Talk:Extraneous and missing solutions: Difference between revisions

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==First Example==

Wow. The first example in this article is absolutely terrible. It's obvious at a glance that solutions where x=2 or x=-2 are not valid. After deriving x=-2, we have the statement, "We arrive at what appears to be a solution rather easily. However, '''something very strange occurs''' when we substitute the solution found back into the original equation..." Actually, nothing strange occurs at all. The solution is not part of the ___domain of potential solutions - which should be obvious before starting. {sigh}. Of course, the second example is even worse. I would personally improve this article if I had any idea what the point of it is. Anyone else? [[User:Tparameter|Tparameter]] ([[User talk:Tparameter|talk]]) 00:58, 19 January 2008 (UTC)

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: --[[User talk:Lambiam|Lambiam]] 06:30, 19 January 2008 (UTC)

::Let me try to define the term. An extraneous solution is a solution. That is first and foremost the most general definition. What kind of solution is it? It has the trait of being "extraneous", which we have to explain. So, I would say that ''an extraneous solution is a solution that is not applicable to a particular problem as it is defined.'' But remember, it ''is'' a solution. [[User:Tparameter|Tparameter]] ([[User talk:Tparameter|talk]]) 07:14, 19 January 2008 (UTC)

:::Unless of course, it is a misnomer. I have just been instructed by a colleague that the term is a misnomer, and actually does not describe solutions at all. To avoid original research, I'm not touching this article, I might make it much worse. [[User:Tparameter|Tparameter]] ([[User talk:Tparameter|talk]]) 07:40, 19 January 2008 (UTC)

::Of course ''every value'' V is ''a'' solution (namely of the equation x = V). The meaning of a term is what people who use the term in general mean when they use it. That is something that cannot always be determined by juxtaposing the usual meanings of components of the term in question. A [[Welsh rabbit]] is not a rabbit that is Welsh and the [[division algorithm]] is not an algorithm. Based on how the term "extraneous solution" is used, it is clear that the terms refers to any solution V of some equation E' such that E', viewed as a proposition, is a valid consequence of an original equation E, but which value V, however, is not a solution of E. Moreover, V has to be found as a solution to E' ''in the process of attempting to solve'' E. It follows that the solutions of E are a subset of those of E'. The solution set of E' consists of the solution set of E plus, possibly, some other solutions, called ''extraneous solutions''.  --[[User talk:Lambiam|Lambiam]] 15:43, 19 January 2008 (UTC)

Lambiam, on your first example, sorry, why isn't 2 a solution? [[User:Tparameter|Tparameter]] ([[User talk:Tparameter|talk]]) 16:08, 19 January 2008 (UTC)

:Oops, I swapped the two sides during editing but made an error; it should have read

::1/(x²+3x+2) = 1/(x²+x−2),

:resulting in an extraneous solution of x = −2.  --[[User talk:Lambiam|Lambiam]] 16:46, 19 January 2008 (UTC)

::So, am I clear then that one form of extraneous solutions are simply "solutions" not in the ___domain of x? [[User:Tparameter|Tparameter]] ([[User talk:Tparameter|talk]]) 17:49, 19 January 2008 (UTC)

:::I don't think so; as I said before interpreting an equation in another ___domain does not amount (as I see it) to transforming it into another equation. In the context of [[elementary algebra]] in which this terminology is used the students probably haven't even been introduced to the notion of solving in other domains than the reals.  --[[User talk:Lambiam|Lambiam]] 00:56, 20 January 2008 (UTC)

::::Isn't it true that in your example above that x = -2 is not in the ___domain of x? [[User:Tparameter|Tparameter]] ([[User talk:Tparameter|talk]]) 14:11, 20 January 2008 (UTC)

:::::By "___domain" I did not mean the [[___domain (mathematics)|___domain]] of a function, but the [[___domain of discourse]], which in this case happens to be a [[___domain (ring theory)|___domain]] in the sense of [[ring theory]]: the ring (and even [[field (mathematics)|field]]) of the [[real number]]s.  --[[User talk:Lambiam|Lambiam]] 03:14, 21 January 2008 (UTC)

:I like the initial example, but I agree that the wording could use revision to make it a bit less condescending and magical. Keep in mind that this page is targeted at a high school algebra audience, who are less likely to inspect an equation for nonsolutions up front (but it can still be formal in tone). [[User:Dcoetzee|Dcoetzee]] 18:13, 31 January 2008 (UTC)

== Is this an example of an extraneous solution? ==

The ___domain is given as the set of all <math>{x : x \in \mathbb{R}, x > 0}</math>.

Given the ___domain, and given

:x² - 3x + 5 = 0

Find x.

Using the [[quadratic formula]], it is easy to find the solutions

:x = 7 and x = -4

where x = 7 is the solution to the problem, and x = -4 is an '''extraneous solution''' because it is not pertinent to the problem. [[User:Tparameter|Tparameter]] ([[User talk:Tparameter|talk]]) 17:46, 19 January 2008 (UTC)

No, this is not a sutiable example of an extraneous solution. Since x = -4 can satisfly the equation x² - 3x + 5 = 0, only does not satisfly the ___domain that sets manually. Note that extraneous solution should focus on the '''naturally-formed ___domain''' but not on the '''manually-formed ___domain'''. Since you can set '''manually-formed ___domain''' at you own taste but '''naturally-formed ___domain''' cannot. '''naturally-formed ___domain''' should comes from the orginal equation. If the solution that does not satisfly the '''naturally-formed ___domain''' is also called extraneous solution, it should be trouble. Since this time all equation may be generate "extraneous solution" if you set an appropriate manually-formed ___domain. The term '''extraneous solution''' should not follow this approach.

If you search '''extraneous solution''' in Yahoo!, you can find many topics talking about '''extraneous solution'''. http://www.jcoffman.com/Algebra2/ch1_5.htm and http://www.mathpath.org/proof/argument.invalid.htm are two appropriate examples.[[User:Doraemonpaul|Doraemonpaul]] ([[User talk:Doraemonpaul|talk]]) 19:51, 18 February 2009 (UTC)

Neither 7 nor -4 satisfy the equation -- regardless of what our domains are. Just plug them into the equation. Do the math and you will see too.

Also, I think it was great that you explained how a solution is not considered extraneous if it only does not satisfy the manually-formed ___domain (as opposed to the naturally-formed ___domain).

The word is "satisfy," by the way -- not "satisfly." [[User:MusicHuman|MusicHuman]] ([[User talk:MusicHuman|talk]]) 14:52, 17 December 2022 (UTC)

== My changes ==

I expanded on this idea a bit, starting with some very simple examples, including discussion of operations that remove solutions, and outlining a more general and (hopefully) more understandable categorization of what operations do what to the solution set. Feel free to revise. [[User:Dcoetzee|Dcoetzee]] 20:31, 31 January 2008 (UTC)

== Possible errors ==

I see a couple of things in this article that don't seem right. Firstly, with the example <math>\sqrt{x}=-1</math>, the article says x=1 is not a solution. That means <math>\surd</math> is being used to mean positive square root (as is common), but in that case the equation has no solutions by definition, so is a rather poor example. Secondly, at the end it says multiplication is surjective by not injective. It's quite clearly not surjective since it's image is just {0}. Am I missing something here, or should I just fix it? --[[User:Tango|Tango]]

([[User talk:Tango|talk]]) 23:24, 1 March 2008 (UTC)

:I've removed the last section because it is mostly incorrect, not easy to fix, and essentially [[WP:OR|OR]]. I wish the article had not been expanded to cover missing solutions, which makes everything a lot more complicated, and as far as I know is not a common term to describe errors that may arise in solving high-school algebra problems. &nbsp;--[[User talk:Lambiam|Lambiam]] 15:45, 2 March 2008 (UTC)

:Removed the incorrect example <math>\sqrt{x}=-1</math>, where the article stated x=1 is not a solution: -1 is most certainly one of the second roots of unity. The missed solution x=1 stems from the fact that roots cannot be treated as "unknowns" that have an explicit value. Taking the n^th root of x is precisely the statement, "at least one of the n roots of x satisfies this equation" -- the "positive square root" operator does not exist for just this reason. Sorry, your calculator lies to you :-) [[Special:Contributions/68.42.7.184|68.42.7.184]] ([[User talk:68.42.7.184|talk]]) 07:32, 23 April 2009 (UTC)

== Extraneous solutions in applied problems ==

There is another type of extraneous solution, that is not mentioned in the article. A solution to an equation arising from an applied problem may be considered extraneous if it is not physically meaningful. A negative length could be an example of this. See http://mathcentral.uregina.ca/QQ/database/QQ.09.02/paul2.html [[Special:Contributions/66.41.7.193|66.41.7.193]] ([[User talk:66.41.7.193|talk]]) 05:00, 29 March 2008 (UTC)

== Example with the imaginary unit, ''i'' ==

Is this a useful example? It got me confused for a while just now:

We know that <math>\frac{1}{i}=-i</math>. However,

<math>\frac{1}{i}=i^{-1}=((-1)^{\frac{1}{2}})^{-1}=\sqrt{(-1)^{-1}}=\pm i</math>

But only <math>-i</math> satisfies the original equation, <math>+i</math> is an extraneous solution.

--[[User:MTres19|MTres19]] ([[User talk:MTres19|talk]]) 19:38, 29 April 2020 (UTC)