January 2018 (UTC)
• Oppose per WP:COMMONNAME as above. There is no universal grammatical rule that requires the additional s. The omission has been taught in British schools for the past sixty years at least. Dbfirs 10:37, 4 January 2018 (UTC)
As I've mentioned in similar discussions elsewhere, the universality of this statement seems to be highly suspect (maybe in some schools, but not in all). cherkash (talk) 02:19, 5 January 2018 (UTC)

The above discussion is preserved as an archive of a requested move. Please do not modify it. Subsequent comments should be made in a new section on this talk page or in a move review. No further edits should be made to this section.