Talk:Associative property

Definition

The definition should be moved to the introduction as it is in other articles such as commutative operation.

Brianjd 04:27, 2004 Nov 14 (UTC)

Operator or Operation?

I would prefer operator to operation, but as I've noticed many articles using operation I hesitate to break consistency by changing over.
—Herbee 16:09, 2004 Mar 1 (UTC)

For the sense intended here, binary operation is the usual term. For example, binary operation is what the AMS call it in the current version of the MSC. --Zundark 17:11, 1 Mar 2004 (UTC)

Move to Associative operation?

Right now, associative operation is just a redirect to this article. I think this article should be moved to associative operation to be consistent with other articles such as commutative operation.

Brianjd 04:32, 2004 Nov 14 (UTC)

Disagree. I believe this page should be called Associative property (also a Redirect to here BTW) because Associativity is not a word. A Majority of sources would show Associative property and almost never is the -ity appended in reliable sources. Cliff (talk) 21:24, 1 April 2011 (UTC)

Either name is ok with me but your move rationale is completely bogus. Associativity is used frequently as a word to describe exactly this concept, as even the most cursory of searches in Google books would reveal. —David Eppstein (talk) 04:50, 7 May 2011 (UTC)

run your Google book search again. Primary results for "associativity" occur in books about programming languages. Primary results for "associative property" and other variants occur in books about mathematics. The real place you should be looking, anyhow, is in mathematical journals. I doubt you'll see much use of "associativity" there. Cliff (talk) 05:28, 7 May 2011 (UTC)

Even if this were true, the programming language meaning is exactly the same as the mathematical meaning, so why shouldn't programming language books count? —David Eppstein (talk) 01:43, 8 May 2011 (UTC)

Of course you will. Try a search on Zentralblatt MATH: 216 papers with "associativity" in the title, and only one with "associative property" in the title (and that one is just a bad translation from French — should be "New proof of the...", not "Proof of a new..."). --Zundark (talk) 08:51, 7 May 2011 (UTC)

Try your search again, 215 articles with associativity in the title, 2440 articles with associative in the title. Of course "associative property" won't occur in the title, because it is an elementary concept that doesn't lend itself to research ,not to mention that it leads to crappy titles. Again, i'm not claiming that "associative property" is the best, but simply that "associativity" is likely the worst. Cliff (talk) 20:49, 7 May 2011 (UTC)

Nobody claimed that "associativity" is more common than "associative", only that "associativity" is commonly used in mathematics. That "associative property" leads to "crappy titles", suitable only for Wikipedia, is not a point I feel inclined to argue with. In fact, it probably leads to crappy sentences as well, but I'm past caring. --Zundark (talk) 22:07, 7 May 2011 (UTC)

The article should be called "associativity" as others have suggested. The term is common in mathematics so Cliff's assertion that it is "not a word" is false. The term avoids the current redundancy in the lead of the form, "The associative property is the property that...". I disagree with Cliff. Jason Quinn (talk) 18:01, 4 July 2012 (UTC)

Inconsistency here ?

There appears to be an inconsistency in this page. If addition is associative, then subtraction is also associative. i.e. (a+b)+c = a+(b+c). Taking a=5, b=-3, c=-2, we have (5-3)-2 = 0 = 5 + (-3-2). If you want to put the parenthesis after the minus sign, as in the example, then you must write 5-(3+2), which is 5-3-2.

I suggest that subtraction as an example of non-associativity be removed. It could be replaced by some other example: e.g. suppose we are in New York and I tell you that you can find a buried treasure by walking 600 km north, 1000 km west and then 600 km south. You will not find the treasure if you walk 1000 km west, 600 km north and then 600 km south!

User:Philip J Kuntz 00:10 2005 Apr 17

yes you will, and you'll have walked 1200 km too much

• I'm afraid the page is correct. Adding -3 *gives the same result* as subtracting 3, but it is not the same operation. Your example uses addition, not subtraction. The example you give, with a=5, b=-3, and c=-2, should read (5 + -3) + -2 = 0 = 5 + (-3 + -2) . This is very different from (5 - 3) - 2 = 0 but 5 - (3 - 2) = 4 .

I'm not sure how well I'm explaining this. Feel free to leave a note on my talk page, and I'd be happy to talk about this. -- Creidieki 19:26, 17 July 2005 (UTC)

Disambiguation

Added reference to CPU cache which discusses associativity as it relates to computer processor architecture.

Intuitive meaning of associativity

I hate to ask such a general question, but could anyone tell me anything about how I should intuitively think about associative versus nonassociative operations? I've been staring at Category Theory for a while now, and the definition of a category requires arrows to be associative. Why? How are associative operations different from nonassociative operations? The article mentions that "order of operations is immaterial", but I don't know that I have a very good intuition for what that means, or why it's important. How do other people think about associativity? -- Creidieki 19:29, 17 July 2005 (UTC)

It is the order of the operations that is immaterial, not the order the symbols occur. I reworded the explanation of associativity to make that clearer. Wellsoberlin (talk) 22:39, 22 November 2008 (UTC)

Definitions are intended to be useful. Is it useful to allow composition of arrows in a category to be non-associative? Probably not, because you can't really prove much about categories without using the associativity, and in any case arrows are intended to behave like functions, and composition of functions is associative. --Zundark 20:11, 17 July 2005 (UTC)

Assignment operator

I've just removed the text

• The assignment operator in many programming languages is right-associative. For example, in the C language

x = y = z;  means  x = (y = z);  and not  (x = y) = z;

In other words, the statement would assign the value of z to both x and y.

The assignment operator is more than a binary operation, and it is beyond the scope of this article. In fact, in the C language the expression x=y does return a value, which is the value of y. In this sense = is associative after all, since "(x=y)=z" = "x=(y=z)" = z! 128.12.181.34 00:30, 23 September 2005 (UTC)

Associativity confusion

This is now a complicated issue. All I would like, is for some one to write down in plain English the meaning of: "\left. \begin{matrix} (x+y)+z=x+(y+z)=x+y+z\quad \\ (x\,y)z=x(y\,z)=x\,y\,z\qquad\qquad\qquad\quad\ \ \, \end{matrix} \right\} \mbox{for all }x,y,z\in\mathbb{R}." and other such incantations on these pages. Although I have done maths beyond secondary level, I have never seen this type of phaseology. I would be most pleased if this type of explanation could be simplified for us plain folk. I am very interested and would be most grateful.

--59.167.9.205 07:52, 6 August 2006 (UTC) Regards BobC

It sounds like you're not seeing the images, and are therefore seeing the raw TeX from the 'alt' attributes. What web browser are you using? --Zundark 08:30, 6 August 2006 (UTC)

Thanks Zundark. I was using Firefox 1.5.0.5. I will tell Mozilla about the problem. I tried Internet Explorer 5 and now I see the light.

Well, I'm using Firefox 1.5 too, and I don't have a problem. I suspect you have Wikipedia images blocked in Firefox for some reason. Try looking at Tools / Options / Content and make sure that "Load Images" is ticked and "for the originating web site only" isn't ticked, and that up.wiki.x.io isn't listed as blocked under Exceptions. --Zundark 07:33, 7 August 2006 (UTC)

Infix notation is confusing

For some time I'm quite against infix notation in both programming languages and in mathematics, since it makes many-many confusions. Almost every time I encounter descriptions of associativity and commutativity I see explanations like here.

Here is an example of what I'm talking of, when we use the prefix notation "add(x,y)" instead of infix "x+y".

commutativity is then:

add(x,y) = add(y,x)

but associativity is:

add(add(x,y),z) = add(add(x,z),y) = add(add(y,z),x)

another bonus example of commutativity after this, with 3 variables this time:

sub(add(x,y),z) = sub(z,add(x,y))

This can be rather confusing with the infix notations. These are the same three lines as above:

x + y = y + x

(x + y) + z = (x + z) + y = (y + z) + x

(x+y)-z = z-(x+y)

I think if someone does not know what associativity is, will not understand it from the above examples as clearly as by prefix notation. Please consider my proposal.

--

No, associativity is add(add(x,y),z) = add(x, add(y,z)). Ralphmerridew (talk) 12:14, 18 August 2008 (UTC)

Series example

I find this example questionable, because it plays fast and loose with the definitions of the partial sums. The series Sum(0, infinity, pow(-1, n)) is not convergent at all, because the limit of the sequence Sn = Sum(0, n, pow(-1,n)) does not exist. The series Sum(0, infinity, (1-1)) is trivially convergent to zero. The second "series" appears to be the expression 1 - Sum(0, infinity, (1-1)) which is not the same expression as the first series. Yaush (talk) 15:15, 12 May 2010 (UTC)

I absolutely agree, there is no immediately obvious way to apply the given definition of associativity to result in the two given 'infinite sums'. The notation 'a+b+c+d+…' doesn't uniquely specify a series in the formal sense, there could be infinitely many finite partial sums of it regarded as summands of the series. One might have to define a generalized form of associativity first or even need infinitary logic. Ninjamin (talk) 13:15, 3 April 2019 (UTC)

The series example has been removed from the article, probably since several years (I have not checked). Nevertheless, the use of associativity for telescoping series deserves to be mentioned in the "See also" section. I have added it. D.Lazard (talk) 14:26, 3 April 2019 (UTC)

Not sure the linear transformation reasoning is complete

Seeing the following,

Because linear transformations are functions that can be represented by matrices with matrix multiplication being the representation of functional composition, one can immediately conclude that matrix multiplication is associative

ei ther I am quite dense or this little prooflet is incomplete -- or both. I'm not suggesting it's wrong, only that the demonstration is too concise. Dratman (talk) 01:51, 4 July 2011 (UTC)

N-ary associativity

The paragraph of the previous versions of the article about this topic is much too technical for this article. I have removed it for this reason, but I have been reverted. I am not convinced that this topic is notable enough for WP. Nevertheless, I have moved it to N-ary associativity (a stub) and I have linked it in the "see also" section. I leave to other editors to rename it, if I have not chosen the right name, to open an WP:AFD if the topic is not notable enough, and to expand the stub. D.Lazard (talk) 22:23, 17 February 2013 (UTC)

It's only a short paragraph, and wasn't doing much harm. I don't think it's too technical at all; it's a very natural generalization of regular associativity. As you say, the new article is very stubby indeed and perhaps fails Notability. But OTOH, an AFD discussion is likely to conclude "merge with associativity". I'll dig around and see if I can improve the stub. Best wishes, Robinh (talk) 19:44, 19 February 2013 (UTC)

I agree that a merge is a possibility. But the first sections of the article are written for readers of very low mathematical level (high school or lower). This generalization is not at all of the same level (recent research article). Thus it must not be in the elementary sections (see manuals of style and, in particular MOS:MATH). A possibility for a merge is to add, just before the section "See also", a section about the generalizations. But WP:NPOV implies to include also paragraphs about the two other generalizations linked to in section "See also". I am not willing to do this work. This is the reason for having chosen the easier way of a stub. But you may go on, if it is not in the elementary sections. D.Lazard (talk) 21:03, 19 February 2013 (UTC)

Right associativity of exponentiation

Is it even meaningful to define operator associativity for an operation denoted by text position? It only makes sense when exponentiation is a symbol operator such as $a\wedge b\wedge c=a\wedge (b\wedge c)\neq (a\wedge b)\wedge c=a\wedge (b\cdot c)$. To interpret $a^{b^{c}}$ as $(a^{b})^{c}$ would be as absurd as interpreting $a^{b+c}$ as $a^{b}+c$ or $M_{ij}$ as $M_{i}j$ AkariAkaori (talk) 22:14, 23 April 2017 (UTC)

True, but even if it is obvious anyway, as long as the statement is correct we can leave it for those who don't see the obvious at first sight. --Yukterez (talk) 06:47, 9 October 2017 (UTC)

Remove Notation for non-associative operations

The section Notation for non-associative operations should be removed from this page. The notions of right and left associativity relate to programming convention and not to math. There is a clean distinction between this article (the associative property in math) and Operator associativity (operator associativity in programming). The programming article is cited as the main article for this section, which seems to blur the separation. Also, the convention of treating $x^{y^{z}}$ as right associative has been marked as citation needed here since March 2016. Rcanand (talk) 17:19, 30 July 2017 (UTC)

That's not true, there are plenty of references related to math that support this, see Bronstein, Khan Academy, Virginia Department of Mathematics, PEMDAS, the list is 10 kilometers long. Of course x-y-z is (x-y)-z and NOT x-(y-z) and so on, but the restriction to real numbers in the section for subtraction and division is superfluid, because the rule doesn't change if the numbers were imaginary or complex, and also the convention does not flip just to avoid division by zero (wich wouldn't work anyway, since if y or z were 0 the division by zero wouldn't go away just by changing the associativity). I'll therefore add some references and remove the restriction to real numbers. --Yukterez (talk) 06:14, 9 October 2017 (UTC)

Revert by user:D.Lazard on 2020-12-19

I didn't assume that this is „such a“ change. But it seems that you didn't get the point that the color changes, which means that it is intended to be a change in contents – (and not in MOS:VAR).
My new proposal brings the contents and avoids the MOS:VAR, which you seem to dislike. –Nomen4Omen (talk) 12:04, 19 December 2020 (UTC)

You do not explain why previous colors must be changed. A change of color is not a change of content; it is a change of style or a cosmetic change. So the rule of MOS:VAR applies. D.Lazard (talk) 12:29, 19 December 2020 (UTC)

Now I'm really somehow surprised. Don't you see that the red color is now in the exponents instead of the mantissa? –Nomen4Omen (talk) 12:35, 19 December 2020 (UTC)

You do not explain here and in you edit summaries why you think that your edits improve the articles. So, you are asking implicitely to watchers of the article to checks your edits in details in order to guess your intentions and to decide whether there are improvements. This is WP:disruptive editing. Although I am unhappy with the time spent because of your lack of explanations, I decided to not revert your last edit of the article. D.Lazard (talk) 13:10, 19 December 2020 (UTC)

Oh, you are so gracious not to revert. And for the case that you didn't see the intended difference I'll try to make it visible:

Previous version:

(1.000₂×2⁰ + 1.000₂×2⁰) + 1.000₂×2⁴ = 1.000₂×2¹ + 1.000₂×2⁴ = 1.001₂×2⁴
1.000₂×2⁰ + (1.000₂×2⁰ + 1.000₂×2⁴) = 1.000₂×2⁰ + 1.000₂×2⁴ = 1.000₂×2⁴
                                                                                          ↑

New version:

                                                                            ↓
(1.000₂×2⁰ + 1.000₂×2⁰) + 1.000₂×2⁴ = 1.000₂×2¹ + 1.000₂×2⁴ = 1.001₂×2⁴
1.000₂×2⁰ + (1.000₂×2⁰ + 1.000₂×2⁴) = 1.000₂×2⁰ + 1.000₂×2⁴ = 1.000₂×2⁴
                                                                            ↑

Thereby I assume that the previous edit wanted to emphasize the difference by placement of the colors (which it didn't — and which I am not absolutely sure of) whereas the colors of new version show the true difference, now in the exponents.

Got it?

(So I hope this being an improvement of the article although it was not easy to explain.) –Nomen4Omen (talk) 13:44, 19 December 2020 (UTC)

Non-associativity of infinite sums

I removed the example claiming that "that infinite sums are not generally associative" since this is highly misleading. Associativity is an algebraic property related to finite sums or products and unrelated to convergence issues of infinite sums or products. (One can define algebraic structures on infinite series, such as the ring of formal power series, and add or multiply finitely many such series together, but these operations are both commutative and associative if the coefficients of the power series belong to a commutative ring, like Z, R or C.) Reader634 (talk) 09:07, 18 June 2021 (UTC)

You are right, but the warning may be useful. So, I have restored the example, and edited it for clarification. By the way, I have also edited the end of the section per WP:TONE. D.Lazard (talk) 10:13, 18 June 2021 (UTC)

It's a useful (and classic) warning about the convergence of series, but not one about a failure of associativity. However, it's not an issue I have any desire to fight about. :) FWIW, I added a simple example of the failure of associativity for the cross product. Reader634 (talk) 10:36, 18 June 2021 (UTC)

Associativity

Why not name or call this article Associativity? Otherwise, it is inconsistent with Binary operation and Binary property (which you can see doesn't even exist)?