text:
Talk:Cauchy sequence

Untitled
Cauchy net redirects here, yet there seems to be nothing about the concept here.... Vivacissamamente

 * This comment was left in 2004, it is no longer true. --JBL (talk) 13:06, 13 December 2018 (UTC)

Untitled

 * Roughly speaking, the terms of the sequence are getting closer and closer together in a way that suggests that the sequence ought to converge. Nonetheless, Cauchy sequences do not always converge.

Some example please --Taw 11 December 2001

added an example, it's kind of kludgy though -- RAE

--

I'd like to put something along the following lines: Cauchy Seqs are initially useful in spaces such as the Reals because they are a test of convergence which doesn't require a value for the potential limit. -- the flip side is that IF all CSs converge then a space is complete.

There's a sort of switch in perception as things move up a level of abstraction which as a mathematician I find self-evident (and interesting), but I suspect non-mathematicians find baffling or even terrifying:
 * theorem: cauchy seqs converge on the Reals
 * abstraction: cauchy seqs on other space, where they might not converge
 * axiom: part of the defn of complete space

Has this been general idea been coverered anywhere in the maths section? -- Tarquin

I don't think it has been covered; it would fit either here or in complete space. AxelBoldt, Wednesday, June 12, 2002

how about:

 * All Cauchy sequences of real or complex numbers converge, hence testing that a sequence is Cauchy is a test of convergence. This is more useful than using the definition of convergence, since that requires the possible limit to be known. With this idea in mind, a metric space in which all Cauchy sequences converge is called complete.
 * Thus R and C are complete; but Q is not. The standard construction of the real numbers involves Cauchy sequences of rational numbers; (something about R being the completion of Q...)

...and something on Mathematical abstraction in general somewhere else. I'll see if I can dig up or remember the proof outlines for "Every convergent sequence is a Cauchy sequence" and "every Cauchy sequence is bounded" -- Tarquin June 12 2002

Definition
How about some sort of formal definition? My elementary analysis textbook states: A sequence $$(s_n)$$ of real numbers is called a Cauchy sequence if $$\forall\epsilon>0\exists N\mbox{ such that }m,n>N\mbox{ implies }\vert s_n-s_m\vert<\epsilon.$$66.71.96.78 17:01, 3 October 2005 (UTC)

 * The formal definition in the article is more general than yours, applying to metric spaces in general rather than specifically to the real line. —Caesura(t) 17:07, 6 December 2005 (UTC)

 * It would also be nice to have Cauchy sequences defined for other absolute values, in particular for p-adic absolute values. Would this be a problem? Gene Ward Smith 09:05, 6 May 2006 (UTC)

 * I guess that could go in the generalization section, as is not really central to the concept of Cauchy sequences as used in analysis. Oleg Alexandrov (talk) 15:48, 6 May 2006 (UTC)

p-adic material out of place
The p-adic material just after the heading Cauchy sequence in a metric space doesn't seem to belong there. McKay 11:18, 16 June 2006 (UTC)
 * I cut it out, together with other fluff. The whole article was a mumbo jumbo of things without clear connections. Oleg Alexandrov (talk) 16:42, 16 June 2006 (UTC)

'All two' or 'any two'?
The first paragraph has again been changed to "all two remaining elements ... ". I don't want to start a revert war here, so I would appreciate other views. My view is that it has to be "any two", as "all two" is both mathematically wrong and grammatically wrong. Madmath789 06:38, 21 June 2006 (UTC)

 * I agree, but "any two" is not very precise. It is the maximum distance between  two of the remaining elements that has to be small. I changed it. McKay 07:57, 21 June 2006 (UTC)

Reference List
The reference list includes two on algebra and one of constructive mathematics. How about a reference to a good analysis text since after all, Cauchy sequences typically are learned as part of analysis, not algebra.

 * Just any analysis text, even if it isn't apparently used as a reference? Why not little Rudin or something? But is that right? There isn't anything, offhand, I can think to add to this article, from a source or otherwise. —vivacissamamente 03:13, 21 October 2006 (UTC)

 * I feel that there should be an analysis text in the references:

I suggest M.Spivak's 'Calculus' as an option. It gives a good treatment of real Cauchy sequences, and of the construction of the reals using Cauchy sequences, as well as other constructions (it's definitely an Analysis text, rather than a Calculus text, despite the title). I'll try to look ISBN, etc, if someone doesn't beat me to it. Messagetolove 13:56, 26 May 2007 (UTC)

 * Have put in Spivak reference ( it's even in Wiki in its own right).

Messagetolove 19:15, 26 May 2007 (UTC)

Recent edits
Here's my revert. Here are the issues.

 * The sequence
 * 0, 1, 1.5, 2, 2.25, 2.5, 2.75, 3, 3.125, 3.25, ..
 * appears convergent to me, in spite of what the example claims

 * There is no need to emphasize that
 * $$d(x_m, x_n) < r$$

implies not only consecutive terms but all remaining terms are getting closer and closer together. It is obvious that we are talking about all terms, since we use different indeces for m and n.

 * Why remove the text about completeness from the def of cauchy sequence? That's the best place to make that point! It is obvious to anybody there that a Cauchy sequence looks as if it is convergent. Then make the reader pay attention right there, rather than moving that text half an article down. Oleg Alexandrov (talk) 15:55, 2 June 2007 (UTC)

 * Completeness has its own section. Especially with the added issue of counterexamples the two issues should not be mixed up.--Patrick 08:45, 3 June 2007 (UTC)

 * It looks to me as if Patrick may have intended to be taking the partial sums of the series obtained by adding 1 once, 1/2 2 times, 1/4 4 times, 1/8 8 times, etc that is, essentially the argument used to prove that the harmonic series diverges (shifted a bit). Clearly, if that is the intended sequence, it is not convergent, but it isn't entirely made clear that this is really what is intended, and I haven't heard that referred to as "harmonic" before. I agree with your other points.
 * Messagetolove 19:06, 2 June 2007 (UTC)

 * Yes, that is what I mean. However, I wrote that this sequence and the sequence of harmonic numbers are counterexamples, I did not write that this sequence is called harmonic. This sequence has easier numbers than the harmonic numbers, so it is easier to see that it diverges, with all natural numbers occurring in it. If the sequence is not clear enough we can add more terms.--Patrick 22:37, 2 June 2007 (UTC)
 * But it is much harder to see the pattern in this sequence, even with an explanation. The harmonic series look simpler to me, one just adds 1/n each time. Also, I am not sure this counterexample is relevant here. I'll add a picture these days, that should make things clearer. Oleg Alexandrov (talk) 23:07, 2 June 2007 (UTC)

I reverted the counterexample of consecutive terms, that one is kind of pointless, and poorly written too (it is not the difference of consecutive terms which goes to zero, it is the distance between them, per the previous paragraph). The interesting counterexample is a Cauchy sequence that is not convergent, and that is below. Oleg Alexandrov (talk) 11:44, 3 June 2007 (UTC)

 * (1) That is not at all an argument for deletion, that is just changing one word.
 * (2) That is not a counterexample, it is a different (equally interesting) issue.
 * Patrick 12:15, 3 June 2007 (UTC)
 * I agree with (1), of course. I don't agree that the thing about consecutive terms is interesting. It is clear enough that the terms are not consecutive from the def. The big deal about Cauchy sequence is relation to completeness. Oleg Alexandrov (talk) 12:24, 3 June 2007 (UTC)

question
Oleg said"> this is clear enough, n is on the horisontal axis, and x_n is on the vertical one. This is a standard way of graphing functions."yes, i see this is true. but, the graph ploted by blue points lies on the "plane", not on the axis. so, the blue points do not show the sequence of x_n, do that? sorry for my poor english, thank you. --218.42.230.29 21:48, 5 August 2007 (UTC)
 * I clarified things a bit saying that what is graphed is not the sequence itself, but its plot. Are things clearer now? Oleg Alexandrov (talk) 00:37, 6 August 2007 (UTC)
 * i thank your kindness. :-) --218.42.230.61 01:41, 6 August 2007 (UTC)

A squiggle I dont know how to say or from what alphabet it comes
I would simply like to point out that there is a squiggle on this page that assumes a fair bit of pre-knowledge that is not documented by a re-link into Wikipedia explaining the character symbol and associated concepts

Generalizing, it would be nice if every mathematical symbol used an article had a link to all that the reader should know about it. Could Wikipedia make this some kind of standard?

Maybe this could be further generalized to some kind of overall standard that any relevant concept with sufficient complexity and obscurity be re-linked back into Wikipedia, at least with a requested article. —Preceding unsigned comment added by Jjalexand (talk • contribs) 12:50, 2 September 2008 (UTC)

Values of exp, sin and cos always irrational?
I think this is not true:

 * The values of the exponential, sine and cosine functions, exp(x), sin(x), cos(x), are known to be irrational for any rational value of x≠0, [...]

Counter-example: $$\cos(60^{\circ}) = \frac{1}{2} \,$$.

(Francisco Albani (talk) 00:31, 22 September 2008 (UTC)).

 * The statement you quoted above is correct in "natural units" which in the case of the trigonometric functions means radians instead of degrees: 60° = 60•2π/360 = π/3, and a rational multiple of an irrational number (π) is still irrational. — Tobias Bergemann (talk) 11:29, 22 September 2008 (UTC)

"m,n > N" vs. "m,n >= N"
Is there any reason "m,n > N" is better than "m,n >= N"? With ">", it seems to me that the first element is always discarded -- which doesn't matter, I guess, but made me wonder if that is necessary or not. 88.65.186.193 (talk) 14:08, 25 April 2009 (UTC)

Since the choice of N is arbitrary so is whether or not you use a strict inequality (or not). For example, if you require that m,n > 199 then m,n >= 200 is just as good. Some authors like it one way, others another. Similarly, when doing a delta-epsilon proof, if two elements {f(y) and f(x)} in a sequence are less than epsilon whenever {x and y} are closer than "delta", then the sequence is convergent... but the same is true if they are less than 5*epsilon or epsilon^2. or sqrt(epsilon) or 1000*epsilon, because the choice is ARBITRARY! Hope that helps.... Brydustin (talk) 03:20, 25 February 2012 (UTC)

 * In short: it means the same but takes one character less... --CiaPan (talk) 07:50, 1 March 2012 (UTC)

Convergence
Why don't we just simply state:
 * a Cauchy sequence convergence in a metric space, though the limit might not in the metric space.

Jackzhp (talk) 22:56, 17 June 2009 (UTC)

 * Because when we say something converges to something else, we implicitly suggest that the limit is in the space? I mean, if the limit is not in the space, where is it? It's not easy to talk about things that are outside the space. -- Taku (talk) 21:39, 18 June 2009 (UTC)

 * That's right. I'd even add some emphasis and say 'because by definition a sequence is convergent if and only if it has a limit,' and a limit of a sequence is defined as an element of the same space to which the sequence's items belong. That's why 'the same' sequence of $$(1/n)_{n\in\mathbb N}$$ is convergent in real numbers (with a limit equal zero) and is divergent in $$\mathbb R^+=(0,\infty)$$ --CiaPan (talk) 12:05, 2 July 2009 (UTC)

General topology
What about generalization into general topology? I suggest a definition: (Cn) is Cauchy iff there is a set such that any its open covering contains an open set S such that there exists an integer N such that for any n>N, Cn is in S.

Is it good? Does it appear somewhere? —Preceding unsigned comment added by 84.229.68.163 (talk) 06:56, 2 July 2009 (UTC)

 * This sounds like the definition of convergence. By definition, a sequence $$x_n$$ converges to x if every open set containing x contains infinitely many $$x_n$$. The problem with generalizing Cauhy sequence is that we need some way to measure the distance between two points; this can be done in a metric space, of course. But in general, not possible. So, we need uniform structure or something else. -- Taku (talk) 11:09, 2 July 2009 (UTC)

 * 'Contains infinitely many' seems too weak for me. Consider the real sequence $$a_n = \sin n$$ It has no limit in the $$(0,1)$$ interval although any open subset of $$(0,1)$$ contains infinitely many terms of $$(a_n)$$. May be we can say 'any open set containing x contains ALL $$x_n$$ for n greater than some k,' that is 'all except at most a finite number ot initial terms.' --CiaPan (talk) 07:14, 3 July 2009 (UTC)

 * You're absolutely right. A simpler counterexample would be "1, 0, 1, 0, 1, 0, ...". It should be "x_n -> x if every neighborhood of x contains all x_n except some finite many terms." -- Taku (talk) 13:43, 3 July 2009 (UTC)

My definition is intended precisely to circumvent this obstacle. The idea is to confine the sequence into an open set "as small as we want" (for sufficiently large n). The difficulty was to define what is "an open set as small as we want" without limit point or metric. What I found appears an elegant solution for this: open covering can be as fine as we want. My definition doesn't require anything undefined in general topology and clearly does generalize topological convergent sequences, but I don't see whether it generalizes metric Cauchy sequences, too. If not, can it be fixed by replacing "any its open covering" with "any its finite open covering"?

By the way, how can one add comment to an arbitrary section or to a comment? (one ugly way to do it is certainly editing) --87.68.41.183 (talk) 15:29, 2 July 2009 (UTC)

English pronunciation of "Cauchy"
In my experience, the name "Cauchy" is pronounced in English as. Should I add this pronunciation to the article, or wait for someone to come along with a reliable source, or at least say, "yeah, I pronounce it that way, too"? —Tanner Swett (talk) 19:28, 21 June 2011 (UTC)
 * I wouldn't know, although one could in principle search YouTube for evidence that people really pronounce it this way, but I've trusted your observation, been bold and added the pronunciation you are suggesting to the article. This way, it is more visible, and the likelihood of people noticing it and complaining or changing it if it is not the most frequent pronunciation is higher. --Florian Blaschke (talk) 15:46, 6 March 2014 (UTC)

Confusing Example
The example of a Cauchy sequence says "The customary acceptance of the fact that any real number x has a decimal expansion is an implicit acknowledgment that a particular Cauchy sequence of rational numbers (whose terms are the successive truncations of the decimal expansion of x) has the real limit x."

But this is followed by "counter example: rational numbers" that says "The rational numbers Q are not complete (for the usual distance): There are sequences of rationals that converge (in R) to irrational numbers; these are Cauchy sequences having no limit in Q."

There is no contradiction in saying that people intuitively understand that a Cauchy sequence of rationals converges to a real number, but a casual reading of the two paragraphs invites confusion. — Preceding unsigned comment added by 208.86.181.160 (talk) 15:10, 19 July 2012 (UTC)

 * What needs some emphasis here is the distinction of domains: we say there are sequences of rationals that converge (in R) to irrational number, so they have no limit in Q and so are divergent in Q.
 * In other words we consider the same sequence of rational numbers being either 'a sequence in R space' (that is a sequence of real numbers, which are by coincidence rational) with a limit in the real space, or 'a sequence in Q space' (a sequence of rational numbers in rational space) with no limit in that space. --CiaPan (talk) 14:52, 22 July 2014 (UTC)

Generalizations to topological groups
I think there is a problem with the definition of the completion of a topological group G when G does not have a countable basis of open neighbourhood of 0. In particular, if K is a field with a Krull valuation, whose value group does not have countable cofinality, then, according to the definition given here, K will be complete (since the only Cauchy sequences in K, as defined here, are eventually constant), contrary to usual terminology. 193.206.101.2 (talk) 14:11, 4 July 2013 (UTC)
 * Another example of the same problem: let G be the additive group of the Banach space $$\ell^1$$, equipped with the weak topology. Then every weakly Cauchy SEQUENCE converges in norm, by Schur's property and the fact that $$L^1$$-spaces are weakly sequentially complete.  However the group (the Banach space) is certainly not weakly complete.  In order to fix this, one should go to the much more complex setting of uniform spaces and Cauchy filters. Bdmy (talk) 18:32, 4 July 2013 (UTC)

arbitrarily
the definition "a sequence whose elements become arbitrarily close to each other" I think to be more clear should say "a sequence whose elements become increasingly arbitrarily close to each other", unless a definition of arbitrarily is provided that includes the notion. because, in plain English, the arbitrary number could be 1 and the integers would then be arbitrarily close to one another. Alternatively, could say the sequence is montonically something or other or some other term. 69.201.168.196 (talk) 13:50, 15 July 2014 (UTC)

 * Even though "arbitrarily close" is an expression that (I'm pretty sure) only mathematicians understand, nobody at all will understand "increasingly arbitrarily close". I think the text is acceptable already because the next sentence spells out what "arbitrarily close" means. "Monotonic" is mathematically incorrect: the differences do not have to continually decrease, but can go up and down provided that they eventually get small and stay small. McKay (talk) 05:34, 16 July 2014 (UTC)

 * No, arbitrarily close doesn't mean one could choose distance equal 1 and declare integers are arbitrarily close. Arbitrarily close means whatever distance you choose (whatever small in this case) the sequence terms eventually get closer to each other than your chosen distance; not only they will get closer than some specific threshold you chose (e.g. 1), but they will get closer than any threshold you can choose. --CiaPan (talk) 14:41, 22 July 2014 (UTC)

Broken LaTeX in this Article
Most of the LaTeX in this article is broken. 174.124.71.135 (talk) 21:51, 20 October 2016 (UTC)

Problem with formal definition and harmonic series example
In the 2nd paragraph of the introductory section of the article, the harmonic series is cited as an example of a series that does not converge, and then a formal definition of a Cauchy sequence is presented, intending to show that the harmonic series does not satisfy this definition.

However, the way this formal definition is presented here, the harmonic series in fact does satisfy it! For any given epsilon > 0, simply choose a value of N such that 1/N < epsilon. Then it is clear that for any pair m,n > N, |1/m - 1/n| < epsilon will indeed hold.

The actual formal definition that is necessary for the Cauchy convergence criterion is correctly stated on the pages for "Convergent series" (section "Cauchy convergence criterion") and "Cauchy's convergence test". In short, it is necessary that for any given epsilon > 0, there exists an N such that the absolute value of *any arbitrarily long sum of terms* beyond the Nth term is less than epsilon. — Preceding unsigned comment added by Rokirovka (talk • contribs) 11:29, 9 May 2018 (UTC)

 * Are you possibly confusing the harmonic sequence and the harmonic series...? The harmonic sequence is Cauchy, as you proove above, and convergent. However, the harmonic series, which is a sequence of partial sums of the harmonic sequence, is not convergent and not Cauchy, either. It does satisfy an intuitive requirement that neighboring terms are arbitrarily close to one another, but it does not satisfy the definition requirement that almost all terms (which means: all terms from some point onwards) are arbitrarily close to each other. --CiaPan (talk) 11:48, 9 May 2018 (UTC)
 * (Forgot to ping. --CiaPan (talk) 11:50, 9 May 2018 (UTC))

 * Perhaps the issue here is the precise meaning of the notation. When I read the notation "a_n", whether in reference to the harmonic sequence or to the harmonic series, I interpret that as "the nth term of the harmonic sequence", that is, "1/n". I do not interpret the notation "a_n" as "the nth partial sum 1 + 1/2 + 1/3 + ... + 1/n", even if the preceding material uses the terminology "harmonic series" rather than "harmonic sequence". I think many other readers will have the same instinctive interpretation of this notation as I do, and thus I think the paragraph as it reads right now will be confusing and unclear to many readers. Rokirovka (talk) 15:06, 9 May 2018 (UTC)

 * But $$a_n$$ does not refer to a harmonic series or sequence at all! The series is shown as an example for what is not sufficient, then the 'more formal' description shows what is necessary. The intervening sentence starting with 'rather' emphasises that the latter is somewhat opposite to the former, hence the symbols used do not describe the same thing. The $$a_n$$ symbol is not associated with the $\sum\frac 1n$ series in the text, neither directly nor implicitly. --CiaPan (talk) 15:18, 9 May 2018 (UTC)

 * I agree with Rokirovka that the paragraph in question is not written very clearly and could lead to confusion (even though it does not contain any unambiguously false statements). --JBL (talk) 15:20, 9 May 2018 (UTC)

 * Yeah, maybe just using a quick example like the sequence $$a_n = \sqrt{n}$$ could work just as well without any need to complicate the issue with series? –Deacon Vorbis (carbon • videos) 15:40, 9 May 2018 (UTC)

I have replaced the paragraph with an example based on $$a_n = \sqrt{n}$$ – Special:Diff/840419862. Please see, verify and fix if necessary. --CiaPan (talk) 19:37, 9 May 2018 (UTC)

Bounded
In this edit you reverted an extension (by ) of boundedness to a more general case, calling it "inappropriate". Can you explain? --JBL (talk) 13:06, 13 December 2018 (UTC)

 * Here is the answer I was going to give:

 * However, now I found that the article Bounded function, linked in the paragraph, explicitly defines boundedness for any metric space with arbitrary choice of any point in the space as an 'origin', from which the distances are measured. So now I stand corrected, and I restore what I reverted before. It's never too late to learn :) Anyway, the 'absolute value' part is still incorrect and I replaced it. --CiaPan (talk) 22:15, 13 December 2018 (UTC)

Infinite sequence
It seems to me that a Cauchy sequence must be an infinite sequence. Is that correct? If so, I think "infinite" should be part of the definition of Cauchy sequence. There can be finite sequences, but a Cauchy sequence surely must be infiite. Dratman (talk) 21:15, 17 July 2021 (UTC)

Dark Mode Compatibility in Wikipedia App
If the official Wikipedia App is supporting dark mode it's strange to see that some articles prefer this recommended style while most articles do it the other way and it works. So how can we solve this problem without this style change? It is definitely no solution to use light mode here because this should work out of box and my edit was the only workaround I could see from other major math articles. 2A02:6D40:21F1:D200:8542:4B01:AC3B:E45C (talk) 07:25, 12 February 2022 (UTC)
 * This is a known issue (see and ) and the correct solution is not to break other things (colon-indenting equations apparently causes issues for screen-readers which, unlike dark mode, is not something readers can easily adjust on their end) but to leave comments or ticket requests in the hope the developers of the app fix the issue sooner rather than later. --JBL (talk) 21:28, 12 February 2022 (UTC)

Fundamental sequence
if you don't object the information about the name "fundamental sequence", why don't you move it then to the place you want it instead of reverting my edit? That is not a valid excuse to revert (see Revert only when necessary). The name "fundamental sequence" is a common name particularly by Russian authors in functional analysis.--Tensorproduct (talk) 20:17, 1 September 2023 (UTC)

 * Because of the three options (1) don't include the information, (2) include it where you put it, and (3) include it somewhere else, I am indifferent between (1) and (3), whereas (2) is clearly worse. You evidently think it is worth including, then be my guest -- in a way that doesn't obviously make the article worse. --JBL (talk) 21:12, 1 September 2023 (UTC)
 * @JayBeeEll Explain how it makes the article worse, that is not an objective statement but your opinion. The name "fundamental sequence" is used in many books and I added one reference.--Tensorproduct (talk) 21:15, 1 September 2023 (UTC)

A Cauchy sequence is also called a fundamental sequence
A Cauchy sequence is also called a fundamental sequence. There are many math books (by big mathematicians such as Israel Gelfand) that use the word fundamental sequence instead of Cauchy sequence. I used as a source a book by another mathematician Heinz-Dieter Ebbinghaus. If I remember correctly, it was the original name before people started to call it a Cauchy sequence. The user User:JayBeeEll keeps on reverting my edit without giving me a proper explanation. The user only says it makes the article worse but does not explain why or even discuss the issue properly. I think this is an important piece of information for an encyclopedia and it should be in the article. And in no way makes it the article worse.--Tensorproduct (talk) 23:26, 1 September 2023 (UTC)
 * As a working mathematician I have never before heard the term "fundamental sequence," but it does seem to appear in some books. (I know that my personal experience is not relevant and that we rely on sources....) The present version with "seldom" seems ok to me, I would understand an editor who thought it should be later in the lead section/made less prominent in some other way, as I do not believe that this term is in common contemporary usage. CapitalSasha ~ talk 03:10, 2 September 2023 (UTC)
 * @CapitalSasha If I remember correctly the term fundamental sequence is actually the real historical name for the Cauchy sequence before people used the word Cauchy sequence. Obviously such sequences were known by people like Leonhard Euler before Cauchy even lived. It seems to me that the term is more common in the literature of German and Russian authors in functional analysis than French authors. However the name appears in some real classical books from Analysis like "generalized functions" by Israel Gelfand and Georgiy Shilov, which is such an important series that the relevance is clearly there (it's the book where Gelfand triples, countably normed space, Gelfand-Shilov spaces, generalized functions etc. are introduced). Anyone studying seriously analysis will eventually come across the term.--Tensorproduct (talk) 04:03, 2 September 2023 (UTC)
 * BTW: At the bottom of the article is a link to the Encyclopedia of Mathematics by Springer which says also "fundamental sequence".--Tensorproduct (talk) 10:11, 2 September 2023 (UTC)
 * Edit nr 2: I forgot to address your claim that "the term is not in contemporary usage". That is not true, see for example "The Method of Rigged Spaces in Singular Perturbation Theory of Self-Adjoint Operators" by Volodymyr Koshmanenko and Mykola Dudkin. That is a book from 2016 and they used this term (and many more do). It is a new book not a new edition of an old one.--Tensorproduct (talk) 13:28, 2 September 2023 (UTC)
 * I think the current phrasing of "seldom" is good, then. CapitalSasha ~ talk 14:13, 2 September 2023 (UTC)
 * All this huffing and puffing, before you bothered to ask: why is it worse? The answer is, it is obviously bad to make the first sentence of an article a meandering journey through etymology and terminology before it gets to substance.  I have fixed it for you; you're welcome.  --JBL (talk) 17:17, 3 September 2023 (UTC)
 * @JayBeeEll I also think it's quite "unfriendly" that you call the input of another mathematician "huffing and puffing". But what I learned here, everybody knows it better. --Tensorproduct (talk) 22:53, 12 January 2024 (UTC)
 * I don't think this talk-page is a very good place for a discussion of personal behavior, but since you started it here, I will respond here. My initial reversion of your addition was polite; if you had, at that moment, opened a discussion on this page (as suggested by WP:BRD) or my user-talk and said "I don't understand what you object to about the placement, could you clarify?" then we would have had a short and constructive discussion, with the same endpoint (the information you added included, in a way that doesn't overburden the lead sentence).  Instead, you reverted me again, and then wrote a long and defensive post that doesn't address in any way my objection.  Could everyone in this interaction have done better?  Probably so.  I think it is a shame that this still bothers you four months later, and I apologize for reverting rather than improving in the first place.  --JBL (talk) 23:57, 12 January 2024 (UTC)
 * @JayBeeEll Well I added a good source by a relatively well-kown mathematician Heinz-Dieter Ebbinghaus, I thought this would be sufficient. Especially since his terminology has some authority in the math community.--Tensorproduct (talk) 21:54, 20 January 2024 (UTC)
 * The parenthetical in this version is awkward and makes getting to the actual substance more difficult. It's also tonally inconsistent: why is a term that is "seldom" used the second most prominent thing about the topic? Out of curiosity, I checked the Google Scholar numbers for both terms; "Cauchy sequence" gives about 64,500 results and "fundamental sequence" about 7,280, including what look to be a fair number of false positives. Such figures can only give the coarsest possible comparison, of course, but a ratio that lopsided is a pretty good indication that one term really is more commonly used than the other. XOR'easter (talk) 22:49, 3 September 2023 (UTC)
 * @JayBeeEll I did ask "why does it make the article worse"? You didn't feel to respond nor having a discussion.
 * I also have never heard that it is policy on Wikipedia to have a specific ordering or place in the article for terminology. There are many math articles that start by listening all the names, here a short list I can think of:
 * Distribution (mathematics)
 * Black–Scholes model
 * Von Neumann algebra
 * Benjamin–Bona–Mahony equation
 * Itô's lemma
 * Optional stopping theorem
 * so it's not policy. Obviously I do not care where the information is placed, I just had a problem with the unjustified reverting you did, you should have made first some research on the terminology if you didn't knew it.
 * @XOR'easter Does it matter which term is more used? The term is still widely used in the mathematics literature. It's common in functional analysis and stochastic calculus. And it's even more common if you look in other languages than English. Maybe the word seldom was a bit exaggerated. And do you really believe the statement "makes getting to the actual substance more difficult"? It's just another name in the bracket (as it is in many other articles) that the reader can skip. But again, I never cared where the term is place.--Tensorproduct (talk) 14:47, 4 September 2023 (UTC)
 * How can an English term be more common in languages other than English? CapitalSasha ~ talk 15:15, 4 September 2023 (UTC)
 * @CapitalSasha I was referring to the term "fundamental" obviously not the "sequence" part. "Fundamental sequence" is in German "Fundamentalfolge", in Russian "fundamental'naya posledovatel'nost" (that is even the Wikipedia name for the article) and in Italian "successione fondamentale". It seems to me that especially Russian authors use the name "fundamental sequence" rather than "Cauchy sequence".--Tensorproduct (talk) 15:53, 4 September 2023 (UTC)
 * Sure, but there are many words in math where the cognate of a foreign-language term is not used in English. For example, what we call a manifold is in French a variété.... Of course, this often leads to the cognate moving to English, as well, as you note. (I believe this is the origin of algebraic variety.) CapitalSasha ~ talk 16:18, 4 September 2023 (UTC)
 * @CapitalSasha But the word fundamental sequence is used in English literature as well (as for example in the books I have mentioned above). Also you can not compare this with "manifold", since "manifold" is a Germanic word and comes from middle English/German, that is why there is no word like that in French (in German there exists also the adjective "mannigfaltig"). Fundamental however comes from Latin and is used in English, French and German too.--Tensorproduct (talk) 17:03, 4 September 2023 (UTC)
 * Um, yes, I do believe what I wrote; otherwise, I wouldn't have written it. XOR'easter (talk) 03:09, 5 September 2023 (UTC)
 * let's try to reduce the temperature here, are there any objections to the current phrasing? CapitalSasha ~ talk 04:53, 5 September 2023 (UTC)
 * For me it's fine this way. I never cared about the placement of the word. JBL talks about "huffing and puffing", while the user could have just simply rephrased it the way the user wants it instead of doing these unjustified reverts. How was I supposed to know how the users wants the phrasing.
 * I find it strange that nobody has heard about this terminology. My research background is probability theory, but I have read a lot about analysis and I have seen many authors using this word.
 * I am not an expert on the history of math but I assume that a Cauchy sequence was also called "fundamental sequence" in the English-speaking world before the name Cauchy sequence was used, since as far as I know, it was the original name for such sequences.--Tensorproduct (talk) 07:47, 5 September 2023 (UTC)

title: Talk:Cauchy sequence


text:
Camille Pissarro

Jacob Abraham Camille Pissarro (, ; 10 July 1830 – 13 November 1903) was a Danish-French Impressionist and Neo-Impressionist painter born on the island of St Thomas (now in the US Virgin Islands, but then in the Danish West Indies). His importance resides in his contributions to both Impressionism and Post-Impressionism. Pissarro studied from great forerunners, including Gustave Courbet and Jean-Baptiste-Camille Corot. He later studied and worked alongside Georges Seurat and Paul Signac when he took on the Neo-Impressionist style at the age of 54.

In 1873 he helped establish a collective society of fifteen aspiring artists, becoming the "pivotal" figure in holding the group together and encouraging the other members. Art historian John Rewald called Pissarro the "dean of the Impressionist painters", not only because he was the oldest of the group, but also "by virtue of his wisdom and his balanced, kind, and warmhearted personality". Paul Cézanne said "he was a father for me. A man to consult and a little like the good Lord", and he was also one of Paul Gauguin's masters. Pierre-Auguste Renoir referred to his work as "revolutionary", through his artistic portrayals of the "common man", as Pissarro insisted on painting individuals in natural settings without "artifice or grandeur".

Pissarro is the only artist to have shown his work at all eight Paris Impressionist exhibitions, from 1874 to 1886. He "acted as a father figure not only to the Impressionists" but to all four of the major Post-Impressionists, Cézanne, Seurat, Gauguin, and van Gogh.

Early years
Jacob Abraham Camille Pissarro was born on 10 July 1830 on the island of St. Thomas to Frederick Abraham Gabriel Pissarro and Rachel Manzano-Pomié. His father was of Portuguese Jewish descent and held French nationality. His mother was from a French-Jewish family from St. Thomas. His father was a merchant who came to the island from France to deal with the hardware store of a deceased uncle, Isaac Petit, and married his widow. The marriage caused a stir within St. Thomas's small Jewish community because she was previously married to Frederick's uncle and according to Jewish law a man is forbidden from marrying his aunt. In subsequent years his four children attended the all-black primary school. Upon his death, his will specified that his estate be split equally between the synagogue and St. Thomas' Protestant church.

When Pissarro was twelve his father sent him to boarding school in France. He studied at the Savary Academy in Passy near Paris. While a young student, he developed an early appreciation of the French art masters. Monsieur Savary himself gave him a strong grounding in drawing and painting and suggested he draw from nature when he returned to St. Thomas.

After his schooling, Pissarro returned to St. Thomas at the age of sixteen or seventeen, where his father advocated Pissarro to work in his business as a port clerk. Nevertheless, Pissarro took every opportunity during those next five years at the job to practice drawing during breaks and after work.

Visual theorist Nicholas Mirzoeff claims that the young Pissarro was inspired by the artworks of James Gay Sawkins, a British painter and geologist who lived in Charlotte Amalie, St. Thomas circa 1847. Pissarro may have attended art classes taught by Sawkins and seen Sawkins's paintings of Mitla, Mexico. Mirzoeff states, "A formal analysis suggests that [Sawkins's] work influenced the young Pissarro, who had just returned to the island from his school in France. Soon afterward, Pissarro began his own drawings of the local African population in apparent imitation of Sawkins," creating "sketches for a postslavery imagination."

When Pissarro turned twenty-one, Danish artist Fritz Melbye, then living on St. Thomas, inspired him to take on painting as a full-time profession, becoming his teacher and friend. Pissarro then chose to leave his family and job and live in Venezuela, where he and Melbye spent the next two years working as artists in Caracas and La Guaira. He drew everything he could, including landscapes, village scenes, and numerous sketches, enough to fill up multiple sketchbooks.

Life in France
In 1855, Pissarro moved back to Paris where he began working as an assistant to Anton Melbye, Fritz Melbye's brother and also a painter. He also studied paintings by other artists whose style impressed him: Courbet, Charles-François Daubigny, Jean-François Millet, and Corot. He also enrolled in various classes taught by masters, at schools such as École des Beaux-Arts and Académie Suisse. But Pissarro eventually found their teaching methods "stifling," states art historian John Rewald. This prompted him to search for alternative instruction, which he requested and received from Corot.

Paris Salon and Corot's influence
His initial paintings were in accord with the standards at the time to be displayed at the Paris Salon, the official body whose academic traditions dictated the kind of art that was acceptable. The Salon's annual exhibition was essentially the only marketplace for young artists to gain exposure. As a result, Pissarro worked in the traditional and prescribed manner to satisfy the tastes of its official committee.

In 1859 his first painting was accepted and exhibited. His other paintings during that period were influenced by Camille Corot, who tutored him. He and Corot both shared a love of rural scenes painted from nature. It was by Corot that Pissarro was inspired to paint outdoors, also called "plein air" painting. Pissarro found Corot, along with the work of Gustave Courbet, to be "statements of pictorial truth," writes Rewald. He discussed their work often. Jean-François Millet was another whose work he admired, especially his "sentimental renditions of rural life".

Use of natural outdoor settings

During this period Pissarro began to understand and appreciate the importance of expressing on canvas the beauties of nature without adulteration. After a year in Paris, he therefore began to leave the city and paint scenes in the countryside to capture the daily reality of village life. He found the French countryside to be "picturesque," and worthy of being painted. It was still mostly agricultural and sometimes called the "golden age of the peasantry". Pissarro later explained the technique of painting outdoors to a student:

 * "Work at the same time upon sky, water, branches, ground, keeping everything going on an equal basis and unceasingly rework until you have got it. Paint generously and unhesitatingly, for it is best not to lose the first impression."

Corot would complete his paintings back in his studio, often revising them according to his preconceptions. Pissarro, however, preferred to finish his paintings outdoors, often at one sitting, which gave his work a more realistic feel. As a result, his art was sometimes criticised as being "vulgar," because he painted what he saw: "rutted and edged hodgepodge of bushes, mounds of earth, and trees in various stages of development." According to one source, such details were equivalent to today's art showing garbage cans or beer bottles on the side of a street. This difference in style created disagreements between Pissarro and Corot.

With Monet, Cézanne, and Guillaumin
In 1859, while attending the free school, the Académie Suisse, Pissarro became friends with a number of younger artists who likewise chose to paint in the more realistic style. Among them were Claude Monet, Armand Guillaumin and Paul Cézanne. What they shared in common was their dissatisfaction with the dictates of the Salon. Cézanne's work had been mocked at the time by the others in the school, and, writes Rewald, in his later years Cézanne "never forgot the sympathy and understanding with which Pissarro encouraged him." As a part of the group, Pissarro was comforted from knowing he was not alone, and that others similarly struggled with their art.

Pissarro agreed with the group about the importance of portraying individuals in natural settings, and expressed his dislike of any artifice or grandeur in his works, despite what the Salon demanded for its exhibits. In 1863 almost all of the group's paintings were rejected by the Salon, and French Emperor Napoleon III instead decided to place their paintings in a separate exhibit hall, the Salon des Refusés. However, only works of Pissarro and Cézanne were included, and the separate exhibit brought a hostile response from both the officials of the Salon and the public.

In subsequent Salon exhibits of 1865 and 1866, Pissarro acknowledged his influences from Melbye and Corot, whom he listed as his masters in the catalogue. But in the exhibition of 1868 he no longer credited other artists as an influence, in effect declaring his independence as a painter. This was noted at the time by art critic and author Émile Zola, who offered his opinion:

 * "Camille Pissarro is one of the three or four true painters of this day ... I have rarely encountered a technique that is so sure."

Another writer tries to describe elements of Pissarro's style:
 * "The brightness of his palette envelops objects in atmosphere ... He paints the smell of the earth."

And though, on orders from the hanging Committee and the Marquis de Chennevières, Pissarro's paintings of Pontoise for example had been skyed, hung near the ceiling, this did not prevent Jules-Antoine Castagnary from noting that the qualities of his paintings had been observed by art lovers. At the age of thirty-eight, Pissarro had begun to win himself a reputation as a landscapist to rival Corot and Daubigny.

In the late 1860s or early 1870s, Pissarro became fascinated with Japanese prints, which influenced his desire to experiment in new compositions. He described the art to his son Lucien:

 * "It is marvelous. This is what I see in the art of this astonishing people ... nothing that leaps to the eye, a calm, a grandeur, an extraordinary unity, a rather subdued radiance ..."

Marriage and children
In 1871 in Croydon, England, he married his mother's maid, Julie Vellay, a vineyard grower's daughter, with whom he had seven children, six of whom would become painters: Lucien Pissarro (1863–1944), Georges Henri Manzana Pissarro (1871–1961), Félix Pissarro (1874–1897), Ludovic-Rodo Pissarro (1878–1952), Jeanne Bonin-Pissarro (1881–1948), and Paul-Émile Pissarro (1884–1972). They lived outside Paris in Pontoise and later in Louveciennes, both of which places inspired many of his paintings including scenes of village life, along with rivers, woods, and people at work. He also kept in touch with the other artists of his earlier group, especially Monet, Renoir, Cézanne, and Frédéric Bazille.

The London years
After the outbreak of the Franco-Prussian War of 1870–71, having only Danish nationality and being unable to join the army, he moved his family to Norwood, then a village on the edge of London. However, his style of painting, which was a forerunner of what was later called "Impressionism", did not do well. He wrote to his friend, Théodore Duret, that "my painting doesn't catch on, not at all ..."

Pissarro met the Paris art dealer Paul Durand-Ruel, in London, who became the dealer who helped sell his art for most of his life. Durand-Ruel put him in touch with Monet who was likewise in London during this period. They both viewed the work of British landscape artists John Constable and J. M. W. Turner, which confirmed their belief that their style of open air painting gave the truest depiction of light and atmosphere, an effect that they felt could not be achieved in the studio alone. Pissarro's paintings also began to take on a more spontaneous look, with loosely blended brushstrokes and areas of impasto, giving more depth to the work.

Paintings
Through the paintings Pissarro completed at this time, he records Sydenham and the Norwoods at a time when they were just recently connected by railways, but prior to the expansion of suburbia. One of the largest of these paintings is a view of St. Bartholomew's Church at Lawrie Park Avenue, commonly known as The Avenue, Sydenham, in the collection of the National Gallery in London. Twelve oil paintings date from his stay in Upper Norwood and are listed and illustrated in the catalogue raisonné prepared jointly by his fifth child Ludovic-Rodolphe Pissarro and Lionello Venturi and published in 1939. These paintings include Lower Norwood Under Snow, and Lordship Lane Station, views of The Crystal Palace relocated from Hyde Park, Dulwich College, Sydenham Hill, All Saints Church Upper Norwood, and a lost painting of St. Stephen's Church.

Returning to France, Pissarro lived in Pontoise from 1872 to 1884. In 1890 he again visited England and painted some ten scenes of central London. He came back again in 1892, painting in Kew Gardens and Kew Green, and also in 1897, when he produced several oils described as being of Bedford Park, Chiswick, but in fact all being of the nearby Stamford Brook area except for one of Bath Road, which runs from Stamford Brook along the south edge of Bedford Park.

French Impressionism
When Pissarro returned to his home in France after the war, he discovered that of the 1,500 paintings he had done over 20 years, which he was forced to leave behind when he moved to London, only 40 remained. The rest had been damaged or destroyed by the soldiers, who often used them as floor mats outside in the mud to keep their boots clean. It is assumed that many of those lost were done in the Impressionist style he was then developing, thereby "documenting the birth of Impressionism." Armand Silvestre, a critic, went so far as to call Pissarro "basically the inventor of this [Impressionist] painting"; however, Pissarro's role in the Impressionist movement was "less that of the great man of ideas than that of the good counselor and appeaser ..." "Monet ... could be seen as the guiding force."

He soon reestablished his friendships with the other Impressionist artists of his earlier group, including Cézanne, Monet, Manet, Renoir, and Degas. Pissarro now expressed his opinion to the group that he wanted an alternative to the Salon so their group could display their own unique styles.

To assist in that endeavour, in 1873 he helped establish a separate collective, called the "Société Anonyme des Artistes, Peintres, Sculpteurs et Graveurs," which included fifteen artists. Pissarro created the group's first charter and became the "pivotal" figure in establishing and holding the group together. One writer noted that with his prematurely grey beard, the forty-three-year-old Pissarro was regarded as a "wise elder and father figure" by the group. Yet he was able to work alongside the other artists on equal terms due to his youthful temperament and creativity. Another writer said of him that "he has unchanging spiritual youth and the look of an ancestor who remained a young man".

Impressionist exhibitions that shocked the critics
The following year, in 1874, the group held their First Impressionist Exhibition, which shocked and "horrified" the critics, who primarily appreciated only scenes portraying religious, historical, or mythological settings. They found fault with the Impressionist paintings on many grounds:
 * The subject matter was considered "vulgar" and "commonplace," with scenes of street people going about their everyday lives. Pissarro's paintings, for instance, showed scenes of muddy, dirty, and unkempt settings;
 * The manner of painting was too sketchy and looked incomplete, especially compared to the traditional styles of the period. The use of visible and expressive brushwork by all the artists was considered an insult to the craft of traditional artists, who often spent weeks on their work. Here, the paintings were often done in one sitting and the paints were applied wet-on-wet;
 * The use of color by the Impressionists relied on new theories they developed, such as having shadows painted with the reflected light of surrounding, and often unseen, objects.

A "revolutionary" style
Pissarro showed five of his paintings, all landscapes, at the exhibit, and again Émile Zola praised his art and that of the others. In the Impressionist exhibit of 1876, however, art critic Albert Wolff complained in his review, "Try to make M. Pissarro understand that trees are not violet, that sky is not the color of fresh butter ..." Journalist and art critic Octave Mirbeau on the other hand, writes, "Camille Pissarro has been a revolutionary through the revitalized working methods with which he has endowed painting". According to Rewald, Pissarro had taken on an attitude more simple and natural than the other artists. He writes:

 * "Rather than glorifying—consciously or not—the rugged existence of the peasants, he placed them without any 'pose' in their habitual surroundings, thus becoming an objective chronicler of one of the many facets of contemporary life."

In later years, Cézanne also recalled this period and referred to Pissarro as "the first Impressionist". In 1906, a few years after Pissarro's death, Cézanne, then 67 and a role model for the new generation of artists, paid Pissarro a debt of gratitude by having himself listed in an exhibition catalogue as "Paul Cézanne, pupil of Pissarro".

Pissarro, Degas, and American impressionist Mary Cassatt planned a journal of their original prints in the late 1870s, a project that nevertheless came to nothing when Degas withdrew. Art historian and the artist's great-grandson Joachim Pissarro notes that they "professed a passionate disdain for the Salons and refused to exhibit at them." Together they shared an "almost militant resolution" against the Salon, and through their later correspondences it is clear that their mutual admiration "was based on a kinship of ethical as well as aesthetic concerns".

Cassatt had befriended Degas and Pissarro years earlier when she joined Pissarro's newly formed French Impressionist group and gave up opportunities to exhibit in the United States. She and Pissarro were often treated as "two outsiders" by the Salon since neither were French or had become French citizens. However, she was "fired up with the cause" of promoting Impressionism and looked forward to exhibiting "out of solidarity with her new friends". Towards the end of the 1890s she began to distance herself from the Impressionists, avoiding Degas at times as she did not have the strength to defend herself against his "wicked tongue". Instead, she came to prefer the company of "the gentle Camille Pissarro", with whom she could speak frankly about the changing attitudes toward art. She once described him as a teacher "that could have taught the stones to draw correctly."

Neo-Impressionist period
By the 1880s, Pissarro began to explore new themes and methods of painting to break out of what he felt was an artistic "mire". As a result, Pissarro went back to his earlier themes by painting the life of country people, which he had done in Venezuela in his youth. Degas described Pissarro's subjects as "peasants working to make a living".

However, this period also marked the end of the Impressionist period due to Pissarro's leaving the movement. As Joachim Pissarro points out:

"Once such a die-hard Impressionist as Pissarro had turned his back on Impressionism, it was apparent that Impressionism had no chance of surviving ..."

It was Pissarro's intention during this period to help "educate the public" by painting people at work or at home in realistic settings, without idealising their lives. Pierre-Auguste Renoir, in 1882, referred to Pissarro's work during this period as "revolutionary," in his attempt to portray the "common man." Pissarro himself did not use his art to overtly preach any kind of political message, however, although his preference for painting humble subjects was intended to be seen and purchased by his upper class clientele. He also began painting with a more unified brushwork along with pure strokes of color.

Studying with Seurat and Signac
In 1885 he met Georges Seurat and Paul Signac, both of whom relied on a more "scientific" theory of painting by using very small patches of pure colours to create the illusion of blended colours and shading when viewed from a distance. Pissarro then spent the years from 1885 to 1888 practising this more time-consuming and laborious technique, referred to as pointillism. The paintings that resulted were distinctly different from his Impressionist works, and were on display in the 1886 Impressionist Exhibition, but under a separate section, along with works by Seurat, Signac, and his son Lucien.

All four works were considered an "exception" to the eighth exhibition. Joachim Pissarro notes that virtually every reviewer who commented on Pissarro's work noted "his extraordinary capacity to change his art, revise his position and take on new challenges." One critic writes:

 * "It is difficult to speak of Camille Pissarro ... What we have here is a fighter from way back, a master who continually grows and courageously adapts to new theories."

Pissarro explained the new art form as a "phase in the logical march of Impressionism", but he was alone among the other Impressionists with this attitude, however. Joachim Pissarro states that Pissarro thereby became the "only artist who went from Impressionism to Neo-Impressionism".

In 1884, art dealer Theo van Gogh asked Pissarro if he would take in his older brother, Vincent, as a boarder in his home. Lucien Pissarro wrote that his father was impressed by Van Gogh's work and had "foreseen the power of this artist", who was 23 years younger. Although Van Gogh never boarded with him, Pissarro did explain to him the various ways of finding and expressing light and color, ideas which he later used in his paintings, notes Lucien.

Abandoning Neo-Impressionism
Pissarro eventually turned away from Neo-Impressionism, claiming its system was too artificial. He explains in a letter to a friend:

 * "Having tried this theory for four years and having then abandoned it ... I can no longer consider myself one of the neo-impressionists ... It was impossible to be true to my sensations and consequently to render life and movement, impossible to be faithful to the effects, so random and so admirable, of nature, impossible to give an individual character to my drawing, [that] I had to give up."

However, after reverting to his earlier style, his work became, according to Rewald, "more subtle, his color scheme more refined, his drawing firmer ... So it was that Pissarro approached old age with an increased mastery."

But the change also added to Pissarro's continual financial hardship which he felt until his 60s. His "headstrong courage and a tenacity to undertake and sustain the career of an artist", writes Joachim Pissarro, was due to his "lack of fear of the immediate repercussions" of his stylistic decisions. In addition, his work was strong enough to "bolster his morale and keep him going", he writes. His Impressionist contemporaries, however, continued to view his independence as a "mark of integrity", and they turned to him for advice, referring to him as "Père Pissarro" (father Pissarro).

Later years

In his older age Pissarro suffered from a recurring eye infection that prevented him from working outdoors except in warm weather. As a result of this disability, he began painting outdoor scenes while sitting by the window of hotel rooms. He often chose hotel rooms on upper levels to get a broader view. He moved around northern France and painted from hotels in Rouen, Paris, Le Havre and Dieppe. On his visits to London, he would do the same.

Pissarro died in Paris on 13 November 1903 and was buried in Père Lachaise Cemetery.

Legacy and influence
During the period Pissarro exhibited his works, art critic Armand Silvestre had called Pissarro the "most real and most naive member" of the Impressionist group. His work has also been described by art historian Diane Kelder as expressing "the same quiet dignity, sincerity, and durability that distinguished his person." She adds that "no member of the group did more to mediate the internecine disputes that threatened at times to break it apart, and no one was a more diligent proselytizer of the new painting."

According to Pissarro's son, Lucien, his father painted regularly with Cézanne beginning in 1872. He recalls that Cézanne walked a few miles to join Pissarro at various settings in Pontoise. While they shared ideas during their work, the younger Cézanne wanted to study the countryside through Pissarro's eyes, as he admired Pissarro's landscapes from the 1860s. Cézanne, although only nine years younger than Pissarro, said that "he was a father for me. A man to consult and a little like the good Lord."

Lucien Pissarro was taught painting by his father, and described him as a "splendid teacher, never imposing his personality on his pupil." Gauguin, who also studied under him, referred to Pissarro "as a force with which future artists would have to reckon". Art historian Diane Kelder notes that it was Pissarro who introduced Gauguin, who was then a young stockbroker studying to become an artist, to Degas and Cézanne. Gauguin, near the end of his career, wrote a letter to a friend in 1902, shortly before Pissarro's death:

 * "If we observe the totality of Pissarro's work, we find there, despite fluctuations, not only an extreme artistic will, never belied, but also an essentially intuitive, purebred art ... He was one of my masters and I do not deny him."

The American impressionist Mary Cassatt, who at one point lived in Paris to study art, and joined his Impressionist group, noted that he was "such a teacher that he could have taught the stones to draw correctly."

Caribbean author and scholar Derek Walcott based his book-length poem, Tiepolo's Hound (2000), on Pissarro's life.

Camille Pissarro is a pivotal character in the historical fiction novels, The Dream Collector, Books I & II by R.w. Meek, depicting his major role among the Impressionists and his open-mindedness toward the Post-Impressionist art of George Seurat, Paul Gauguin and Vincent van Gogh.

The legacy of Nazi-looted Pissarros
During the early 1930s throughout Europe, Jewish owners of numerous fine art masterpieces found themselves forced to give up or sell off their collections for minimal prices due to anti-Jewish laws created by the new Nazi regime. Many Jews were forced to flee Germany starting in 1933, and then, as the Nazis expanded their hold over all of Europe, Austria, France, Holland, Poland, Italy and other countries. The Nazis created special looting organizations like the Reichsleiter Rosenberg Taskforce whose mission it was to seize Jewish property notably valuable artworks. When those forced into exile or deported to extermination camps owned valuables, including artwork, they were often sold to finance the Nazi war effort, sent to Hitler's personal museum, traded or seized by officials for personal gain. Several artworks by Pissarro were looted from their Jewish owners in Germany, France and elsewhere by the Nazis.

Pissarro's Shepherdess Bringing Home the Sheep (La Bergère Rentrant des Moutons") was looted from the Jewish art collectors Yvonne et Raoul Meyer in France in 1941 and transited via Switzerland and New York before entering the Fred Jones Jr Museum at the University of Oklahoma.   In 2014, Meyer's daughter, Léonie-Noëlle Meyer filed a restitution claim which resulted in years of court battle. The lawsuit resulted in the recognition of Meyer's ownership and its transfer to France for five years, coupled with an agreement to shuttle the painting back and forth between Paris and Oklahoma every three years after that.   However, in 2020 Meyer filed suit in a French court to challenge the accord.  After Fred Jones Jr Museum sued Meyer requesting heavy financial penalties, the Holocaust survivor abandoned her effort to recover the Pissarro, saying, "I have no other choice.

Pissarro's Picking Peas (La Cueillette) was looted from Jewish businessman Simon Bauer, in addition to 92 other artworks seized in 1943 by the Vichy collaborationist regime in France.

Pissarro's Sower And Ploughman, was owned by Dr Henri Hinrichsen, a Jewish music publisher from Leipzig, until 11 January 1940, when he was forced to relinquish the painting to Hildebrand Gurlitt in Nazi-occupied Brussels, before being murdered in Auschwitz in September 1942.

Pissarro's "Le Quai Malaquais, Printemps", owned by German Jewish publisher Samuel Fischer, founder of the famous S. Fischer Verlag, passed through the hands of infamous Nazi art looter Bruno Lohse.

Pissarro's Le Boulevard de Montmartre, Matinée de Printemps, owned by Max Silberberg, a German Jewish industrialist whose renowned art collection was considered "one of the best in pre-war Germany", was seized and sold in a forced auction before Silberberg and his wife Johanna were murdered in Auschwitz.

In the decades after World War II, many art masterpieces were found on display in various galleries and museums in Europe and the United States, often with false provenances and labels missing. Some, as a result of legal action, were later returned to the families of the original owners. Many of the recovered paintings were then donated to the same or other museums as a gift. One such lost piece, Pissarro's 1897 oil painting, Rue St. Honoré, Apres Midi, Effet de Pluie, was discovered hanging at Madrid's government-owned museum, the Museo Thyssen-Bornemisza. In January 2011 the Spanish government denied a request by the US ambassador to return the painting. At the subsequent trial in Los Angeles, the court ruled that the Thyssen-Bornemisza Collection Foundation was the rightful owner. In 1999, Pissarro's 1897 Le Boulevard de Montmartre, Matinée de Printemps appeared in the Israel Museum in Jerusalem, its donor having been unaware of its pre-war provenance. In January 2012, Le Marché aux Poissons (The Fish Market), a color monotype, was returned after 30 years.

During his lifetime, Camille Pissarro sold few of his paintings. By the 21st century, however, his paintings were selling for millions. An auction record for the artist was set on 6 November 2007 at Christie's in New York, where a group of four paintings, Les Quatre Saisons (the Four Seasons), sold for $14,601,000 (estimate $12,000,000 – $18,000,000). In November 2009 Le Pont Boieldieu et la Gare d'Orléans, Rouen, Soleil sold for $7,026,500 at Sotheby's in New York.

In February 2014 the 1897 Le Boulevard de Montmartre, Matinée de Printemps, originally owned by the German industrialist and Holocaust victim Max Silberberg (de), sold at Sotheby's in London for £19.9M, nearly five times the previous record.

In October 2021 Berlin's Alte Nationalgalerie restituted Pissarro's "A Square in La Roche-Guyon" (1867) to the heirs of Armand Dorville, a French Jewish art collector whose family was persecuted by the Nazis and whose paintings had been sold at a 1942 auction in Nice that was overseen by the Commissariat Général aux Questions Juives. The museum then purchased the Pissarro back.

A family of painters
Camille's son Lucien was an Impressionist and Neo-impressionist painter as were his second and third sons Georges Henri Manzana Pissarro and Félix Pissarro. Lucien's daughter Orovida Pissarro was also a painter. Camille's great-grandson, Joachim Pissarro, became Head Curator of Drawing and Painting at the Museum of Modern Art in New York City and a professor in Hunter College's Art Department. Camille's great-granddaughter, Lélia Pissarro, has had her work exhibited alongside her great-grandfather. Another great-granddaughter, Julia Pissarro, a Barnard College graduate, is also active in the art scene. From the only daughter of Camille, Jeanne Pissarro, other painters include Henri Bonin-Pissarro (1918–2003) and Claude Bonin-Pissarro (born 1921), who is the father of the Abstract artist Frédéric Bonin-Pissarro (born 1964).

The grandson of Camille Pissarro, Hugues Claude Pissarro (dit Pomié), was born in 1935 in the western section of Paris, Neuilly-sur-Seine, and began to draw and paint as a young child under his father's tutelage. During his adolescence and early twenties he studied the works of the great masters at the Louvre. His work has been featured in exhibitions in Europe and the United States, and he was commissioned by the White House in 1959 to paint a portrait of U.S. President Dwight Eisenhower. He now lives and paints in Donegal, Ireland, with his wife Corinne also an accomplished artist and their children.

List of paintings

 * The Banks of the Oise near Pontoise 1873, Indianapolis Museum of Art
 * Pont Boieldieu in Rouen, Rainy Weather, 1896, Art Gallery of Ontario
 * Steamboats in the Port of Rouen, 1896, Metropolitan Museum of Art
 * Le Boulevard de Montmartre, Matinée de Printemps, view from window, 1897, private collection
 * Hay Harvest at Éragny, 1901, National Gallery of Canada
 * Self-portrait, 1903, Tate Gallery, London

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Camille Pissarro


text:
Cardiology diagnostic tests and procedures

The diagnostic tests in cardiology are methods of identifying heart conditions associated with healthy vs. unhealthy, pathologic heart function.

History
Obtaining a medical history is always the first "test", part of understanding the likelihood of significant disease, as detectable within the current limitations of clinical medicine. Yet heart problems often produce no symptoms until very advanced, and many symptoms, such as palpitations and sensations of extra or missing heart beats correlate poorly with relative heart health vs disease. Hence, a history alone is rarely sufficient to diagnose a heart condition.

Auscultation
Auscultation employs a stethoscope to more easily hear various normal and abnormal sounds, such as normal heart beat sounds and the usual heart beat sound changes associated with breathing versus heart murmurs.

Blood tests
A variety of blood tests are available for analyzing cholesterol transport behavior, HDL, LDL, triglycerides, lipoprotein little a, homocysteine, C-reactive protein, blood sugar control: fasting, after eating or averages using glycated albumen or hemoglobin, myoglobin, creatine kinase, troponin, brain-type natriuretic peptide, etc. to assess the evolution of coronary artery disease and evidence of existing damage. A great many more physiologic markers related to atherosclerosis and heart function are used and being developed and evaluated in research.

(*) due to the high cost, LDL is usually calculated instead of being measured directly source: Beyond Cholesterol, Julius Torelli MD, 2005 ISBN 0-312-34863-0

Electrocardiogram
Electrocardiography (ECG/EKG in German vernacular. Elektrokardiogram) monitors electrical activity of the heart, primarily as recorded from the skin surface. A 12 lead recording, recording the electrical activity in three planes, anterior, posterior, and lateral is the most commonly used form. The ECG allows observation of the heart electrical activity by visualizing waveform beat origin (typically from the sinoatrial or SA node) following down the bundle of HIS and ultimately stimulating the ventricles to contract forcing blood through the body. Much can be learned by observing the QRS morphology (named for the respective portions of the polarization/repolarization waveform of the wave, P,Q,R,S,T wave). Rhythm abnormalities can also be visualized as in slow heart rate bradycardia, or fast heart rate tachycardia.

Fasegram
A Fasegraphy allows expanding the system of Electrocardiography diagnostic features, based on the evaluation of the speed characteristics of the process, and thereby increasing the sensitivity and specificity of ECG-diagnostics.

Fasegraphy allows determining the initial features of changes in the cardiac muscle, even on a single-channel ECG, which are underestimated in traditional ECG diagnostics.

Holter monitor
A Holter monitor records a continuous EKG rhythm pattern (rarely a full EKG) for 24 hours or more. These monitors are used for suspected frequent rhythm abnormalities, especially ones the wearer may not recognize by symptoms. They are more expensive than event monitors.

Event monitor
An event monitor records short term EKG rhythm patterns, generally storing the last 2 to 5 minutes, adding in new and discarding old data, for 1 to 2 weeks or more. There are several different types with different capabilities. When the wearer presses a button on the monitor, it quits discarding old and continues recording for a short additional period. The wearer then plays the recording, via a standard phone connection, to a center with compatible receiving and rhythm printing equipment, after which the monitor is ready to record again. These monitors are used for suspected infrequent rhythm abnormalities, especially ones the wearer does recognize by symptoms. They are less expensive than Holter monitors.

Cardiac stress testing
Cardiac stress testing is used to determine to assess cardiac function and to disclose evidence of exertion-related cardiac hypoxia. Radionuclide testing using thallium or technetium can be used to demonstrate areas of perfusion abnormalities. With a maximal stress test the level of exercise is increased until the person's heart rate will not increase any higher, despite increased exercise. A fairly accurate estimate of the target heart rate, based on extensive clinical research, can be estimated by the formula 220 beats per minute minus patient's age. This linear relation is accurate up to about age 30, after which it mildly underestimates typical maximum attainable heart rates achievable by healthy individuals. Other formulas exist, such as that by Miller (217 - (0.85 × Age)) and others. Achieving a high enough heart rate at the end of exercise is critical to improving the sensitivity of the test to detect high grade heart artery stenosis. High frequency analysis of the QRS complex may be useful for detection of coronary artery disease during an exercise stress test.

Electrophysiology study
The electrophysiology study or EP study is the end all of electrophysiological tests of the heart. It involves a catheter with electrodes probing the endocardium, the inside of the heart, and testing the conduction pathways and electrical activity of individual areas of the heart.

Medical imaging
Cardiac imaging techniques include coronary catheterization, echocardiogram, intravascular ultrasound, retinal vessel analysis and the coronary calcium scan.


text:
Carlo Collodi

Carlo Lorenzini (24 November 1826 – 26 October 1890), better known by the pen name Carlo Collodi, was an Italian author, humourist, and journalist, widely known for his fairy tale novel The Adventures of Pinocchio.

Early life
Collodi was born in Florence on 24 November 1826. His mother Angiolina Orzali Lorenzini was a seamstress from Collodi, the town from which he later took the pen name, and his father Domenico Lorenzini was a cook. Both parents worked for the marchese Ginori Lisci. Carlo was the eldest child in the family and he had ten siblings; seven died at a young age. He spent most of his childhood in the town of Collodi where his mother was born. He lived there with his maternal grandmother. After attending primary school, he was sent to study at a theological seminary in Colle Val d’Elsa. An account at the seminary shows that the marchese had offered financial aid, but the boy found that he did not want to be a priest so he continued his education at the College of the Scolopi Fathers in Florence. In 1844, he started working at the Florentine bookstore Libreria Piatti, where he assisted Giuseppe Aiazzi, a prominent Italian manuscript specialist.

Career
During the Italian Wars of Independence in 1848 and 1860, Collodi served as a volunteer with the Tuscan Army. His active interest in political matters can be seen in his earliest literary works, as well as in the founding of the satirical newspaper Il Lampione in 1853. This newspaper was censored by order of the Grand Duke of Tuscany. In 1854, he published his second newspaper, Lo scaramuccia ("The Controversy"). Lorenzini's first publications were in his periodicals. A debut came in 1856 with the play Gli amici di casa and parodic guidebook Un romanzo in vapore, both in 1856. By 1860, he published his first notable work called Il signor Alberi ha ragione! (Mr. Alberi Is Right!), which outlined his political and cultural vision of Italy. This is the text where Lorenzini started using the Collodi pseudonym, which was taken from his mother's hometown.

Collodi had also begun intense activity on other political newspapers such as Il Fanfulla; at the same time he was employed by the Censorship Commission for the Theatre. During this period he composed various satirical sketches and stories (sometimes simply by collating earlier articles), including Macchiette (1880), Occhi e nasi (1881), and Storie allegre (1887).

Collodi became disenchanted with Italian politics afterwards, so he turned to children's literature and his first works involved translating French fairy tales into Italian. In 1875, for instance, he completed Racconti delle fate, a translation of French fairy tales by Charles Perrault. In 1876, Collodi wrote Giannettino (inspired by Alessandro Luigi Parravicini's Giannetto), the Minuzzolo, and Il viaggio per l'Italia di Giannettino, a pedagogic series which explored the unification of Italy through the ironic thoughts and actions of the character Giannettino.

Lorenzini became fascinated by the idea of using an amiable, rascally character as a means of expressing his own convictions through allegory. In 1880, he began writing Storia di un burattino (Story of a Marionette), also called Le avventure di Pinocchio, which was published weekly in Giornale per i bambini. Pinocchio was adapted into a 1940 film by Disney that is considered to be one of Disney's greatest films.

Collodi died suddenly in Florence on 26 October 1890 at the age of 63 and is interred at Cimitero Monumentale Delle Porte Sante in Florence. The National Carlo Collodi Foundation was established in 1962 to promote education and the works of Collodi, and Pinocchio Park, which was opened in 1956 in the town of Collodi and remains a popular attraction today.

title: Carlo Collodi


text:
Constructible number

In geometry and algebra, a real number $$r$$ is constructible if and only if, given a line segment of unit length, a line segment of length $$|r|$$ can be constructed with compass and straightedge in a finite number of steps. Equivalently, $$r$$ is constructible if and only if there is a closed-form expression for $$r$$ using only integers and the operations for addition, subtraction, multiplication, division, and square roots.

The geometric definition of constructible numbers motivates a corresponding definition of constructible points, which can again be described either geometrically or algebraically. A point is constructible if it can be produced as one of the points of a compass and straight edge construction (an endpoint of a line segment or crossing point of two lines or circles), starting from a given unit length segment. Alternatively and equivalently, taking the two endpoints of the given segment to be the points (0, 0) and (1, 0) of a Cartesian coordinate system, a point is constructible if and only if its Cartesian coordinates are both constructible numbers. Constructible numbers and points have also been called ruler and compass numbers and ruler and compass points, to distinguish them from numbers and points that may be constructed using other processes.

The set of constructible numbers forms a field: applying any of the four basic arithmetic operations to members of this set produces another constructible number. This field is a field extension of the rational numbers and in turn is contained in the field of algebraic numbers. It is the Euclidean closure of the rational numbers, the smallest field extension of the rationals that includes the square roots of all of its positive numbers.

The proof of the equivalence between the algebraic and geometric definitions of constructible numbers has the effect of transforming geometric questions about compass and straightedge constructions into algebra, including several famous problems from ancient Greek mathematics. The algebraic formulation of these questions led to proofs that their solutions are not constructible, after the geometric formulation of the same problems previously defied centuries of attack.

Geometrically constructible points
Let $$O$$ and $$A$$ be two given distinct points in the Euclidean plane, and define $$S$$ to be the set of points that can be constructed with compass and straightedge starting with $$O$$ and $$A$$. Then the points of $$S$$ are called constructible points. $$O$$ and $$A$$ are, by definition, elements of $$S$$. To more precisely describe the remaining elements of $$S$$, make the following two definitions: Then, the points of $$S$$, besides $$O$$ and $$A$$ are:
 * a line segment whose endpoints are in $$S$$ is called a constructed segment, and
 * a circle whose center is in $$S$$ and which passes through a point of $$S$$ (alternatively, whose radius is the distance between some pair of distinct points of $$S$$) is called a constructed circle.
 * the intersection of two non-parallel constructed segments, or lines through constructed segments,
 * the intersection points of a constructed circle and a constructed segment, or line through a constructed segment, or
 * the intersection points of two distinct constructed circles.

As an example, the midpoint of constructed segment $$OA$$ is a constructible point. One construction for it is to construct two circles with $$OA$$ as radius, and the line through the two crossing points of these two circles. Then the midpoint of segment $$OA$$ is the point where this segment is crossed by the constructed line.

Geometrically constructible numbers
The starting information for the geometric formulation can be used to define a Cartesian coordinate system in which the point $$O$$ is associated to the origin having coordinates $$(0,0)$$ and in which the point $$A$$ is associated with the coordinates $$(1, 0)$$. The points of $$S$$ may now be used to link the geometry and algebra by defining a constructible number to be a coordinate of a constructible point.

Equivalent definitions are that a constructible number is the $$x$$-coordinate of a constructible point $$(x,0)$$ or the length of a constructible line segment. In one direction of this equivalence, if a constructible point has coordinates $$(x,y)$$, then the point $$(x,0)$$ can be constructed as its perpendicular projection onto the $$x$$-axis, and the segment from the origin to this point has length $$x$$. In the reverse direction, if $$x$$ is the length of a constructible line segment, then intersecting the $$x$$-axis with a circle centered at $$O$$ with radius $$x$$ gives the point $$(x,0)$$. It follows from this equivalence that every point whose Cartesian coordinates are geometrically constructible numbers is itself a geometrically constructible point. For, when $$x$$ and $$y$$ are geometrically constructible numbers, point $$(x,y)$$ can be constructed as the intersection of lines through $$(x,0)$$ and $$(0,y)$$, perpendicular to the coordinate axes.

Algebraically constructible numbers
The algebraically constructible real numbers are the subset of the real numbers that can be described by formulas that combine integers using the operations of addition, subtraction, multiplication, multiplicative inverse, and square roots of positive numbers. Even more simply, at the expense of making these formulas longer, the integers in these formulas can be restricted to be only 0 and 1. For instance, the square root of 2 is constructible, because it can be described by the formulas $$\sqrt2$$ or $$\sqrt{1+1}$$.

Analogously, the algebraically constructible complex numbers are the subset of complex numbers that have formulas of the same type, using a more general version of the square root that is not restricted to positive numbers but can instead take arbitrary complex numbers as its argument, and produces the principal square root of its argument. Alternatively, the same system of complex numbers may be defined as the complex numbers whose real and imaginary parts are both constructible real numbers. For instance, the complex number $$i$$ has the formulas $$\sqrt{-1}$$ or $$\sqrt{0-1}$$, and its real and imaginary parts are the constructible numbers 0 and 1 respectively.

These two definitions of the constructible complex numbers are equivalent. In one direction, if $$q=x+iy$$ is a complex number whose real part $$x$$ and imaginary part $$y$$ are both constructible real numbers, then replacing $$x$$ and $$y$$ by their formulas within the larger formula $$x+y\sqrt{-1}$$ produces a formula for $$q$$ as a complex number. In the other direction, any formula for an algebraically constructible complex number can be transformed into formulas for its real and imaginary parts, by recursively expanding each operation in the formula into operations on the real and imaginary parts of its arguments, using the expansions
 * $$(a+ib)\pm (c+id)=(a \pm c)+i(b \pm d)$$
 * $$(a+ib)(c+id)=(ac-bd) + i(ad+bc)$$
 * $$\frac{1}{a+ib}=\frac{a}{a^2+b^2} + i \frac{-b}{a^2+b^2}$$
 * $$\sqrt{a+ib} = \frac{(a+r)\sqrt{r}}{s} + i\frac{b\sqrt{r}}{s}$$, where $$r=\sqrt{a^2+b^2{}_{\!}}$$ and $$s=\sqrt{(a+r)^2+b^2}$$.

Algebraically constructible points
The algebraically constructible points may be defined as the points whose two real Cartesian coordinates are both algebraically constructible real numbers. Alternatively, they may be defined as the points in the complex plane given by algebraically constructible complex numbers. By the equivalence between the two definitions for algebraically constructible complex numbers, these two definitions of algebraically constructible points are also equivalent.

Equivalence of algebraic and geometric definitions
If $$a$$ and $$b$$ are the non-zero lengths of geometrically constructed segments then elementary compass and straightedge constructions can be used to obtain constructed segments of lengths $$a+b$$, $$|a-b|$$, $$ab$$, and $$a/b$$. The latter two can be done with a construction based on the intercept theorem. A slightly less elementary construction using these tools is based on the geometric mean theorem and will construct a segment of length $$\sqrt{a}$$ from a constructed segment of length $$a$$. It follows that every algebraically constructible number is geometrically constructible, by using these techniques to translate a formula for the number into a construction for the number.

In the other direction, a set of geometric objects may be specified by algebraically constructible real numbers: coordinates for points, slope and $$y$$-intercept for lines, and center and radius for circles. It is possible (but tedious) to develop formulas in terms of these values, using only arithmetic and square roots, for each additional object that might be added in a single step of a compass-and-straightedge construction. It follows from these formulas that every geometrically constructible number is algebraically constructible.

Algebraic properties
The definition of algebraically constructible numbers includes the sum, difference, product, and multiplicative inverse of any of these numbers, the same operations that define a field in abstract algebra. Thus, the constructible numbers (defined in any of the above ways) form a field. More specifically, the constructible real numbers form a Euclidean field, an ordered field containing a square root of each of its positive elements. Examining the properties of this field and its subfields leads to necessary conditions on a number to be constructible, that can be used to show that specific numbers arising in classical geometric construction problems are not constructible.

It is convenient to consider, in place of the whole field of constructible numbers, the subfield $$\mathbb{Q}(\gamma)$$ generated by any given constructible number $$\gamma$$, and to use the algebraic construction of $$\gamma$$ to decompose this field. If $$\gamma$$ is a constructible real number, then the values occurring within a formula constructing it can be used to produce a finite sequence of real numbers $$\alpha_1,\dots, \alpha_n=\gamma$$ such that, for each $$i$$, $$\mathbb{Q}(\alpha_1,\dots,\alpha_i)$$ is an extension of $$\mathbb{Q}(\alpha_1,\dots,\alpha_{i-1})$$ of degree 2. Using slightly different terminology, a real number is constructible if and only if it lies in a field at the top of a finite tower of real quadratic extensions, $$\mathbb{Q} = K_0 \subseteq K_1 \subseteq \dots \subseteq K_n,$$ starting with the rational field $$\mathbb{Q}$$ where $$\gamma$$ is in $$K_n$$ and for all $$0< j\le n$$, $$[K_j:K_{j-1}]=2$$. It follows from this decomposition that the degree of the field extension $$[\mathbb{Q}(\gamma):\mathbb{Q}]$$ is $$2^r$$, where $$r$$ counts the number of quadratic extension steps.

Analogously to the real case, a complex number is constructible if and only if it lies in a field at the top of a finite tower of complex quadratic extensions. More precisely, $$\gamma$$ is constructible if and only if there exists a tower of fields $$\mathbb{Q} = F_0 \subseteq F_1 \subseteq \dots \subseteq F_n,$$ where $$\gamma$$ is in $$F_n$$, and for all $$0<j\le n$$, $$[F_j:F_{j-1}]= 2$$. The difference between this characterization and that of the real constructible numbers is only that the fields in this tower are not restricted to being real. Consequently, if a complex number a complex number $$\gamma$$ is constructible, then the above characterization implies that $$[\mathbb{Q}(\gamma):\mathbb{Q}]$$ is a power of two. However, this condition is not sufficient - there exist field extensions whose degree is a power of two, but which cannot be factored into a sequence of quadratic extensions.

To obtain a sufficient condition for constructibility, one must instead consider the splitting field $$K=\mathbb{Q}(\gamma,\gamma',\gamma'',\dots)$$ obtained by adjoining all roots of the minimal polynomial of $$\gamma$$. If the degree of extension is a power of two, then its Galois group $$G=\mathrm{Gal}(K/\mathbb{Q})$$ is a 2-group, and thus admits a descending sequence of subgroups $$G = G_n \supseteq G_{n-1} \supseteq \cdots \supseteq G_0 = 1,$$ with $$|G_k| = 2^k$$ for $$0\leq k \leq n.$$ By the fundamental theorem of Galois theory, there is a corresponding tower of quadratic extensions $$\mathbb{Q} = F_0 \subseteq F_1 \subseteq \dots \subseteq F_n = K,$$ whose topmost field contains $$\gamma,$$ and from this it follows that $$\gamma$$ is constructible.

The fields that can be generated from towers of quadratic extensions of $$\mathbb{Q}$$ are called  of $$\mathbb{Q}$$. The fields of real and complex constructible numbers are the unions of all real or complex iterated quadratic extensions of $$\mathbb{Q}$$.

Trigonometric numbers
Trigonometric numbers are the cosines or sines of angles that are rational multiples of $$\pi$$. These numbers are always algebraic, but they may not be constructible. The cosine or sine of the angle $$2\pi/n$$ is constructible only for certain special numbers $$n$$: Thus, for example, $$\cos(\pi/15)$$ is constructible because 15 is the product of the Fermat primes 3 and 5; but $$\cos(\pi/9)$$ is not constructible (not being the product of Fermat primes) and neither is $$\cos(\pi/7)$$ (being a non-Fermat prime).
 * The powers of two
 * The Fermat primes, prime numbers that are one plus a power of two
 * The products of powers of two and any number of distinct Fermat primes.

Impossible constructions
The ancient Greeks thought that certain problems of straightedge and compass construction they could not solve were simply obstinate, not unsolvable. However, the non-constructibility of certain numbers proves that these constructions are logically impossible to perform. (The problems themselves, however, are solvable using methods that go beyond the constraint of working only with straightedge and compass, and the Greeks knew how to solve them in this way. One such example is Archimedes' Neusis construction solution of the problem of Angle trisection.)

In particular, the algebraic formulation of constructible numbers leads to a proof of the impossibility of the following construction problems:

 * Doubling the cube
 * The problem of doubling the unit square is solved by the construction of another square on the diagonal of the first one, with side length $$\sqrt2$$ and area $$2$$. Analogously, the problem of doubling the cube asks for the construction of the length $$\sqrt[3]{2}$$ of the side of a cube with volume $$2$$. It is not constructible, because the minimal polynomial of this length, $$x^3-2$$, has degree 3 over $$\Q$$. As a cubic polynomial whose only real root is irrational, this polynomial must be irreducible, because if it had a quadratic real root then the quadratic conjugate would provide a second real root.

 * Angle trisection
 * In this problem, from a given angle $$\theta$$, one should construct an angle $$\theta/3$$. Algebraically, angles can be represented by their trigonometric functions, such as their sines or cosines, which give the Cartesian coordinates of the endpoint of a line segment forming the given angle with the initial segment. Thus, an angle $$\theta$$ is constructible when $$x=\cos\theta$$ is a constructible number, and the problem of trisecting the angle can be formulated as one of constructing $$\cos(\tfrac{1}{3}\arccos x)$$. For example, the angle $$\theta=\pi/3=60^\circ$$ of an equilateral triangle can be constructed by compass and straightedge, with $$x=\cos\theta=\tfrac12$$. However, its trisection $$\theta/3=\pi/9=20^\circ$$ cannot be constructed, because $$\cos\pi/9$$ has minimal polynomial $$8x^3-6x-1$$ of degree 3 over $$\Q$$. Because this specific instance of the trisection problem cannot be solved by compass and straightedge, the general problem also cannot be solved.

 * Squaring the circle
 * A square with area $$\pi$$, the same area as a unit circle, would have side length $$\sqrt\pi$$, a transcendental number. Therefore, this square and its side length are not constructible, because it is not algebraic over $$\Q$$.

 * Regular polygons
 * If a regular $$n$$-gon is constructed with its center at the origin, the angles between the segments from the center to consecutive vertices are $$2\pi/n$$. The polygon can be constructed only when the cosine of this angle is a trigonometric number. Thus, for instance, a 15-gon is constructible, but the regular heptagon is not constructible, because 7 is prime but not a Fermat prime. For a more direct proof of its non-constructibility, represent the vertices of a regular heptagon as the complex roots of the polynomial $$x^7-1$$. Removing the factor $$x-1$$, dividing by $$x^3$$, and substituting $$y=x+1/x$$ gives the simpler polynomial $$y^3+y^2-2y-1$$, an irreducible cubic with three real roots, each two times the real part of a complex-number vertex. Its roots are not constructible, so the heptagon is also not constructible.

 * Alhazen's problem
 * If two points and a circular mirror are given, where on the circle does one of the given points see the reflected image of the other? Geometrically, the lines from each given point to the point of reflection meet the circle at equal angles and in equal-length chords. However, it is impossible to construct a point of reflection using a compass and straightedge. In particular, for a unit circle with the two points $$(\tfrac16,\tfrac16)$$ and $$(-\tfrac12,\tfrac12)$$ inside it, the solution has coordinates forming roots of an irreducible degree-four polynomial $$x^4-2x^3+4x^2+2x-1$$. Although its degree is a power of two, the splitting field of this polynomial has degree divisible by three, so it does not come from an iterated quadratic extension and Alhazen's problem has no compass and straightedge solution.

History
The birth of the concept of constructible numbers is inextricably linked with the history of the three impossible compass and straightedge constructions: doubling the cube, trisecting an angle, and squaring the circle. The restriction of using only compass and straightedge in geometric constructions is often credited to Plato due to a passage in Plutarch. According to Plutarch, Plato gave the duplication of the cube (Delian) problem to Eudoxus and Archytas and Menaechmus, who solved the problem using mechanical means, earning a rebuke from Plato for not solving the problem using pure geometry. However, this attribution is challenged, due, in part, to the existence of another version of the story (attributed to Eratosthenes by Eutocius of Ascalon) that says that all three found solutions but they were too abstract to be of practical value. Proclus, citing Eudemus of Rhodes, credited Oenopides (c. 450 BCE) with two ruler and compass constructions, leading some authors to hypothesize that Oenopides originated the restriction. The restriction to compass and straightedge is essential to the impossibility of the classic construction problems. Angle trisection, for instance, can be done in many ways, several known to the ancient Greeks. The Quadratrix of Hippias of Elis, the conics of Menaechmus, or the marked straightedge (neusis) construction of Archimedes have all been used, as has a more modern approach via paper folding. Although not one of the classic three construction problems, the problem of constructing regular polygons with straightedge and compass is often treated alongside them. The Greeks knew how to construct regular $n$-gons with $$n=2^h$$ (for any integer $$h\ge 2$$), 3, 5, or the product of any two or three of these numbers, but other regular $n$-gons eluded them. In 1796 Carl Friedrich Gauss, then an eighteen-year-old student, announced in a newspaper that he had constructed a regular 17-gon with straightedge and compass. Gauss's treatment was algebraic rather than geometric; in fact, he did not actually construct the polygon, but rather showed that the cosine of a central angle was a constructible number. The argument was generalized in his 1801 book Disquisitiones Arithmeticae giving the condition for the construction of a regular $n$-gon. Gauss claimed, but did not prove, that the condition was also necessary and several authors, notably Felix Klein, attributed this part of the proof to him as well. Alhazen's problem is also not one of the classic three problems, but despite being named after Ibn al-Haytham (Alhazen), a medieval Islamic mathematician, it already appears in Ptolemy's work on optics from the second century.

Pierre Wantzel proved algebraically that the problems of doubling the cube and trisecting the angle are impossible to solve if one uses only compass and straightedge. In the same paper he also solved the problem of determining which regular polygons are constructible: a regular polygon is constructible if and only if the number of its sides is the product of a power of two and any number of distinct Fermat primes (i.e., the sufficient conditions given by Gauss are also necessary). An attempted proof of the impossibility of squaring the circle was given by James Gregory in Vera Circuli et Hyperbolae Quadratura (The True Squaring of the Circle and of the Hyperbola) in 1667. Although his proof was faulty, it was the first paper to attempt to solve the problem using algebraic properties of $\pi$. It was not until 1882 that Ferdinand von Lindemann rigorously proved its impossibility, by extending the work of Charles Hermite and proving that π is a transcendental number. Alhazen's problem was not proved impossible to solve by compass and straightedge until the work of Jack Elkin.

The study of constructible numbers, per se, was initiated by René Descartes in La Géométrie, an appendix to his book Discourse on the Method published in 1637. Descartes associated numbers to geometrical line segments in order to display the power of his philosophical method by solving an ancient straightedge and compass construction problem put forth by Pappus.

title: Constructible number