# Talk:Number/Archive 2

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## intro is a mess.

The first sentence is unwieldy and bordering on nonsensical, not a good start to the article. It currently reads: "A number is a mathematical object used to quantify (count and measure) and to represent quantity, in several forms, of which the most primitive, primary and simplest one is the symbolic signification of a particular, invariable, constant quantity." Cliff (talk) 12:43, 20 April 2011 (UTC)

I agree. I have reverted to the previous version, before recent changes by Faus (talk · contribs), which reads "A number is a mathematical object used to count and measure" - much simpler and clearer. Gandalf61 (talk) 13:30, 20 April 2011 (UTC)
...and more inexact. A number is not a mathematical object used to count and measure, but it's rather a mathematical object in a relation of a representational kind with quantity. So you are deleting the wrong clause of the definition, (I left untouched the first, wrong part of the opening definition for not upsetting anyone), but hey, I'm not going to waste my time in permanently monitoring this article, so don't worry, you win. --Faus (talk) 20:43, 21 April 2011 (UTC)
I believe what is there is far better than the proposed change to 'a relation of a representational kind'. Dmcq (talk) 22:10, 21 April 2011 (UTC)
However, what you believe (or I believe) is not the important thing; the important thing is the correct definition of number. But wait, this is Wikipedia... ok then... --Faus (talk) 09:58, 25 April 2011 (UTC)
Faus, an encyclopedia is not a resource for mathematical definitions, but as a resource for people who want to find things out. Imagine a person who might search for the word "number" in an encyclopedia. It is less likely that they're looking for a mathematical definition, but a more general one that will help them understand and not cause further confusion. We're not saying that there is no room in the article for the "correct" definition, but that the intro is not the place for it. Cliff (talk) 20:57, 25 April 2011 (UTC)
Well if it is so much more accurate despite sounding so abstruse where please would I find a source corresponding to a definition of a number as 'a relation of a representational kind'? Dmcq (talk) 22:07, 25 April 2011 (UTC)
As always. 23:58, 25 April 2011 (UTC) — Preceding unsigned comment added by Cliff (talkcontribs)

Is arithmatic a British spelling Or just a wrong one? Cliff (talk) 17:43, 15 June 2011 (UTC)

It was a spelling mistake introduced by Cpiral (talk · contribs) on June 13. I have reverted to the previous version of the lead, before Cpiral's changes on June 11 and June 13. This previous version was both shorter and clearer than its replacement. Gandalf61 (talk) 08:55, 16 June 2011 (UTC)
Restoring the previous version seems best at this point. I did like the inclusion of talk about how the definition of number is changing with new research. Cliff (talk) 21:13, 16 June 2011 (UTC)

The new lead, with paragraphs for definition, linguistics, and operations was an improvement in the following ways:

• Puts the multiple ideas pertaining to linguistics in their own paragraph, out of the definition that is the article's actual main content
• Mentions the "reals" with the other basic number classifications mentioned
• Moves "arithmetic" from the paragraph on operations to the paragraph on definition
• Introduces the idea of a digit and positional notation (in the paragraph on linguistics)

It was mostly a copy edit but the whole time I was hoping to place subtle hints about number theory, because some numbers are not used to count and measure. They serve the theory of numbers themselves. To do this I added to the first, defining paragraph the ideas of "operational definition" and the continuing development of number systems, and of number theory (also known as "arithmetic"). I added a link to the theory of number systems (which is what the article is mostly about). I rewrote a sentence that mixed the ideas of "number types" ("negatives" and "zero") and "number systems". Minor fixes: "label" is not the right word; "serial number" and "phone number" are probably overlinked. Cheers. — CpiralCpiral 02:23, 17 June 2011 (UTC)

Cpiral, since your edit I had been thinking about how to fix your work while keeping some of the ideas. I think the main problem was that you had done too much, so were unable to ensure the quality of writing expected. If you'd like to slow down and make smaller improvements that can be edited by others, we can build a better lead as a group. Or we can do it here. I'd like to help, but your lead was confusing rather than enlightening. Cliff (talk) 03:48, 17 June 2011 (UTC)
I did too much. Parts or links did not flow. Thank you.
• "the linguistic use of numbers as word-like or letter-like symbols"
• "...what numbers can be, partly by research and development in pure mathematics."
The kicker is the footnote problem. It made it seem like to much. It was unclear and confusing why such a thing might be said there at the beginning. See next section "intro review". — CpiralCpiral 17:38, 17 June 2011 (UTC)

## Review of a new intro

A number is a mathematical object applied in counting and measurement.[1] The history of mathematics provides us an "operational definition" of number, for number's properties are still expanding. A number system is well defined set of numbers and their arithmatic.

Negative numbers appeared around the year zero. Nine hundred years later zero was officially a number used in formal calculations. The fundamental number systems are the naturals (the numbers we count with), the integers (whole numbers), the rationals (fractions), the reals, and the complex numbers. Number theorists are still working to understand what a number can be. Number systems and their histories are summarized here.

The mathematical [[mathematical notation|mathematical notation] for representing numbers is the numeral system. This numeral system extends into linguistics where numbers serve as word-like or letter-like symbols, such as a telephone number or a serial number, or other codes such as an ISBN. The word number has several uses: abstract object, symbol, or word; but this article concerns itself with the numbers that use digits to represent mathematical objects in a positional notation, unlike the "numerals" in linguistics.

Each of the number systems described below defines its numeric operations. The most basic operations are the unary operation, which inputs a single number and outputs a single number, and the binary operation, which inputs two numbers and produces a single output number. The operations of addition, subtraction, multiplication, division, and exponentiation are all binary; but the integer operation "successor" produces, singly, "number plus one". The successor of -10 is -9.

1. ^ Numbers do not serve only to count and measure. Pure, unapplied numbers help develop number systems that find application in physics, chemistry, biology, computing, engineering, coding and cryptography, random number generation, acoustics, communications, graphic design and even music and business. Nikolay Lobachevsky says "There is no branch of mathematics, however abstract, which may not some day be applied to phenomena of the real world."
This is just dreadful. I don't think I'll boither contributing to trying to rescue it. I'll simply oppose it. Dmcq (talk) 20:03, 17 June 2011 (UTC)
Agree with Dmcq. There is too much in that proposed intro that is either ungrammatical, poorly worded, unclear or just plain wrong. Oppose. Gandalf61 (talk) 10:03, 18 June 2011 (UTC)

After reading the current intro, I wondered if language differences should be noted after the description of the "common use" of the word number. For example, German-English dictionaries often translate Nummer and Zahlen to number, but 'Nummer' is used in a labeling sense (house numbers, telephone numbers, ranking "We're number 1"), but 'Zahlen' tends to refer to numbers in a mathematical sense. Not certain if this happens in other languages. Thelema418 (talk) 09:41, 1 August 2012 (UTC)

## Mainly for missing definition

Arthur Rubin deleted my additions on prime numbers with his wrong(!) subjective reasoning, last night.

Addition and multiplication is defined perfectly well for the prime numbers, but this set is not closed under these two (binary) operations. That is the truth!

But what seems now suddenly far more important: in the article this is not demanded to call a number set a "number system": to be closed under addition and multiplication. Therefore single persons like Arthur Rubin, can interpret this as their likeness, subjectivly now. Thus I think this MUST be defined in the article to meet some standards if this is the only reason to delete the information on the "number system" of prime numbers. :-(

Moreover it looks not good, to double mention the postive integers in the eye-catching table. Did you forgot also to double the non-negative integers? Or should I call you also the positive reals to insert? *gg* These obvious misfits are a reason why such an article is no longer considered to be of high quality. Sorry. 2001:638:504:C00E:214:22FF:FE49:D786 (talk) —Preceding undated comment added 15:47, 20 September 2012 (UTC)

You are right - the positive integers do not belong in the table either as they break the progression in which each set is a superset of the set above and a subset of the set below. I have removed that row from the table. Gandalf61 (talk) 15:57, 20 September 2012 (UTC)
Would people please leave their logic at home when editing Wikipedia - see WP:OR. It doesn't matter if you think the positive integers don't form a number system or not. The sources are what matters and many sources do include them as a system of numbers rather than just a set of numbers like the primes. I will consider it vandalism to continue on this course. Dmcq (talk) 16:22, 20 September 2012 (UTC)
I still don't see why the positive integers need a separate row in the table, as they coincide with one of the definitions of natural numbers, and they don't have a separate section in the text below the table. But if you insist on reinstating them then at least consider placing the positive integers row at the top of the table - since they are the "smallest" set listed - rather than in a random place half way down. Gandalf61 (talk) 16:30, 20 September 2012 (UTC)
I missed that about the natural numbers, yes I agree they don't need to be in twice. Sorry I saw that as a pointy extension of the previous argument about prime numbers. Dmcq (talk) 18:00, 20 September 2012 (UTC)

## Definition of i

I believe that defining i as "the square root of negative one" is a wrong approach. (e.g. because ${\displaystyle i^{2}=-1=(-i)^{2}}$ - so it is not unambiguous) Shouldn't be the complex numbers defined as ℝ×ℝ with following operations:

• (a; b) + (x; y) := (a + x; b + y);
• (a;b)⋅(x;y):=(ax-by; bx+ay);

and than denote i:=(0,1) ?

Chilkes (talk) 22:21, 26 February 2012 (UTC)

There are some advantages in defining complex numbers in that way, but for the purposes of this particular article I think the informal definition "the square root of negative one" is fine. The interested reader can get more details by reading the linked articles such as imaginary unit. Gandalf61 (talk) 11:31, 27 February 2012 (UTC)
i is the negative square root of minus one, just like 1 is the square root of 1 and −1 is the negative square root. Dmcq (talk) 00:15, 27 June 2012 (UTC)

No. Sorry, but i is neither positive nor negative. The symbol i represents one of the square roots of −1. Once that symbol is introduced, it is easy to show that −i is the other. Rick Norwood (talk) 12:07, 27 June 2012 (UTC)

i is not a number and cannot be defined as a number. It is exactly a 2 dimensional vector with the operations you stated. Incompetent mathematicians think of it as a number because the real part can be used with "real numbers" (also non-existent). Complex theory is inherently unsound, but then again, those who advocated its use were not very smart mathematicians - much like the mathematicians you encounter today. 98.194.121.63 (talk) 20:41, 11 July 2012 (UTC)

You are still pushing the "new calculus", which has no place in Wikipedia unless published. However, this statement, although clearly seen to be false by any competent mathematician, is not obviously wrong to a layman, so further discussion might be productive.
But probably not. — Arthur Rubin (talk) 21:32, 11 July 2012 (UTC)
i can be very well defined as a number. You first should ask your self: what could be a number? And this is a philosophic question -- probably far too tricky to be solved in agreement on wikipedia audience. :)
But now to the original problem. I cite: "the square root of negative one". The basic problem lies in ill-definition / usage of english words, IMHO! There is a unique square root function (per definition a function must be unique) and a certain value in its range is called the square root of ... (e.g. -1). On the other hand the solution(s) of the equation "x^2= -1" are called square root(s) of -1 and there are two different. As often, the real problem lies in bad or not existing precise definition of wordings. — Preceding unsigned comment added by 2001:638:504:C00E:214:22FF:FE49:D786 (talk) 16:06, 20 September 2012 (UTC)
I agree with what was said so far and would like to add that the formulation "i is one of the two solutions of the equation x^2= -1", although it may at first not seem to solve the problem, accurate. It is not important, "which" of the two solutions you define to be i. The other, as Rick states earlier, can be shown to be -i then, meaning that if i is a solution (whatever i is), then -i must also be a solution. One of the effects that this symmetry has is that complex solutions to polynomial equations always appear in conjuagte pairs. By the way, 98.194.121.63's statement sounds very similar to what was said in the past about negative numbers (e.g. negative solutions to polynomial equations) when they were still uncommon. --Doubaer (talk) 11:42, 26 June 2014 (UTC)

## What is a number?

Close "the new calculus". Nothing to see here. Move along.
The following discussion has been closed. Please do not modify it.

Popular misconceptions:

1. Number came before measure. FALSE

2. Ratios came from natural numbers. FALSE

3. Irrational numbers are numbers. FALSE

4. Real numbers exist. FALSE

5. Complex numbers are numbers. FALSE

All one has to do is study Euclid's Elements from which all these misconceptions are easily debunked. http://thenewcalculus.weebly.com/uploads/5/6/7/4/5674177/magnitude_and_number.pdf 98.194.121.63 (talk) 16:52, 26 June 2012 (UTC)

What does that have to do with this article, even if they were at all plausible. Which they are not. — Arthur Rubin (talk) 00:42, 27 June 2012 (UTC)
Rubin, the fact that you have to ask proves you are not a mathematician. What does Euclid have to do with number? Answer: Everything.

98.194.121.63 (talk) 17:54, 8 July 2012 (UTC)

I'd forgotten you were here. I wish I could still forget. As for your assertions, they are not published anywhere, so cannot be used as a reference in Wikipedia. If they referred to a published work which said those things, that work might be usable, but, as there is no such published work ....
Don't feed the trolls. Rick Norwood (talk) 12:01, 27 June 2012 (UTC)
Norwood, you are a troll and it shows. 98.194.121.63 (talk) 17:54, 8 July 2012 (UTC)

### Blacklisting URLS?

How number is derived: https://www.filesanywhere.com/fs/v.aspx?v=8b6e62875e636d7bb399 197.79.38.87 (talk) 11:01, 8 September 2014 (UTC)

## If the symbol for a number is called a numeral, what is the word for a number called?

Quite simply, the heading says it all. -- 16:09, 13 February 2013 (UTC)

It is also called a numeral - see numeral system - "A numeral system (or system of numeration) is a writing system for expressing numbers" - and numeral (linguistics) - "In linguistics, a numeral is a word class (part of speech) designating numbers". This ambiguity sometimes causes confusion. Gandalf61 (talk) 09:48, 14 February 2013 (UTC)
Why should there be a special word? One can either say list ten animals or list ten words for animals. There isn't a special word for word for animal. Dmcq (talk) 14:11, 14 February 2013 (UTC)
Probably, historically, mathematicians were fussier than biologists. Today the distinction between number and numeral is seldom made. However, if you want to get technical, there is a difference between the word "cat" and the animal "cat". Thus a cat has whiskers, but "cat" has three letters (and no whiskers). Rick Norwood (talk) 15:26, 14 February 2013 (UTC)
After some research, I have concluded that numbers such as "one" "two" "three" are cardinal and ones such as "first" "second" "third" are ordinal. Further than that, in word form is verbose and otherwise it is in numeric form. -- 16:59, 14 February 2013 (UTC)
I'll mention (because it came up in an article edit) that a "numeral" is generally understood to be a single symbol (or "digit"), while a "number" is written using one or more numerals/symbols/digits. So, the only way "symbol for a number" makes sense is when the number in question can be represented by a single symbol/numeral/digit (i.e. 0-9 in base-10, Arabic). —[AlanM1(talk)]— 07:26, 8 March 2013 (UTC)

Alan, I do not understand what your point is. In the heading clearly states "a number" which is singular in of itself. I was simply trying to determine what the noun for the verbose/long name of a number is. 12:44, 8 March 2013 (UTC)

No, a numeral may have more than one symbol. 123 is a numeral. XVII is a Roman numeral. Rick Norwood (talk) 15:40, 8 March 2013 (UTC)

I disagree. M-W defined numeral (noun) as "1: a conventional symbol that represents a number" (singular symbol). OED, though, says "a figure, symbol, or group of figures or symbols denoting a number. a word expressing a number." I still think my usage is more common. "123" is a number, consisting of three numerals. "XVII" is a number composed of 4 Roman numerals (i.e. one might write "Convert 17 to Roman numerals, not a Roman numeral). Time to find some published sources, I guess. —[AlanM1(talk)]— 00:52, 9 March 2013 (UTC)

M-W is using the word "symbol" in the common sense that would be understood if I said "cat" is a symbol for a feline. There is no intent there to limit the symbol to a single character. OED is more careful, noting that a numeral can be a group of symbols. 123 is a numeral, which expresses in three digits the same number as CXXIII. The word for a single symbol numeral is "digit".Rick Norwood (talk) 20:09, 30 October 2014 (UTC)

## Arabic versus Veyselic Numbers

Arabic numbers are written LTR, most significant digits are on the left, however Veyselic Numbers are arabic numbers written from right to left, RTL, like number ten is 01 (or in arabic .1). More info can be found in http://wiki.alquds.edu/?query=User_talk:Veyselperu Thanks 213.74.174.138 (talk) 08:57, 22 August 2013 (UTC)

As many elementary math books note, what we call Arabic numbers are really Arabic numerals. And Verselic Numbers are really numerals as well. The quantity indicated does not change whether the digits are written right to left or left to right. Both use base ten.Rick Norwood (talk) 20:12, 30 October 2014 (UTC)

## Is number theory really called "arithmetic".

The article says the word "arithmetic" is also used for number theory. That's not a usage I've ever heard? Is anybody aware of such a usage?Rick Norwood (talk) 12:29, 3 November 2014 (UTC)

Try this: Serre, Jean-Pierre Cours d'arithmétique. (French) Collection SUP: "Le Mathématicien", 2 Presses Universitaires de France, Paris 1970 188 pp. Tkuvho (talk) 12:48, 3 November 2014 (UTC)
It would probably be better to write "the word arithmetic has also been used for number theory". There are many old references that use "arithmetic" or "higher arithmetic". See the last paragraph of the lead of Number theory for details. D.Lazard (talk) 12:52, 3 November 2014 (UTC)
I think "has been used" is an understatement. The term is still in widespread use and also appears in the names of other subjects like arithmetic geometry. --Sammy1339 (talk) 15:30, 3 November 2014 (UTC)

Thanks.Rick Norwood (talk) 14:27, 3 November 2014 (UTC)

## Factual accuracy of the History section

I added a disputed tag to that section of the article. Sources are woefully inadequate and there are many dubious claims. For example, the proof that ${\displaystyle {\sqrt {2}}}$ is not rational is called an "existence proof," the apocryphal story about Hippasus seems to be described as fact, Lambert's proof that ${\displaystyle \pi }$ is irrational is weirdly listed as the first result about transcendental numbers, about Cantor's work it is falsely claimed "this was the first mathematical model that represented infinity by numbers and gave rules for operating with these infinite numbers," it is claimed that Tartaglia had a hand in discovering imaginary numbers (which as far as I know he never used - they were briefly mentioned in Cardano's Ars Magna whose only connection to Tartaglia was that it plagiarized him, and not on that subject), and generally I can't verify hardly anything in this section for extreme lack of references. I'm planning on making a lot of changes to this article when I have time, but this will be the last section I will work on, because I regard it as lowest priority. (Math before math history - top priority in my view is a formal definition of the main number systems, which is entirely lacking right now.) To anyone who worked on this section: please provide references if you can. To allow an opportunity for this I'm going to leave the section untouched for a week or two, but without references I'm inclined to outright remove most of it (replacing what I can of course), since I'd rather not include all these questionable claims and I'm not qualified to rewrite the whole section by myself. Also please note that mathematics textbooks are not reliable sources for mathematics history before the 20th century - for example, in just one paragraph, Jacobson's famous graduate textbook Basic Algebra manages not only to bungle the history of the cubic equation but to get Tartaglia's name wrong. And of course things we all heard in lectures - which seems to be the source of some of the information here - are even less reliable. --Sammy1339 (talk) 17:40, 3 November 2014 (UTC)

Good luck with your project.Rick Norwood (talk) 21:57, 3 November 2014 (UTC)
I don't think wholesale deletion of the section is appropriate. The work on any given wiki page is incremental. There is material there that's legitimate; other passages need to be worked on. One could try identifying the contributor who contributed the questionable passages by examining the page's history, and inviting him to provide references, etc. Tkuvho (talk) 15:00, 4 November 2014 (UTC)
@Tkuvho:The section has been there in a similar state for years and is almost totally unsourced, and about every third sentence is dubious. The editors who contributed this material may not be around anymore, but I absolutely encourage anybody who can to provide references. --Sammy1339 (talk) 15:51, 4 November 2014 (UTC)

## Sammy1339's edit

Compare "A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, and so forth. A notational symbol that represents a number is called a numeral. In addition to their use in counting and measuring, numerals are often used for labels (as with telephone numbers), for ordering (as with serial numbers), and for codes (as with ISBNs). In common usage, the term number may refer to a symbol, a word, or a mathematical abstraction."

with

"A number is one of several related types of mathematical objects. The original examples are the natural numbers 1, 2, 3, and so forth, which are commonly used for counting and ordering. Among many other uses, two of the most important are measurement and the description of probability. Fractions are well-suited to these purposes. Numerals, which represent numbers symbolically, are often used for labels (as with telephone numbers) and codes (as with ISBNs). In common usage, the term number may refer to a symbol, a word, or a mathematical abstraction."

Rick Norwood (talk) 15:15, 5 November 2014 (UTC)

## Lede

The rearrangement of the lead by Sammy1339 has been reverted by Rick Norwood, with edit summary rv Good faith edit. Number should be an entry level article. To rewrite it in a more abstract style is not an improvement.

I do not understand this edit summary, and I do not see any "more abstract style": the main difference between the two leads, lies in the order of the sentences. The advanced mathematics (real and complex numbers) have been pushed by Sammy toward the end of the lead, while the common usage of the word number and the basic arithmetic operations are moved toward the beginning. Both changes improve readability for the layman.

Beside this reordering, the main change that I have noted id the change of

A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, and so forth.

into

A number is one of several related types of mathematical objects. The original examples are the natural numbers 1, 2, 3, and so forth, which are commonly used for counting and ordering. Among many other uses, two of the most important are measurement and the description of probability. Fractions are well-suited to these purposes.

I do not see anything "more abstract" in the latter formulation. The difference lies only in the introduction of the fundamental idea that there are several kinds of numbers, and the mentions of fractions and probabilities. This is certainly not too abstract for the layman, as its suffices to open a newspaper encounter these two concepts. IMO, the mention in the lead of the existence of several types of numbers and the fact that fractions are numbers is fundamental. This could help to avoid confusions such the one that led to the strange discussion at Talk:Integer#Percentages.

In summary, I disagree with Rick, when he says that Sammy's edit is not an improvement: remaining at the same starting level, it better follows the guidelines of MOS:LEAD, by sorting the sentences by increasing level of technicality. Moreover, this is done by increasing the accuracy of the wordings. Therefore, I'll reinstall Sammy's lead, and suggest to discuss further improvements from this version. D.Lazard (talk) 14:55, 5 November 2014 (UTC)

While I was typing the section above, D. Lazard restored Sammy1339's version and added this section. I'm going to move the two versions here for ease of comparison.Rick Norwood (talk) 15:18, 5 November 2014 (UTC)
Version One

Compare "A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, and so forth. A notational symbol that represents a number is called a numeral. In addition to their use in counting and measuring, numerals are often used for labels (as with telephone numbers), for ordering (as with serial numbers), and for codes (as with ISBNs). In common usage, the term number may refer to a symbol, a word, or a mathematical abstraction."

Version Two

A number is one of several related types of mathematical objects. The original examples are the natural numbers 1, 2, 3, and so forth, which are commonly used for counting and ordering. Among many other uses, two of the most important are measurement and the description of probability. Fractions are well-suited to these purposes. Numerals, which represent numbers symbolically, are often used for labels (as with telephone numbers) and codes (as with ISBNs). In common usage, the term number may refer to a symbol, a word, or a mathematical abstraction.

Here is why I prefer Version One. It begins with a simple declarative sentence that tells the reader what they want to know. The first sentence of version two does not tell the reader anything. If the first sentence doesn't tell the reader anything, they are unlikely to read the second sentence. The only change in the second sentence is the use of 1 instead of 1. The problem with the third sentence in version two is that, so far, only natural numbers have been introduced, which are not used for measurement and probability. I know the sentence doesn't actually say 1, 2, 3 are used for measurement and probability, but one sentence following the other suggests a connection which is not intended. Version Two then says "fractions are well-suited to these purposes". If fractions really were well-suited to measurement and probability, then we should introduce them before the two examples of their use. Actually, real numbers are better suited than fractions, especially for measurement. No problem with the other two sentences.
I'll pause here for comments. Rick Norwood (talk) 16:07, 5 November 2014 (UTC)
I too prefer the first version. It's the only version that agrees with the guidelines for the first sentence, that it should wherever possible say what the subject is. So "A number is a..." should be how it starts whatever the definition that follows. It is less technical, mentioning natural numbers and numeral but no other mathematical concepts, while the second version adds fractions and probability, not only more technical but unnecessarily specific – numbers appear in almost all areas of maths and singling out those two is odd. Also stating that probability (with measurement) is important is a value judgement which seems even more dubious – one could argue that using numbers in engineering and physics is as important, or using them in arithmetic is even more so.--JohnBlackburnewordsdeeds 16:57, 5 November 2014 (UTC)
I would like the first sentence to say what the subject is, too, but there is no brief definition. The first line in the first version is simply wrong. Real numbers are not used for counting, and complex numbers are not generally used for measuring or labeling, neither are any of these things really defining properties of numbers. So I changed it for the sake of accuracy. --Sammy1339 (talk) 20:38, 5 November 2014 (UTC)
There's nothing wrong with it as a brief definition. You use numbers to count, measure and label many things. They are used for other things too, but those are perhaps the most obvious and familiar applications. These different applications often require different numbers; natural numbers for counting, real numbers for most measurements, etc. But they're all numbers.--JohnBlackburnewordsdeeds 21:01, 5 November 2014 (UTC)
But complex numbers are not for any of those things, so it's not a definition so much as a selection of things some numbers are used for, which is why it doesn't belong in the first sentence where it is likely to be misinterpreted as defining the concept of a number. --Sammy1339 (talk) 21:22, 5 November 2014 (UTC)

Complex numbers are used for measuring. Electrical impedance is one example. In any case, the first sentence in Version Two is empty of content, since almost any word can replace "numbers". An integral is one of several related types of mathematical objects. A ring is one of several related types of mathematical objects. A function is one of several related types of mathematical objects. The first sentence should tell the reader something they want to know.Rick Norwood (talk) 22:03, 5 November 2014 (UTC)

Well, various physical quantities are represented by complex numbers, but these are aggregated from real data (like resistance and reactance in your example, or phase and probability amplitude for a particle in quantum mechanics). It seems to me that the results of measurements are always real, but even that might be a little naive. We can probably have a philosophical discussion about what measurement is. More to the point though, the issue of whether numbers can be used in this way says nothing about what they are - the original first sentence was hardly descriptive. If I wanted to start with a definition, I would say "A number is an element of the complex number field," and obviously that is too technical, besides still being incomplete. I can't think of a good way to express what a number is to a layperson in less than two or three paragraphs. --Sammy1339 (talk) 22:48, 5 November 2014 (UTC)
After having read above comments, my opinion is that both versions have some advantages. Also, both versions omit the most important fact about the context of the topic: numbers are at the basis of almost all mathematics. Thus, I have tried to take everything into account, and this leads to the following proposition for the first paragraph.
Numbers are mathematical objects that are at the basis of almost all mathematics, and, among many uses, are used to count and measure. The original examples are the natural numbers 1, 2, 3, and so forth, which are well suited for counting. For measuring, one generally use rational numbers, which are commonly represented by fractions. A notational symbol that represents a number is called a numeral. In addition to their use in counting and measuring, numerals are often used for labels (as with telephone numbers), for ordering (as with serial numbers), and for codes (as with ISBNs). In common usage, the term number may refer to a symbol, a word, or a mathematical abstraction.
D.Lazard (talk) 23:41, 5 November 2014 (UTC)
Sorry if I seem contrarian, but I definitely can't agree with the statement that "numbers are the basis of almost all mathematics." Even in the earliest days they were not needed for synthetic geometry, which made up the better part of higher mathematics until the Renaissance, and today they have only a minor role if any in most of abstract algebra, most of topology, most of mathematical logic ... in fact I've rarely heard the word "number" in a math talk. They're everywhere in analysis of course, but even then, they can easily be replaced by more general constructions; it's just that most analysts don't care about that sort of thing. Anyway the super-specialness of numbers is a popular misconception I definitely don't want this page to perpetuate. It's enough to say that numbers have huge importance in mathematics - but that's not a defining feature either. --Sammy1339 (talk) 00:38, 6 November 2014 (UTC)
Trying to resolve this issue I looked at the 1911 Britannica definition, which unfortunately is not very descriptive either: "a word generally expressive of quantity, the fundamental meaning of which leads on analysis to some of the most difficult problems of higher mathematics." And still not too accurate - are complex numbers expressive of quantity? Are negative numbers, even? I agree with the concern that the first sentence in the current version is rather empty and I'm really trying to think of a way to improve it, but the term simply might not admit a succinct definition, as apparently was noticed 100 years ago. --Sammy1339 (talk) 01:16, 6 November 2014 (UTC)

Now we have three proposals. I'll call D.Lazard's proposal Version Three. I like it better than Two, less than One. It has the virtue of saying something in the first sentence to the people who visit this page and are not mathematicians. I agree with Sammy1339 that "the basis of almost all mathematics" is an overstatement. Euclid proved theorems about numbers, but they were not as basic to Euclid's Elements as points and lines. And choosing the rational numbers instead of the real numbers for measurement is non-standard. Is the square root of two not the measure of the hypotenuse of a right triangle whose base and height are one? Is pi not the measurement of the ratio of circumference to diameter?

It seems to me that all the things we call "numbers" are either subsets or extensions of the real numbers, but that doesn't really solve our problem. We seem to be at an impasse. I hope more people weigh in on this important question.Rick Norwood (talk) 01:19, 6 November 2014 (UTC)

My rationale for naming fractions instead of real numbers as "suitable for measurement" is that actual measurements are approximate, leaving the reals no practical advantage. (i.e. I meant measurement in the sense of rulers, not in the sense of Lebesgue.) The purpose of the reals is doing calculus, but that's a rather abstract place to start. The definition of numbers, of course, is the second-to-last paragraph. It could be moved to the top, but then this conflicts with the convention of proceeding from more concrete to more abstract. I'm also a little concerned about giving top billing to "number" as understood in pure mathematics, a rare and perverse occupation which shouldn't be allowed to dominate a concept that rightly belongs to everybody. --Sammy1339 (talk) 01:34, 6 November 2014 (UTC)
Oh and by the way, p-adic numbers are neither subsets nor extensions of the reals, but per my comment above perhaps nobody should care. Anyway "real number" is much too esoteric the first sentence anyhow - most undergrads don't know what it means. --Sammy1339 (talk) 01:40, 6 November 2014 (UTC)
Rick Norwood I agree with your ordering: three is an improvement on two but I still prefer one. There's nothing incorrect in three but I much prefer one's straightforward and direct definition. Sure it leaves out all sorts of mathematical detail and subtlety but that's not needed in the first sentence, which has to be readily understood by all levels of readership, i.e. all ages and abilities.--JohnBlackburnewordsdeeds 03:59, 6 November 2014 (UTC)

I had hoped others would weigh in, but that hasn't happened, so essentially we have three versions. Two only have the support of the person who wrote them. Version One is older and has two supporters. I propose we go back to Version One, at least until we get some definite consensus. Rick Norwood (talk) 22:54, 7 November 2014 (UTC)

I posted what I felt was a compromise (I still prefer the current version to that one) but JohnBlackburne reverted it. Personally I'd rather move on and revisit this some other time. --Sammy1339 (talk) 23:27, 7 November 2014 (UTC)
While we're trying to decide among three versions it isn't really helpful to modify it to yet another version without discussion. Per Rick Norwood the only version there's any consensus over is the first one, which was the original. so we should return to that, until we arrive at consensus on a new version.--JohnBlackburnewordsdeeds 01:39, 8 November 2014 (UTC)

Good.Rick Norwood (talk) 13:17, 8 November 2014 (UTC)

## Hatnote

An editor is starting edit warring for including in the hatnote a direct link to Book of Numbers. He has already been reverted by two different users, and he reverts immediately these reverts. This direct link in inappropriate for at least three reasons. Firstly, it is aimed for "disambiguating an article name that is not ambiguous" (see WP:NAMB). Secondly, this article is linked to in Number (disambiguation), to which refers the hatnote {{other uses}}. Thirdly, it breaks the policy of neutral point of view, by emphasizing without good reasons a particular use of the word "number". For these three reasons, I'll revert this editor again. D.Lazard (talk) 18:22, 30 October 2014 (UTC)

Note that this page is named "Number" (singular) rather than "Numbers". So strictly speaking it is not an issue of disambiguation at all. Tkuvho (talk) 18:30, 30 October 2014 (UTC)
It is not quite that simple, as "Numbers" redirects here. Sławomir Biały (talk) 18:50, 30 October 2014 (UTC)
Further, Number (disambiguation) disambiguates both "number" and "numbers". --50.53.38.33 (talk) 20:05, 30 October 2014 (UTC)
There are two redirects in play: Numbers redirects to "Number", and Numbers (disambiguation) redirects to Number (disambiguation). --50.53.38.33 (talk) 20:34, 30 October 2014 (UTC)
This hatnote does not belong here. I fully agree with D.Lazard's original edit. There's no such hatnote in our Book article either. - DVdm (talk) 20:16, 30 October 2014 (UTC)
I agree with all of that, except it's not so much a POV issue as undue weight. The Book of Numbers is hardly an alternative primary topic; it may seem so to some Christians but only some, and Christians are only a small minority of English speakers worldwide. Rather than put undue weight on that use it should just link to the disambiguation page, where it's rightly listed as just one item among many.--JohnBlackburnewordsdeeds 20:18, 30 October 2014 (UTC)
• I don't necessarily disagree, except that it had to have been added for a reason. I'm currently going through the history one revision at a time (I'm up to Revision as of 17:49, May 6, 2010) trying to find when it was added and by whom. My guess is that it was a heavily searched for topic and because "Numbers" (short for "The Book of Numbers") was redirected to this page, people were having difficulty in finding it. Please be patient, and as soon as I have found the edit (and hopefully some discussion about this someplace), I'll report back and we can move forward. In the mean time, please leave the page in the condition it was in BEFORE the BOLD edit was made to remove this hatnote until AFTER this DISCUSSion has concluded and there is consensus for the change. Thank you for your patience. :) Happy editing in the mean time! — {{U|Technical 13}} (etc) 20:24, 30 October 2014 (UTC)

I made my edit (and several other edits) before I read your comment, else I would have waited. Still, I can't imagine anybody searching for the Book of Numbers having trouble with the disambiguation page.Rick Norwood (talk) 20:36, 30 October 2014 (UTC)

I moved Book of Numbers to the top of the "Literature" list on Number (disambiguation). --50.53.38.33 (talk) 20:46, 30 October 2014 (UTC)

One solution might be to have "numbers" redirect to the disambiguation page instead of here.Rick Norwood (talk) 20:53, 30 October 2014 (UTC)

Having Numbers redirect to Number (disambiguation) is a good idea. --50.53.38.33 (talk) 21:06, 30 October 2014 (UTC)
• (edit conflict) So, I finally found it... I suppose I should have worked down from the most recent edits instead of up from the oldest as it was about six weeks ago it was introduced in this edit by Red Slash which was immediately REVERTed in this revision by D.Lazard and then rereverted by Red Slash in this change citing Wikipedia:Hatnote in his edit summary. Reading that editing guideline, I see there are a couple subsections that seem to support it should be there and I see other sections that maybe suggest otherwise. It's not entirely clear to me at this time which section of that page Red Slash was suggesting supports inclusion. One potential compromise I can possibly think of is a discussion of whether or not Numbers should redirect to Number, Book of Numbers, or Number (disambiguation). I'm tempted to think that perhaps it should redirect to the disambig. I'd like to hear if RS can clarify his position on why it should be included and I'd like to hear others opinions about fixing what I consider to be the biggest part of the problem, which is the redirect. — {{U|Technical 13}} (etc) 21:19, 30 October 2014 (UTC)

I don't think changing the target of Numbers makes sense. Almost always the singular and plural of a title go to the same place. Otherwise editors will find themselves repeatedly getting it wrong, writing e.g. Numbers or Numbers interchangeably and expecting then to do the same thing so not checking where they actually go, and then being surprised when they go to different places. The guideline is WP:POFRED, and one of the main reasons for creating redirects to an article is from its plural.--JohnBlackburnewordsdeeds 21:32, 30 October 2014 (UTC)

Numbers (TV series) is a closer title match and gets about twice as many page views as Book of Numbers. It's best to leave all uses for the disambiguation page. I noticed that Red Slash promotes Jesus on their user page. The hatnote shouldn't link to a religious topic which may not even be the main use of "Numbers" as a title. I think Numbers should continue to redirect to the singular as we usually do. PrimeHunter (talk) 21:52, 30 October 2014 (UTC)
• Now that I've researched and found the actual chain of events, I've removed the hatnote from the article as that is the proper condition per the BRD process. I apologize if it seemed I was for having that content there, I honestly am as unbiased as can be about it as long as process is followed. I also apologize for putting a couple of you in a knee-jerk "quick-revert" situation, but I didn't realize at the time that removing the content was actually the revert and the BRD process wasn't properly being followed to begin with. Anyways, now to discuss what to do with Numbers. I believe that changing the redirect to the disambiguation page is the best option here to prevent undue preference of any one term over another and I don't see there as being a clear primary topic. While "numbers" may be a plural form of "number" it's also a proper noun for at least a TV series (which is no longer in production) and a book of the bible. It's also apparently a magazine, a band, multiple music album titles, and a few other things. So, all of that said, lacking a definite PRIMARYTOPIC and to prevent UNDUE weight being attributed to any of the topics, I'd say the best option is to change the redirect to point to the disambiguation page. — {{U|Technical 13}} (etc) 22:54, 30 October 2014 (UTC)
JohnBlackburne: "… writing e.g. Numbers or Numbers interchangeably …"
To be clear, the wiki markup is: [[Number]]s and [[Numbers]].
--50.53.38.33 (talk) 07:22, 31 October 2014 (UTC)
The problem is that many editors expect the singular and plural to lead to the same article, and if they write a sentence with a plural then they often wikilink the whole word. For example, Distributive property currently includes: "allows a type theory to add concepts like [[numbers]]". PrimeHunter (talk) 13:23, 31 October 2014 (UTC)
JohnBlackburne: "… editors will find themselves repeatedly getting it wrong …"
Redirects are for the convenience of readers, not editors — editors should be testing their links before saving. Anyway, there is a bot that notifies editors who link to a disambiguation page. --50.53.46.109 (talk) 17:52, 31 October 2014 (UTC)

### Redirect

I think it is clear from the comments above that while not everybody agrees, if we put it to a vote then linking numbers to number (redirect) is the solution with the most support. Nobody is likely to type in "numbers" looking for this page, and if they do the redirect will quickly lead them here. Not that it is a big deal one way or the other, but I don't think always linking a plural to a singular is Wikipedia official policy. If I'm wrong, I'm sure somebody will supply the appropriate quotation. Rick Norwood (talk) 23:07, 30 October 2014 (UTC)

I linked to the guideline above, and it's overwhelmingly the practice. I can't think of any instance where the singular goes to an article and the plural goes to another. The only instances I've come across are separate disambiguation pages, one for singular and one for plural, because together they'd be too long. If you still think it should be changed probably the best place is at WP:RFD, which is not just for deletion but discussion.--JohnBlackburnewordsdeeds 01:33, 31 October 2014 (UTC)
How about Blues? Tkuvho (talk) 09:17, 31 October 2014 (UTC)
Good example. There are pages for Blue, Blues, Blue (disambiguation), and Blues (disambiguation). --50.53.46.109 (talk) 14:13, 31 October 2014 (UTC)
Yes, good example. In that case there are two primary topics, the colour and the music type ("The Blues"). There are probably others. But that is an exceptions where the plural has a distinct meaning. Number and numbers are the singular and plural of the same thing, the plural doesn't have a distinct meaning.--JohnBlackburnewordsdeeds 14:37, 31 October 2014 (UTC)
Well, but it does — it means the Book of Numbers, for example; that's the whole reason this is being discussed. "Blues" can also be a straightforward plural of "blue", so the cases are actually reasonably parallel, I think.
I support retargeting numbers to somewhere other than this page, probably number (disambiguation). --Trovatore (talk) 14:51, 31 October 2014 (UTC)
Root is an article. Roots and Roots (disambiguation) are redirects to Root (disambiguation). See also, Statistic and Statistics. --50.53.46.109 (talk) 14:56, 31 October 2014 (UTC)
Dot and Dots are different disambiguation pages. --50.53.46.109 (talk) 15:21, 31 October 2014 (UTC)
Compare Job (article), Jobs (dab), and Job (disambiguation). --50.53.46.109 (talk) 15:38, 31 October 2014 (UTC)
Pen is an article, and Pens is a disambiguation page. --50.53.46.109 (talk) 20:12, 31 October 2014 (UTC)
Sun is an article, and Suns is a disambiguation page. --50.53.46.109 (talk) 20:18, 31 October 2014 (UTC)

This seems pretty definitive. I'll make the change if nobody else does. But I'll wait until tomorrow.Rick Norwood (talk) 22:45, 31 October 2014 (UTC)

Retargeting numbers to numbers (disambiguation) seems to make more sense than retargeting it to number (disambiguation). Tkuvho (talk) 20:38, 1 November 2014 (UTC)
(ec)The problem with that is that numbers (disambiguation) is already a redirect. I do think that it might make sense to split them, though. I think the current setup makes it too hard for readers looking for the Book of Numbers.
Here's the thing: Usually we consider two criteria, the first being what readers are likely to mean when they enter a term in the search box, and the second being what editors are likely to mean when they wikilink a term.
I think the first criterion is likely to point fairly strongly to the Book of Numbers (and perhaps also to Numb3rs the TV show). There isn't too much reason to type the s if you're looking for number.
The second criterion, on the other hand, is more aligned with what John Blackburne has been saying; my intuition is that most editors typing [[numbers]] are likely to intend to link to this article.
But here's the thing: they probably shouldn't be. Extremely general topics like this one are a bit paradoxical; they're very important articles, but they're usually poor targets for wikilinks. There aren't very many contexts where it's natural for a reader to want to say "oh, at this point I want to go read about numbers in general". We should usually discourage wikilinks to extremely general topics, except occasionally from other articles about extremely general topics, of which there aren't very many. --Trovatore (talk) 20:47, 1 November 2014 (UTC)

That would not accomplish anything, since numbers (disambiguation) automatically redirects to number (disambiguation). However, I can't figure out how to change where "numbers" automatically redirects. Does anyone know how to do this?Rick Norwood (talk) 20:41, 1 November 2014 (UTC)

Trovatore, I am not sure what the final conclusion of your train of thought above was but if it is that number (disambiguation) and numbers (disambiguation) should be split, I agree. There are several items there that appear in the plural in addition to the book of Numbers. Incidentally, the book of Numbers is occasionally referred to as "Numbers" just as one might say "Genesis" rather than "book of Genesis". Tkuvho (talk) 21:10, 1 November 2014 (UTC)

I've restored the Numbers redirect to point to this page and created an RfD for it. I should have noticed it before but the list of exceptions to the plural rule our SPA IP found is hardly definitive. There are 8,870 redirects in Category:Redirects from plurals, so the few found are about 0.1% of them. There are very few exceptions, and they usually only exist where there's good reason, such as the singular not having a plural (such as Sun) or the singular and plural meaning quite different things (such as statistic and statisctics). But in this case Numbers is just the plural of Number, the primary topic.--JohnBlackburnewordsdeeds 10:23, 4 November 2014 (UTC)

There are not that many examples of the plural form not redirecting to the singular because adding the final "s" usually results in the meaning of the plural rather than a different meaning. In those cases when the meaning is different, it is appropriate to redirect accordingly. Redirecting Numbers to Numbers (disambiguation) (a new page) is a good illustration of WP:IAR. Other editors are invited to comment. Tkuvho (talk) 10:41, 4 November 2014 (UTC)

I've already given my opinion above, as have a number of others, so there is no need to repeat it. If JohnBlackburne is the only person here who knows how to change a redirect, then I guess what he wants wins. By the way, "suns" is a perfectly good plural. See, for example, "A Thousand Splendid Suns", by the author of "The Kite Runner". And "Numbers" is a common name for a book of the Bible, so it has a meaning unrelated to "number".Rick Norwood (talk) 13:07, 4 November 2014 (UTC)

I am not sure what you mean by your question about changing a redirect. You edit a redirect page the way you would edit any other page, though I don't recommend any further changes until there is a consensus. Note that there is a separate discussion page now for these redirects at https://wiki.alquds.edu/?query=Wikipedia:Redirects_for_discussion/Log/2014_November_4 Tkuvho (talk) 13:34, 4 November 2014 (UTC)
For the record, I'm against a redirect, and therefore think this article ought to have a hatnote. --Sammy1339 (talk) 14:27, 4 November 2014 (UTC)
The article has a hatnote already. The issue is what kind of hatnote it should have. Tkuvho (talk) 14:58, 4 November 2014 (UTC)

Try to vote at the bottom of https://wiki.alquds.edu/?query=Wikipedia:Redirects_for_discussion/Log/2014_November_4#Numbers Tkuvho (talk) 16:04, 11 November 2014 (UTC)

## archiving the talk page

Resolved

The posts on this page go back to 2007. I know there is a way to archive the older ones, but I don't know what it is. If anyone knows how to do this, or how to set up auto-archiving of threads that have not changed in more than a year, it would probably be an improvement. Rick Norwood (talk) 13:13, 11 December 2014 (UTC)

• Hey Rick, you are right. I've set up archiving and archived some of the oldest messages into the first archive to it's maxarchivesize. A bot will come clean up the rest of them that are more than a year old in 24-48 hours. It can be adjusted after that if the page is still too long. Happy editing! — {{U|Technical 13}} (etc) 15:00, 11 December 2014 (UTC)

## Suggested correction

English language 66.87.82.174 (talk) 01:36, 27 January 2015 (UTC)

Context? — Arthur Rubin (talk) 02:26, 27 January 2015 (UTC)

## Rational/irrational

Surely the picture next to the real numbers is wrong, as any real number must be either rational or irrational, and the little Venn diagram shows the rationals and irrationals as disjoint subsets of the reals when one should be the complement of the other?

i.e. the label for "irrational" should go in the whole space that's in the reals but not in the rationals, NOT have its own little circle inside the reals (leaving a large section of the reals that is apparently neither rational nor irrational). —Preceding unsigned comment added by 81.159.20.37 (talk) 23:54, 7 April 2009 (UTC)

Agreed, I came here to make the same comment 202.36.179.66 (talk) 07:14, 3 July 2009 (UTC)
I removed the incorrect image. It had been up there for long enough. It is not only wrong because it implies there are real numbers that are neither rational or irrational, it also could mislead one about the relative "sizes" of the sets (i.e. irrational numbers are far more numerous than rational ones, etc.) —Preceding unsigned comment added by 202.36.179.66 (talk) 02:13, 20 July 2009 (UTC)
Sorry, but the diagram is still wrong; it still implies that there are real numbers of a different type, neither rational nor irrational, with all that free space around both sets. Maybe using a "squarish" diagram you could put the irrational side-by-side with the rational, this one containing all the pertaining number sets. Then draw a thick line around both, forming a rectangle containing both and just it, meaning the real numbers set. Just my ten cents as a Math teacher.189.62.116.193 (talk) — Preceding undated comment added 21:03, 22 February 2014 (UTC)
What diagram are you talking about? If you mean this one, the criticism is valid, but the image doesn't appear in the article, so I'm not sure what the problem is. If you think it's so misleading that it shouldn't be hosted by WP at all, on the grounds that someone might stumble on it and be misled, well, you can make that argument at Wikipedia:Files for deletion. --Trovatore (talk) 21:53, 22 February 2014 (UTC)
It was included on the List of types of numbers article. I removed it from there until a better one can be made.--Theodore Kloba (talk) 21:26, 13 March 2015 (UTC)

something i thought of https://i.ibb.co/1zSRq2d/kuva.png 2001:14BB:1CD:2E04:1553:E147:CAEA:B56E (talk) 12:20, 9 December 2021 (UTC)

## ?Problems with i ?

Just when I thought it was save in the lee of Euler ...

Evidently, there are problems with the roots. Already square roots, as defined with the reals must be cropped to make sound functions, leaving the trace of ambiguity wrt +/- as roots for equations, they do not exist there for negative arguments, yes, and they invite to antinomies, as elaborated in the article, if not handled with utmost care. I really do not get it, where there might be problems with my edits, besides with wikipedians. :) -Purgy (talk) 13:45, 28 November 2016 (UTC)

Ah, I see, my arguments are not even heard, but simply twinkled away. Talk pages are only for the lower classes, obviously mislead lines are ignored, and sensible remedies (dis-)qualified as personal opinion. I revert one last time, and repeat my petition to discuss here, and not via apodictic edit comments. -Purgy (talk) 14:04, 28 November 2016 (UTC)

I want to explicitely confirm that avoiding the misleading sqrt(-1) really is an improvement, beyond personal opinion. It definitely takes the complex numbers first, before one can talk about complex (square) roots. -Purgy (talk) 14:10, 28 November 2016 (UTC)

(edit conflict) Please read WP:BRD. I agree that some of the sentences that you have edited need improvement. But, in most cases, you replace a badly written sentence by something that is worse or misleading or containing a personal POV. As several independent sentences are concerned by the same edit, it is too time consuming for other editors to improve your edits, and the only solution, recommended by WP:BRD, is too keep the old version until a consensus is obtained by discussing here. Thus please, explain the motivation of your edit. D.Lazard (talk) 14:25, 28 November 2016 (UTC)
I think the article would be better off with the sqrt(-1) in the lead removed okay. It would be much better to make a clear case for what you think is actually wrong rather than attacking people for reverting your badly phrased English. I do not think this is an article where 'extreme care' in dealing with the mathematics would be helpful, it should be more of a popular article and made easy to read, though if one can avoid problems so much the better. I think something like '..., and complex numbers, which extend the real numbers by including the imaginary unit ${\displaystyle i}$ such that its square ${\displaystyle i^{2}=-1}$." might do the trick and I'll try putting that in. Dmcq (talk) 17:18, 28 November 2016 (UTC)
It never came to mind that would have attacked someone by my phrasings, but rather -not righteously!- felt myself attacked by D.Lazard's bureaucratic triptychon of "worse, misleading, PPOV", while I had already presented my arguments before. I want to apologize for any offense, and for not being able to make my case clear, which I -unintentionally!- caused by my suboptimal use of the English language. BTW, I was firstly reverted by Paul August for "problematic" content, not inapt language use.
On the case itself, it appears evident to me that you share at least a slight preference for i² = -1 with me, and therefore I want to hint to the rather disputable introduction of a "formal square root", which replaced my "squares to -1" on one occasion. Is "to square to" really that bad a verb? I heard it on many occasions in lectures.
I share the opinion that this is not an article where unconditional math rigidity were a sine qua non, but -as said- the sqrt(-1) is -to me evidently, and as demonstrated in the article itself- more capable of leading to misconceptions, than the offered alternative, even when not that much wide spread in all communities. Under the given circumstances I do not consider it very fruitful to discuss the respective advantages and flaws. Just let me know if otherwise. I consider this a quite readable article with interspersed nice remarks for an interested non-professional.
For the time being I cannot imagine that i² = -1 makes the article harder to read than in the sqrt-variant, but I will refrain from editing in this improvement. -Purgy (talk) 09:28, 29 November 2016 (UTC)
I agree that it is better to avoid, as far as possible, to write ${\displaystyle {\sqrt {-1}}.}$ However we must remember that this article is intended for readers, that may know nothing about algebra, equations, factorization of polynomials (introduced in one of your edits). For a beginner, and historically, complex number have been introduced for making always possible the arithmetic operation of computing square roots, exactly as integers and rational numbers were introduced for making always possible subtraction and division. From this point of view, the old version with ${\displaystyle {\sqrt {-1}}}$ is better than the version of Dmcq: "which extend the real numbers by including the imaginary unit ${\displaystyle i}$ such that its square ${\displaystyle i^{2}=-1}$". Therefore, I'll replace it by "which extend the real numbers by adding a square root of −1". This avoids both ${\displaystyle {\sqrt {-1}}}$ and unnecessary technicalities. D.Lazard (talk) 10:51, 29 November 2016 (UTC)
I most humbly apologize in advance, but it is simply not true that I introduced the notion "factorization of polynomials" into this article. To the contrary, my effort was to connect the original occurrence of this term more obviously to the notion of "algebraic closure", which caused the original mentioning of factorization. Imho, connecting "not algebraically closed" with missing solutions of algebraic equations (I introduced this!), is pedagogically more straight forward, than immediately associating it with factorizability (the original occurrence).
It is in some pervert way amusing that all three people involved in contributing here, agree on some notation to be best avoided, but nevertheless, the one, who claims this most misleading view especially for this here article, supersedes the two others and forces some traditional High School level(?) view into the article. There are no reasons made explicit, why newbies would understand some rubbish better than correct, equally simple definitions. Paul August, who started this dispute by considering my edits not "badly phrased", but "problematic"(sic!) did not comment on this at all, yet. An other remarkable haut gout of this debate is the mother tongue of the ("sqrt of -1" instead of "sqrt(-1)")-advocate: FRENCH! The french mathematicians, world famous for their exquisite formal power and access to math, rotate in their respective graves, when reading about "formal square roots" to introduce complex numbers, and the claim i := sqrt(-1) were "meaningful" in beginners' education and "easier to grasp" than i² := -1.
This article was once proposed for featuring it calling it good, now it is under some neighbourhood watch. -Purgy (talk) 09:18, 30 November 2016 (UTC)
Please stop the personal attacks mixed with apologies and stick to the subject matter.
If a student sees i^2=-1 I don't see why they wouldn't immediately turn that into sqrt(-1) in their minds so whilst I can see there are problems with sqrt(-1) I don't see that avoiding any mention of it will fix anything. As to High School level, the obvious question is why is closure so important and useful that one adds unreal quantities, isn't it like adding the inverse of zero? So I don';t see a reason why the idea of closure should be added before sqrt(-1) in an article about numbers. Dmcq (talk) 10:02, 30 November 2016 (UTC)
OK, no more efforts to signal a cooperative mindset:
• Please, stop that libel and slander by repeatedly, unsourcedly accusing me of attacking persons.
• I strongly claim that the reverted formulations in the article are certainly not below the linguistic average in Wikipedia (I do not refer here to my angered texts in the talks), and do not deserve the standard rebuffing verdict "badly phrased".
• As one can easily verify, my starting statement rotates about arguments against sqrt(-1) and pro i²=-1. That the consensus, the latter being the better introduction, nevertheless lead to rubbish (formal square root!) prevailing, without any reflexions on why not change, naturally and necessarily drew my replies to the embedding environment.
• Even when senselessly repeated, it is not true that I introduced the term "algebraic completeness" in this article.
I want to avoid looking like trolling against the petrified Wikipedia establishment (too much rock), so I will stop commenting here (until further notice, at my discretion). Nevertheless, I really(!) cordially invite anyone to my talkpage for an exchange on a fair footing. -Purgy (talk) 10:48, 1 December 2016 (UTC)
I am sorry you take that attitude. I will not interact with you except where necessary unless you confine your comments to the subject matter. Dmcq (talk) 13:05, 1 December 2016 (UTC)

Since you're sorry, take this! Nothing is new in the biggest part, all is compiled from above:

Just when I thought it was save in the lee of Euler ... (to throw in a reputable source)
Arguments against roots:
- as defined with the reals, (they) must be cropped to make sound functions (doing awy with the lower branch of the relation)
- leave a trace of ambiguity wrt +/- (whereever they appear)
- they do not exist within the reals for negative arguments (why should they for i ?)
- invite to antinomies, if not handled with utmost care. (as elaborated in the article)
- It definitely takes the complex numbers first, before one can talk about complex (square) roots

Arguments for introducing i² = -1
- avoiding the misleading sqrt(-1) really is already an improvement (squaring is a well behaved function)
- this sensible remedy (is)(dis-)qualified as "personal opinion" (as defence, appealing to Euler above)
Questioning the reasons for reverting (is this already considered inappropriate???)
- Where are the (postulated) problems with my edits?
---
All the above was written even before(!) you tried to insert the i² = -1
---
Immediately afterwards I
- critisized the rather disputable introduction of a "formal square root" (not defined for reals, still not applicable to complex numbers)
- stated agreement to certain views (I share the opinion ...) on the article,
- reinforced the above mentioned drawbacks of the roots-formulation and
- documented my inability to perceive any increase of conceptual hardship with the i² = -1 variant, which is just postulated without any supporting argument.
---
As something new I add in reply to your opinion on students, that I certainly would not require students to forgo a (to be introduced) complex square root, but I would not squeeze in inappropriate tools, like the real square root, to fit in an undefined environment. I prefer to admit to the reader that at the stage of introducing i, the concept of a complex root is not yet formalized. I prefer the caveat to the "trust me"-promises, and the complex root is fundamentally different to the real root in some way.

BTW, how to deal with the untenable claim that I had introduced the notions of "factorizing of polynomials" and of "algebraic closure" in the article?

Besides some timing reference, and my personal need for sincerity, this is strictly on the subject matter, at least imho. -Purgy (talk) 10:22, 2 December 2016 (UTC)

I just did a Google of "how should complex numbers be introduced" and got some interesting results. There seemed to be no great preference for i or sqrt first in the individual introductions, but do tend to show both i^2+1=0 or i^2=-1 and sqrt(-1) early on, so I think the most relevant one I can point to for ideas about this is at stackexchange How can I introduce complex numbers to precalculus students? plus some of the links it references. I tried also "How best to define a complex number" and got a different set. Wikipedia is not supposed to be a text book and is more on the define side though it is not a dictionary either. I do think there is a preference for throwing a couple of definitions and examples at people rather than trying to give just a single definition and exclude others. Dmcq (talk) 11:10, 2 December 2016 (UTC)
Since it does not seem to be over yet (please, see below), I ask for your appreciation of me withholding my comments to your search for the time being. -Purgy (talk) 17:05, 2 December 2016 (UTC)
Your reverted edit in the article includes "which would allow for a factorization of the associated polynomial in linear factors". Your claim that you "had not introduced the notions of factorizing of polynomials" is therefore untenable. The only mention of "algebraic closure" in the recent discussion is yours in the summary of this edit [1]. In the same edit, you have introduced the phrase a solution to the algebraic equation, which refers explicitly to algebra and equation. This was this phrase that I had in mind, when writing However we must remember that this article is intended for readers, that may know nothing about algebra, equations, .... Thus, please, stop complaining about inexistent accusations. D.Lazard (talk) 12:11, 2 December 2016 (UTC)
OMG!, can't you be satisfied that your consensually deprecated version still dominates the article, and for whom might it be necessary to give a line of evidence that I'm correct, and you insist on battling?
Looking a few paragraphs below my partially reverted edit you will find the sentence:
In abstract algebra, the complex numbers are an example of an algebraically closed field, meaning hat every polynomial with complex coefficients can be factored into linear factors.

I do not know when this sentence was introduced, but it definitely takes precedence to my edit, in which I tried to put the also already existing sentence
The real numbers are not, however, an algebraically closed field, because they do not include the square root of minus one.

in a better formal interrelation, closing a chain of notions from "alg. closed" -> "missing solution" -> "factoring of polynomials" -> "alg. closed" by editing it to:
The real numbers are not, however, an algebraically closed field, because they do not include a solution to the algebraic equation x²+1=0, often addressed as the square root of minus one, which would allow for a factorization of the associated polynomial in linear factors.

Please note, how delicately I tried to edit, and still, I never said a single word against the reversion of these -imho very well explainable and explained(!)- edits, even when I strongly disagree with the given arguments for the reversion, but I protested and I still protest against the claim that I had introduced the notions of factorization and closure into this article.
Therefore,
- my claims hold wrt "factorization of polynomials (introduced in one of your edits)" by D. Lazard, and wrt "I don't see a reason why the idea of closure should be added" by Dmcq.
- I never disputed having introduced the term "solution to the algebraic equation"
- I never "complained about inexistent accusations", so I cannot stop this
I do, however, strongly doubt that there are many readers of this article, "who know nothing about algebra, equations, ...." There may well be many readers who have not heard about Gauß Fundamental Theorem, and about "closure" in general context, but as said already, I consider these remarks as little highlights, not impeding it's readability, but generating curiosity.
I also wonder why a concept, consensually deprecated, still subjugates the article, and why the obvious flaw of "formal square root" is not considered as detrimental, both to Wikipedia's reputation, as to readers' possible gain from this article, and why this is therefore not reverted. -Purgy (talk) 16:43, 2 December 2016 (UTC)

## Skimming through stackexchange

When looking at the efforts there, I could not destill that i²=-1 and i=sqrt(-1) were treated on equal footing, not even close. There is one particularly nice article, hailed to more than 700 upvotes, which (also!) does not mention the sqrt in introducing the complex numbers. The whole collection of remarks is bantered by the laudable search for a most visually accessible introduction of the complex numbers. Obviously this lead to representing them in the plane, with multiplcation as rotating and stretching, identifying the i-fold as a 90°-rotation, not in the minimum relying on sqrts. What a nice coincidence when heuristics and pure math intentions meet in one view : i²=-1 !

It would be ridiculous to claim that i²=-1 would not call for the question about a square-root-function in the complex domain, applicable even to -1, strongly disallowed with the real square root, but I consider the answer to this to be a welcome consequence of introducing i via its square, and not the rhizom of its introduction. Up to now I missed the chance to get to know why a sqrt-introduction of i could be seen as methodologically, didactically, or mathematically advantageous.

Looking a bit beyond the currrent horizon, at the construction of the quaternions, there the famous Hamilton carvings i²=j²=k²=-1 show up, documenting a principle of construction. Is this a reason for hiding i²=-1 behind i=sqrt(-1)?

I cannot assume that sufficient qualified reliable sources do not prefer the i²=-1, I have not seen any evidence of disadvantages, nor of any advantage of the sqrt, I even meant to perceive consensus on the sqrt being problematic, so, please, why is there still this strange claim alive that the number-article must preserve the sqrt-introduction, and keep up unsourced constructions of "formal square roots", and what are the "problems" announced with the first revert of my edits here. -Purgy (talk) 11:21, 6 December 2016 (UTC)

What was upvoted more than 700 times? You're not confusing with bronze badges are you? Please be specific about what you are referring to rather than expecting me to search around trying to figure it out from the long list of possibilities. Referring to specific instances to illustrate what you are saying would be very helpful. Dmcq (talk) 12:14, 6 December 2016 (UTC)
Following the first link in your reference to here, and scrolling to the second(?) entry, leads to the mentioned article, which I assume to have this number of upvotes. Sorry, I do not know that much about stackexchange to be aware of bronze badges, their looks and places and meanings.
I'm certainly prepared to show you all the ways to places I know, and, honestly, I expected you had been there, already, and I certainly did not expect, that you would search for it. May I, in exchange, also ask for places, offering answers to my questions for preferring the sqrt-introduction, and to be more specific about the items I should illustrate? -Purgy (talk) 14:22, 6 December 2016 (UTC)

After I was forced to give a line of evidence against libelous claims of me " complaining about inexistent accusations", I hoped to get rid of this dead freight by starting a new section, reporting about impressions of some linked to place, and collecting some material in the intent to contribute to an improvement of this article. -Nada!- I had to take it that I have to "be specific" about places I assumed to be known, and that I'm not allowed "to expect" someone would search, even when I didn't do this in the slightest, and that I should submit "specific instances to illustrate what I were saying" to make my suggestions -perhaps then!- a candidate to be considered gratiously for further contemplation. But, unlucky me, offering the required link, cheekily setting a personal pronoun in italics (to make evident, which link was addressed! -yes, these italics), re-generated too much negative vibes ... (see above)

NO! I am not going to expose myself any further to a never ending story of offense and humiliation, just for trying to be allowed to discuss improvements of this article. I declare that I am prepared and willing to try to contribute to an improvement of this article, and discuss this with everyone (really everyone!), who is willing to do so on equal footing (not just: it's me, who's right), but not as a petitioner, being talked down and rebuffed for intransparent reasons. Recall, that I still wait for being given a rational reason, why my primary edits were considered problematic, and therefore reverted first hand. -Purgy (talk) 13:10, 7 December 2016 (UTC)

## Number article names

The articles about the numbers from 0 to 9 are called 0 to 9. However, the article about the number 10 is called 10 (number). I believe that it should be called 10. Is it possible to change the name of the article? Bobby Jacobs (talk) 23:25, 5 March 2017 (UTC)

I noticed that 10 (number) was just moved to 10. However, there are more number articles that should be moved. Bobby Jacobs (talk) 15:05, 8 March 2017 (UTC)
Just for reference: there was an RFC about the titles of these articles, see Talk:AD 1/Archive 1#RFC: Should articles "1" to "100" be about numbers instead of years?. -- Tea2min (talk) 15:12, 8 March 2017 (UTC)

## Irrational Numbers

Why are irrational numbers not included in the 2nd paragraph. The phrases "In mathematics, the notion of number has been extended over the centuries to include 0, negative numbers, rational numbers such as 1/2 and −2/3, real numbers such as √2 and π," implies that the rational numbers are the real numbers whereas the real numbers include the rational numbers, the irrational numbers, the natural numbers, the negative integers and 0. Ruskin (talk) 12:17, 3 April 2017 (UTC)

To include the irrational numbers in that sentence you would need to answer the question–what are the irrational numbers an extension of? Since there is no answer to this question, the irrationals don't belong in a discussion about extending number sets. To put this into perspective consider this. There are rational numbers which are not integers, yet you are not claiming that they should be included as a separate set of numbers in this progression. --Bill Cherowitzo (talk) 17:48, 3 April 2017 (UTC)

## Help on edits of History

Could someone knowledgeable, please, check the authenticity and the removal of a picture by edits starting here? Purgy (talk) 12:18, 12 April 2017 (UTC)

The main changes I see were first to combine two paragraphs on Bhramagupta, and second the (I would guess accidental) deletion of a valuable image. Earlier, someone deleted the whole section on zero, but that was quickly reverted. I think the section looks ok now, but I'll reread it and see if I catch anything. Rick Norwood (talk) 12:31, 12 April 2017 (UTC)

## Strong reservations

I want to express distinct objection to the last edits by Gireen, at least as far as content beyond references (which I cannot judge) is concerned. In my opinion the statements of what is defined as what require revision, and especially the claim of "irrationals" being "decimals" is flawed. Perhaps some guardian angels should have a look at this. Purgy (talk) 05:46, 23 May 2017 (UTC)

Specifically on the edit to the irrationals section, I think you read my revision incorrectly. The (cited) definition I gave for irrational numbers was "Real numbers that cannot be represented as the fraction ${\displaystyle a \over b}$ for any integers ${\displaystyle a}$ and ${\displaystyle b}$". I then, later in the paragraph, said that "A decimal represents an irrational number if and only if it has an infinite number of digits and does not eventually repeat", which is not only true, it can be logically derived from the sentence you reverted the document back to ("A decimal represents a rational number if and only if it has a finite number of digits or eventually repeats forever, after any initial finite string of digits"). All I did was turn the sentences from "P is sufficient for Q" to "¬P is necessary for ¬Q". If you have a big problem with that statement you should consider editing the current document aswell, because it says the same thing.
Regardless, what else did you see that you disagreed with, so I can amend the original edit?— Preceding unsigned comment added by Gireen (talk (talkcontribs) 01:14, 24 May 2017 (UTC)
I agree with Purgy Purgatorio, and I have reverted your edit. Feel free to reinsert the references that you have added, if they are relevant for the present state of the article. Here are the detailed reasons of my revert:
• Natural numbers ... are formally defined ...: Normally, the normal user of this article ignore the meaning of "formally defined". Moreover, you introduce a circular definition, as whole numbers are usually defined from natural numbers.
• Proof in section "Rational numbers"": useless here, as this section is a summary of another article, and confusing, as possibly hiding the main points of this summary
• Section "Real numbers": circular definition again, the reals are defined in terms of rationals and irrationals, and later the irrationals are defined in terms of reals and rationals.
D.Lazard (talk) 06:52, 24 May 2017 (UTC)
Edit conflict, perhaps some duplications contained:
Line 52: Not that I am a fan of " most familiar numbers", but "formally defined as as the set of ... whole numbers" without "formal" definition of whole numbers is definitely worse to me. (In my world the whole numbers are constructed from the naturals.) Removing the parens around "sometimes called ... " gives even more, instead of less importance to a deprecated naming, assumedly perpetuated in elementary school. Supressing a "reason" for including the 0, (strangely) imported from a group structure and not inherent to the naturals per se, is imho no improvement to this article.
Line 81: Your footnote mistakes the "definition" of "equality of rationals" for a theorem in need of a proof.
Line 91: Assuming the "irrationals" to get a hold on the reals by combining them with the rationals is circular reasoning, therefore I object to this edit. Again, "measuring numbers" are not my particular favourites, but they do not pretend to be "formally" defined. You are right that the propositions about non-rationals and irrationals within the reals are equivalent, but even though I did not oppose to this very edit, I do not consider it an improvement. "Irrational" is more of a naming convention but a useful definition, they are a continuum and there are many variants to be irrational ;) . I deny the a priori ability of decimals for "having infinitely many digits". This requires considering limits and gets deeply involved in the definition of reals. I strongly plead for restricting all these didactic troubles associated with the equality 1=0.999... within this rather elementary article to an absolute minimum. I consider "infinitely repeating" as less embarrassing than "infinitely many digits". Decimals are best left outside of abstract considerations, imho.
Line 102: The deleted sentence is correct and more appropriate than referring to non-repeating or infinitely many decimals.
I hope you take from this that it is tedious to selectively revert and to discuss ex post some perhaps inappropriately bold edits, and that it is advisable to discuss such things in advance. Finally, I want to remark that the reversion by D.Lazard is certainly not based on personal amity.
Please, do not amend your original edits, they are refuted. Purgy (talk) 08:28, 24 May 2017 (UTC)

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## Introduction of i

Now that a bot sent some animosity to oblivion, may I ask if re-discussing a revamp of the introduction of ${\displaystyle i}$ in this article as e.g.

${\displaystyle i}$, a new number: a square root of −1 modified from "${\displaystyle i={\sqrt {-1}}}$" 16:19, 9 December 2017 (UTC)
vs.
${\displaystyle i^{2}=-1}$

might be reasonable?

I do not want to appear as insisting on a specific version of introduction in this article, I just suggest this according to my personal conviction of what I estimate as superior from an encyclopedic-mathematical view. Purgy (talk) 10:29, 9 December 2017 (UTC)

I do not see the issue. I agree that the notation ${\displaystyle {\sqrt {-1}}}$ is problematic, but it does not appear in the article. Nevertheless, I have added a link to Square root, where the first sentence asserts that the sentences "i is a square root of –1" is synonymous to "the square of i is –1". Thus the presentation in the article ("adding a square root of −1" and "a new number: a square root of −1") is conform with the historical origin, and coherent with the general terminology used in WP. Thus, the only thing that deserve a discussion is to decide whether a caveat is useful for warning that the use of ${\displaystyle {\sqrt {-1}}}$ is not common and possibly confusing. D.Lazard (talk) 12:15, 9 December 2017 (UTC)
First, I beg pardon for having sloppily abbreviated the current verbiage of introducing ${\displaystyle i}$. I modified my statement above.
Of course, if ${\displaystyle i}$ were introduced to my likings, I would not see primarily any issue either, and evenly of course, I do know that the current phrasings (including the "formal square root"), to which I oppose, are your copyright. As for your links one can easily check that I also tried, fully unsuccessfully, to put the legen–wait for it–dary imaginary unit under scrutiny. This is not to deny any historical roots, but to respect a more contemporary view on this non-unity ${\displaystyle i}$, not distinguishable from an other, set aside object, the square of which is also -1. Yes, I know I am not congruent with the current "general terminology" in WP, which I consider -pardon- outdated in this respect.
It is my agenda to avoid the hidden ambiguity in a non-existent/yet undefined/multivalued, ... "${\displaystyle {\sqrt {-1}}}$", no, "square root of −1" as much as possible, and make the inherent bi-valence apparent by ${\displaystyle i^{2}=-1}$, and work from this solid base to the interesting questions, like e.g., roots of unity. I am not interested in discussing much about caveats before this is correctly under the belt. Imho, the use of ${\displaystyle {\sqrt {-1}}}$ is neither uncommon nor confusing, if carefully introduced employing ${\displaystyle i^{2}=-1}$, instead of juggling with roots, imported from reals and imposed on negatives in an abusive manner, just calling them formal.
It is self-evident that me alone is no consensus, and if you see no issue, and nobody else does, besides me, then we can most simply wait for the bot archiving this, too. Cheers, Purgy (talk) 16:19, 9 December 2017 (UTC)
I understand better where is the point for you. This seems to be related to the reason for which I qualifies the notation ${\displaystyle {\sqrt {-1}}}$ as problematic. There are many ways for explaining that. One of the simplest is that this notation does not allows distinguishing between the phrases "a square root" and "the square root", or, worse, suggests the implicit use of the second phrase. The confusion between these two phrase is common: one read often "${\displaystyle {\sqrt {2}}}$ is the square root of two" (wrong) instead of "${\displaystyle {\sqrt {2}}}$ is the positivesquare root of two" (correct). Another explanation, more technical, is that ${\displaystyle {\sqrt {\;^{\;}}}}$ suggests a function of the complex plane to itself, and that there is no square root function that is continuous over the whole complex plane.
Thus the sentences "x is a square root of y" and "y is the square of x" are strictly equivalent. This what I meant when linking to Square root, and writing that I do not see the issue.
Nevertheless, there is, here, a rather deep natural question, which took more than a century to be understood (roughly, from de Moivre to Galois): which square root of –1 is chosen for i? The answer of this question would be misplaced in this article . Nevertheless the right answer is that any square root of –1 is convenient, and this does not matter, as every algebraic formula that is true for i is also true for –i. This is this property that makes the importance of the conjugation. It generalization to the roots of an irreducible polynomial (every algebraic formula that is true for one root is true for all roots) is sometimes called Galois principle.
You may be right that ${\displaystyle {\sqrt {-1}}}$ is commonly used. But the use of ${\displaystyle {\sqrt {\;^{\;}}}}$ with an argument that is not a positive real number is highly confusing and should be carefully avoided. I thin that you are confused because of not having understood this. D.Lazard (talk) 18:45, 9 December 2017 (UTC)
I can easily agree with you that parts of me preferring "${\displaystyle i^{2}=-1}$" to any "formulation involving roots", when introducing i, are based on all these subtleties you mention yourself. But I cannot view the suggested discrimination between "explicit use of the surd symbol" and just "addressing it verbally" as a viable work around. My preference relies also on the "century of mathematical understanding", which adds another scent to the ways of introducing i. I also agree on what you refer to as Galois principle, and also on the importance of conjugation, the first probably being too aloof to be mentioned here, but the second should be addressed as something new that can be done to (complex) numbers, besides converting them to their additive inverses.
However, I do not agree on the sentences "x is a square root of y" and "y is the square of x" being "strictly equivalent". In introducing the complex numbers it is to me of fundamental importance to characterize x as resulting from a square, based on the fundamental principles of multiplication, opening exactly two possibilities, and not only stating that x were "a" square root, with this notion still lacking any meaning within the complex realm. This requirement is connected to my statement that I prefer the "explicit" bi-valence to a "hidden" ambivalence, and is also based on the fact that at this point along the introduction the "square root" has no mathematical meaning for y, yet. I suppose, at this very point you were inspired to introduce the "formal square root", with equally undefined content. An additional small point of disagreement is that I do not feel confused, not even a little bit, about a careful extension of the notion of roots within reals to the roots within complex numbers. To the contrary, as you can see partly from me opposing against the "formal root".
I think introducing i via roots may be historically correct, but is nowadays mathematically outdated, and I also think that the step from "carefully avoiding surds" to "introducing via square" is highly recommendable within WP. Purgy (talk) 10:34, 10 December 2017 (UTC)

## Transfinite numbers

Triggered by recent edits: should there be a statement in this paragraph, similar to

Ordinal and cardinal numbers have been introduced at the end of 19th century by Georg Cantor, when he initiated the study and the classification of infinite sets. D. Hilbert called this a paradise, mathematicians never would allow to be expelled from.

(Suggestion mostly of the first sentence more precisely 17:06, 3 August 2018 (UTC) by D.Lazard). Purgy (talk) 08:36, 3 August 2018 (UTC)

This discussion began on my talk page, about a revert I did on this page. My suggestion is only the first sentence, and is aiming to provide an historical context to this generalization of the concept of numbers. I oppose to include the second sentence in this article, because the section does not contains enough information for allowing the reader to have an opinion about Hilbert's comment. Also, this comment makes sense only if replaced in its historical context, which is out of scope here. D.Lazard (talk) 08:59, 3 August 2018 (UTC)
I just made a suggestion, which easily can be discarded. I oppose, however, that readers would "need an opinion" about Hilbert's statement, and claim that this citation expresses Hilbert's appreciation without any knowledge of historical context or, even more rare, about the deep implications of Cantor's notions. I hope I did nothing offensive with dragging the suggestion here, I just liked it. Purgy (talk) 17:06, 3 August 2018 (UTC)

I am not favoring or opposing inclusion of one or both of these sentences, I just want to improve the grammar. I suggest the following:

The mathematical distinction between ordinal numbers and cardinal numbers was introduced near the end of the 19th century by Georg Cantor, when he initiated the study and classification of infinite sets. Many mathematicians considered this a discover of great importance to all of mathematics. For example, David Hilbert said, "No one shall expel us from the paradise that Cantor has created. reference: "On the Infinite" in Mathematische Annalen 95, (1926).

Rick Norwood (talk) 13:02, 3 August 2018 (UTC)

I think it was more than just the distinction between them ... Purgy (talk) 17:06, 3 August 2018 (UTC)

## transcendental numbers

Transcendental numbers are first introduced as a sub-class of complex numbers. Is this correct? --Richardson mcphillips (talk) 16:17, 12 November 2018 (UTC)

Not exactly. As far as I know, real transcendental numbers have been considered first. Although complex numbers were well known when transcendental numbers were first considered (19th century), the focus was set on real transcendental numbers. A possible reason was that, for a complex transcendental number, either the real part or the imaginary part must be a real transcendental number. So the study of transcendence is basically a problem of real analysis. D.Lazard (talk) 16:49, 12 November 2018 (UTC)
In my first answer, I understood "first" as referring to history. Reading the article again, it appears that it refers to the first place in the article. The definition given in the article is correct as being the one that is generally given in modern mathematics. Restricting transcendental numbers to real ones would also be correct, because a definition is simply a convention, and thus cannot be wrong. But this would complicates phrasing in some situations. For this reason the definition of the article is more convenient. D.Lazard (talk) 20:54, 12 November 2018 (UTC)

## Extension to complex numbers

@D.Lazard: With regard to your revert, on which you commented "Badly formatted and wrong (many complex numbers are not purely imaginary)", I edited the sentence

In mathematics, the notion of number has been extended over the centuries to include 0, negative numbers, rational numbers such as 1/2 and −2/3, real numbers such as √2 and π, and complex numbers, which extend the real numbers by adding a square root of −1.

because, as I read it, it meant that the set of real numbers was extended by the result of adding a square root of -1, with no multiplier, to each real number. That is, alongside 7.85 there would be 7.85 + i, and maybe also 7.85 - i, but not 7.85 + 2i. The use of "add" here is confusing, I think you'll admit, when the other references to the same concept are framed in terms of sets in a way even a layperson can understand, using "extend" and "include", while in a mathematical context "add" most obviously refers to _+x, not _∪{x}. And since √-1 has two possible values, i and -i, the singular "a square root of -1" can easily be taken to mean "only one of the square roots of -1".

I certainly didn't intend to imply that complex numbers were only imaginary, and I'm afraid I don't see what caused you to think I did.

As to the formatting, you are quite right: I typoed a period for a space in the string «term "number"». --Please {{Ping}} me to discuss. --Thnidu (talk) 03:55, 25 March 2019 (UTC)

To editor Thnidu: The intended meaning of "adding" is clearly "adding to the set of real numbers". I agree that this formulation is confusing. Your formulation is also confusing, suggesting that every complex number is the product of i and a real number. I'll edit the sentence as a tentative for clarify this. D.Lazard (talk) 04:12, 25 March 2019 (UTC)
To editor D.Lazard: Your revision is much clearer and, I think, satisfactory without further editing. --Thnidu (talk) 04:37, 25 March 2019 (UTC)

## What is done with numbers?

IMO the article needs to begin and reorganize around what is done with numbers, including (but not limited to) counting, indexing, sorting (sorting and indexing are two different methods for sequentially ordering data in tables), measuring, etc. I would like to see at least a 3-4 sentence 'explanation' of the uses of numbers that lists uses and a concept of 'number' (number theory). MaynardClark (talk) 18:20, 18 May 2020 (UTC)