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I'd like to raise the issue of constant misuse of parentheses in the U.S., and I'm wondering if anybody else shares the same feelings, has had the same issues, knows any history behind it, and can offer some thoughts on the subject. Let me explain what I'm talking about.

In elementary algebraic notation, i.e. when writing algebraic expressions, parentheses mean two things: order of operations and function notation; http://mathworld.wolfram.com/Parenthesis.html seems to agree with me. (There are all the other things, such as interval notation, matrices, etc, but I'm not discussing those here.) But here in the U.S. it is used all the time to enclose each factor in multiplication, so $5\cdot2$ would be written as $(5)(2)$. And while it's technically speaking not wrong, as it still means the same thing, it's redundant and pedagogically terribly harmful, imho. This way of writing multiplication is so persistent from early on and then everywhere, that students grow up believing that that's what parentheses mean: that their purpose is to enclose each individual quantity in multiplication. Not for grouping, not for function notation, but mostly just for that. Almost any of my students here (and I've tried asking a whole class, so it's kinda experimentally confirmed) will write the product of $2$ and $x+1$ as "$(2)x+1$".

In my teaching I always try to correct my students' bad writing habits. Here's a recent example. Solving an example on the board during my "College Trigonometry" class, I actually came to the need of setting up a product of something like (don't remember exactly) $2$ and $\sin x+1$, which I intentionally wrote as "$2\cdot\sin x+1$". I turned to my class and said: "By the way, this is wrong. Can you correct what I wrote?" One of the best students in the class immediately said: "It misses parentheses."

"Halleluja! My efforts haven't been futile," I thought to myself... until he continued: "$2$ must be in parentheses because it's multiplied."

It would have been just a little notational nuisance, had it not been a cause of so many real mistakes. The most obvious one is when people don't distribute multiplication, since there's nothing indicating the need to distribute. I mean, as a quick example, when instead of $2(x+1)$ someone wrote $(2)x+1$, and then they need to plug e.g. $x=3$, they'll get $7$ instead of $8$. Another one, that I believe stems from the same misuse of parentheses, is misreading function notation. Again, I've done this literally with thousands of my students, and most read "$f(x)$" as "$f$ times $x$" or "$\sin(x)$" as "sine times $x$".

I only have the experience of working in two countries. In Russia, my homecountry, nobody ever writes $(2)(5)$. (And mind you, its educational system has a truckload of problems of its own, but at least this is not one of them.) In the U.S., it's not just about the students, but it's all over the books, textbooks, and in almost everybody's math writing, including teachers and professors — which is probably why all students pick up this habit. I honestly don't know about the rest of the world.

So, back to the questions that I asked in the beginning. Can anybody explain where this writing tradition comes from? Does anyone agree that it causes some real problems in students' learning of math?

Update: Let me clarify my concerns a little bit. As others pointed out below, $(2)(5)$ is weird but not wrong. And I agree: it's not wrong mathematically, but it is wrong pedagogically. And although it still irritates me (to put it mildly), that's not the problem in itself. But $(2)x+1$ is actually wrong when it stands instead of $2(x+1)$. I apologize for not being clear enough, but here's what I meant: I believe that seeing $(2)(5)$ results in students' writing $(2)x+1$, and that's wrong, and that's the problem.

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    $\begingroup$ Wow, you learn something new every day! I have the reverse (far less serious) problem - my undergrad students are unwilling to use () for multiplication unless there are two or more terms, meaning some calculations have to be completed before they can be written down. $\endgroup$ – Jessica B Oct 24 '16 at 6:45
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    $\begingroup$ I totally agree and I had to do this same exercise in my college algebra class last week. Keep in mind that many U.S. teachers and textbooks explicitly teach that parentheses mean multiplication. I sometimes feel like Don Quixote fighting against it. Blog post of mine here on the issue: madmath.com/2013/10/are-parentheses-multiplication.html $\endgroup$ – Daniel R. Collins Oct 24 '16 at 13:13
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    $\begingroup$ The problem isn't (2)(5), it's (2)x+1 instead of (2)(x+1). The latter is correct and is consistent with the example. $\endgroup$ – Wayne Oct 25 '16 at 21:21
  • $\begingroup$ It looks like I'm going to be in the minority here but I've been teaching algebra at the college level for almost ten years with multiple textbooks and I've never seen this issue. $\endgroup$ – G. Allen Oct 28 '18 at 20:17
  • $\begingroup$ Extra parentheses cannot change the order of operations, but missing ones can. In your example the parentheses around x+1 are missing. Computer devs using extra parentheses all the time when they are not sure in the precedence. $\endgroup$ – Rusty Core Oct 29 '18 at 17:19
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To answer the ultimate question ("Can anybody explain where this writing tradition comes from?"): It's explicitly taught that way by many U.S. instructors and textbooks.

Examples: From the otherwise excellent Martin-Gay Prealgebra & Introductory Algebra (sec 1.5):

The × is called a multiplication sign... The symbols ∙ and () can also be used to indicate multiplication.

Shaum's Outlines Elementary Algebra (sec 1-4):

The symbols for the fundamental operations are as follows... Multiplication: ×, (), ∙, no sign

A research presentation for the NCTM actually took not thinking that parentheses are multiplying as a sign of student misunderstanding (Welder, "Using Common Student Misconceptions in Algebra to Improve Algebra Preparation"):

Misconceptions: Bracket Usage -- Beginning algebra students tend to be unaware that brackets can be used to symbolize the grouping of two terms (in an additive situation) and as a multiplicative operator

I could go on (and probably more U.S. teachers teach this even more adamantly; I've never met anyone who opined otherwise). To answer the other question ("Does anyone agree that it causes some real problems in students' learning of math?"), then I would say: Yes, absolutely.

The most common mistake I see in this vein is that if one is taught both that "parentheses mean multiplying" and in the order of operations that "parentheses come first", then that combination results in many students thinking that juxtaposed multiplying is meant to happen first, e.g.: They will simplify $2(3)^2$ as $6^2 = 36$ and so forth.

Others will think that parentheses are entirely interchangeable with the dot notation and juxtaposition, e.g: when substituting $x = -2$ in the expression $3x^2$, they're prone to writing things like $3 \cdot -2^2$ or $3 - 2^2$, which they may or may not recover from on the next line. Even just writing $3 \cdot -2$ is nonstandard, of course, but if corrected they'll grouse about it as being easier than using parentheses.

As noted in a comment, I've written about this on my blog in the past on MadMath. I think it would be tremendous if we could flip this around and start teaching that it is juxtaposition that indicates multiplying, not the parentheses themselves (and therefore reserve more focus for parentheses grouping and giving precedence to any operation contained inside, including unary negation).

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    $\begingroup$ Thank you for the excellent answer! (And sorry I couldn't thank you earlier - had some crazily busy days.) I knew that it comes from school, but I thought it's a silent tradition. I'm shocked to learn that it's so blatantly stated in textbooks and accepted as the norm! And I read your blog post now - excellent piece! I see all the time many of the same problems that you mention. I'm not sure I agree what the biggest problem is: in my experience, the order of operations errors, like the one you described, do happen, but not as frequently as some other errors. But overall this is a huge problem. $\endgroup$ – zipirovich Oct 28 '16 at 23:17
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It is actually wrong to say that parenthesis means multiplication. In $(2)(5)$ it is the lack of an operator between the parenthesis that implies multiplication, NOT the parenthesis. The parenthesis are "needed" because $25$ means the number twenty-five, the parenthesis are purely for grouping.

"No operator means multiplication" is an extremely common convention that is much more useful to learn than "parenthesis means multiplication", which is just plain wrong, since it only means multiplication when there is either a multiplication operator (obviously) or no operator, which the first rule already covers.

So yes, I agree that teaching students something that is not true and unnecessary very likely causes problems for students learning math

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    $\begingroup$ Exactly this. The emphasis should be on parenthesis indicating "grouping" not "multiplication". Multiplication is simply the default operator when two groups are placed next to each other, which is seen in other formulations, like $2x$ or $5\sin(x)\cos(2x+1)$, etc. $(2)(5)$ is a bit redundant, but if I were working through a problem by hand, I would almost certainly use $2(5)$ rather than $2\times5$ or $2\cdot 5$. The latter two could be too easily mistaken for something else. $\endgroup$ – pwcnorthrop Oct 25 '16 at 13:52
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    $\begingroup$ @pwcnorthrop $2(5)$ also looks strange to me but at least it uses the minimum amount of parenthesis if you insist on going down that road and makes it clearer that the parenthesis is for grouping and have no magical multiplicative properties:) I always use $2\cdot 5$ for multiplying numbers and I'm not sure what it would be confused with, $2.5$ perhaps? Never had that issue personally. What would you confuse $2\times 5$ with? $\endgroup$ – smernst Oct 25 '16 at 20:46
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    $\begingroup$ I don't have very good handwriting, so $\cdot$ and $.$ look identical, as do $\times$ and $x$ (though I will use $\times$ when doing scientific notation). Heck, I use a script $s$ because I've gotten them confused with $5$'s before. And if I have had problems, imagine my poor teachers! $\endgroup$ – pwcnorthrop Oct 25 '16 at 21:18
  • $\begingroup$ @pwcnorthrop You can use $2*5$ to avoid mixing "$\times$" with "$x$" or "$\cdot$" with "$.$". BTW, for me $2(5)$ looks inconsistent and is too similar to $f(5)$. When reading such a text I would wonder if there is some convention used that allows me interpret numbers as functions, for example if one defines $f+g$ as $(f+g)(x) = f(x) + g(x)$, then writing $f+2$ may suggest that $2$ can be interpreted as a function given by $2(x) = 2$. $\endgroup$ – dtldarek Oct 27 '16 at 12:02
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    $\begingroup$ In any case, my primary concern of marking such formulations wrong is that it reinforces the idea in students' minds that math is an arbitrary set of rules that will vary from year-to-year and teacher-to-teacher. Admittedly, this is not necessarily the OP's fault as apparently students a taught that '$()$' means multiplication, which is insane and incredibly distressing. $\endgroup$ – pwcnorthrop Oct 27 '16 at 14:29
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I disagree that it is "terribly harmful".

Do not prevent them from writing $(2)(5)$. Instead prevent them from writing things that are actually wrong.

Thinking that $\sin x$ is $\sin$ times $x$ can happen (and does happen) even without parentheses.

I agree $(2)(5)$ looks strange, but if they can write $(4-2)(4+1)$, what is wrong with $(2)(5)$?

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    $\begingroup$ To answer your last question: the expression $(2)(5)$ is not wrong mathematically, but it's wrong pedagogically, because it promotes a misconception about the meaning of parentheses, which then leads to other real mistakes. $\endgroup$ – zipirovich Oct 24 '16 at 13:55
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    $\begingroup$ On another note, don't even get me started on $\sin(x)$ versus $\sin x$. I find far more people unprepared for trigonometric functions versus doing trigonometry because of this one notation than I can shake a stick at. Versus $(2)(5)$ which is not really ambiguous, even if unhelpful. $\endgroup$ – kcrisman Oct 24 '16 at 14:52
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    $\begingroup$ I thought (2)(5) was used to avoid confusion with 25 (twenty five), especially in situations where the dot could be overlooked or not seen very well (in some fonts), or misunderstood as a period, or when a raised dot symbol isn't available, or when the "times" symbol could cause one to think the letter "x" is intended. For what it's worth, I don't recall EVER having a student get confused on something as a result of this notation, although I suppose some might have and I never learned about it. $\endgroup$ – Dave L Renfro Oct 24 '16 at 15:20
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    $\begingroup$ I'll add that if something like (2)(5) occurs in the middle of a series of steps to show the student's work, it should not be marked even stylistically bad. E.g. "Solve (4-2)(4+1). Show your work." should result in the student offering any of "(4-2)(4+1) = 2⋅5 = 10", "(4-2)(4+1) = 2∗5 = 10", or "(4-2)(4+1) = 2×5 = 10", "(4-2)(4+1) = (2)⋅(5) = 10", "(4-2)(4+1) = (2)∗(5) = 10", "(4-2)(4+1) = (2)×(5) = 10", or "(4-2)(4+1) = (2)(5) = 10". $\endgroup$ – Jed Schaaf Oct 24 '16 at 17:25
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    $\begingroup$ But grouping is not itself a mathematical operation. Additions, multiplications, roots, sines, integrals, etc. are. The grouping symbols, (), are used either to indicate an order of evaluation that falls outside the normal order of operations or to show explicitly the order of evaluation in a formula that may be confusing. And what is confusing for students just learning math may be much simpler than what may be confusing for those of us who are intimately familiar with it. Extra parentheses should never be "wrong" if they don't change the formula. "Unnecessary" ≠ "wrong". $\endgroup$ – Jed Schaaf Nov 17 '16 at 17:20
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This problem starts in elementary school. In fact, I am one of the elementary school teachers that has started students along this path that you complain about. I have taught my 5th and 6th grade students that once they start using variables they should no longer use the multiplication sign since it might be mistaken for the variable x.

I teach my students that while the book might use a dot for multiplication the students should never write $5\cdot2$ since they might confuse it with $5.2$ From experience I can assure you that elementary students will not be able to distinguish between the multiplication dot that they write and the decimal point that they write. In a printed book it is much clearer.

As is standard in textbooks and the literature that I've read, I teach my students that they can use parentheses for multiplication symbols. I have tried to teach order of operations in great depth so that they don't think that $(2)5+1$ is anything other than $10+1$. In fact my students know that

$(5)2+1=10+1=11$ because they know that $5\cdot2+1=10+1=11$

I would suggest that a greater effort be made to teach order of operations in depth. Most of the elementary teachers that I know don't understand order of operations and many are still stuck on PEMDAS and are convinced that addition comes before subtraction since A comes before S. They believe:

$8-3+5=8-8=0$ and not $8-3+5=5+5=10$

I would also suggest that * be adopted as a multiplication symbol. It is easy for students to draw and widely used in spreadsheets. It is much easier for students to use than the multiplicative dot and would eliminate the problems of parentheses.

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    $\begingroup$ Thank you for your honest answer! I really appreciate it. But the point I want to make is that down the road this does a lot more harm than good. Somehow, in their minds the idea of the order of operation in relation to parentheses gets almost completely lost. I mean, if you ask them, they'll tell you that parentheses come first. But in their minds it's like a trivia fact that has nothing to do with their own doing of math. When they write expressions, they can't use parentheses to show proper order of operations because they use them only for multiplication. $\endgroup$ – zipirovich Oct 28 '16 at 23:38
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    $\begingroup$ I can't think of any activity in the curriculum that teaches them how to use parentheses in this way so you have a valid point. My point is that this is part of textbooks and curriculum and starts in elementary school. $\endgroup$ – Amy B Oct 31 '16 at 13:35
  • $\begingroup$ Then I guess that's the root of the problem! We need better textbooks and curricula that don't teach wrong things. There's definitely a way to do it, because lots of other countries somehow manage to get it right! (Even though they may be not so right on lots of other things. Wouldn't it be awesome to get the best of all different worlds combined in one place?) Back to reality, it is distressing that textbooks teach our kids wrong. I guess we're stuck fighting our own little battles... $\endgroup$ – zipirovich Oct 31 '16 at 14:58
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Another one, that I believe stems from the same misuse of parentheses, is misreading function notation

To me, this is the more serious problem that occurs. I don't necessarily have people saying "f(x) is f times x" but I do have people that then note $f(3)=6$ deduce that the function is multiplication by two. Once you start composition of functions (this even happened to me today), we get the question of $f(g(x))=f(x)g(x)$...

But as to a "solution", I have a different point of view on this. I agree that the ambiguity is terribly annoying and can be harmful; but natural languages have much more ambiguity, and we somehow get around that. Because we have a limited number of symbols and ways to indicate typical operations, and because much notation is ingrained due to many years of use, it is often better to teach what people will actually see in "real life" and provide good warnings.

As a non-parenthesis example, I will take $\frac{df}{dx}$. No one would suggest (I hope) that, historically speaking, this notation has not been very successful, and that it not only reminds us that derivatives come from difference quotients, but even (properly handled) allow very convenient manipulations for e.g. solving differential equations. Nonetheless, from the point of view of the question here, this is horrible notation! And some misconceptions arise from it. So we teach when you can use this or when not (at least if we are mathematicians and not physicists).

Another example is exponential notation, where I often have students who are quite confused why in $e^{x^2}$ the "inside function" is $x^2$, and exactly how things are related. Better, from this point of view, is $\exp(x^2)$. But there are other natural advantages to exponential notation, so we use it, and - hopefully - point out the ambiguity and give good exercises or activities to highlight the "right" answers.

So to bring this back to parentheses, I think that the "right" solution is to acknowledge this ambiguity and try to resolve it. One way in which I often do this is to explicitly write $\cdot$ or $\times$ as appropriate; another way is to have activities which highlight this difference and then really engage each student individually on it. This is pretty labor-intensive, and won't work for every situation. But simply avoiding the parentheses altogether - for now - is worse, because students simply will not interpret $3+2x \cdot x+1$ as anything other than $4+2x^2$, as as you point out grouping is what they are for. In fact, I usually have the opposite problem of getting them to use parentheses to indicate this is $(3+2x)\cdot (x+1)$. So that is what I recommend, in this context. Seeing $(2)(5)$ on occasion is a small price to pay for them understanding the grouping bit; there isn't really any other interpretation possible, at least.

All that said, if you have good ideas of how to resolve this systematically from your previous experience, so that no one writes $(2)(5)$ but still knows what to do with $x^2(x+1)$, it would probably be a good thing. (And hopefully that doesn't include replacing $2\cdot 5$ with $2\, . 5$ as the British do!) I'm just not sure how to do that, and I have to teach the students I actually have.

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    $\begingroup$ I agree with you. I added an update to my original post. And I wish I knew a solution... but alas, I don't. I always correct my students when they write $(2)(5)$, but I explain to them that in this case it's bad style and I don't take off points. I fight with it because the same students will make and do make actual mistakes, when I do have to take off points (you know, students tend to pay more attention when it's about points on the test). My approach is that consistently promoting good style will help them also in expressions where it does matter. $\endgroup$ – zipirovich Oct 24 '16 at 15:34
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    $\begingroup$ @zipirovich I would caution against "always" marking that as stylistically bad. If the multiplication was performed accurately, and especially if "(2)(5)" occurs in the middle of a series of steps that show the student's work in solving the problem, then it should not be marked as anything except possibly a place to reduce the number of symbols needed to express the formula. $\endgroup$ – Jed Schaaf Oct 24 '16 at 17:30
  • $\begingroup$ "No one would suggest (I hope) that, historically speaking, this notation has been very successful..." I think there's a missing not that should be in there? $\endgroup$ – Daniel R. Collins Oct 25 '16 at 0:00
  • $\begingroup$ @DanielR.Collins - No. I think OP (kcrisman) means (overly simplified): "No one would suggest it has been successful.", or "Everyone would agree it has not been successful." It very easily confuses the beginner/novice. $\endgroup$ – Kevin Fegan Oct 25 '16 at 16:33
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    $\begingroup$ Oops! Definitely missing a "not" there, not sure how it escaped me. From the context it's clear that I mean this. (Leibniz notation is a well-known example in math history circles of notation making the difference, similar to Gauss' congruence notation.) That doesn't mean I disagree that it easily confuses the beginner, as I also point out. It is a very successful, useful notation that easily confuses the beginner, if that makes sense. $\endgroup$ – kcrisman Oct 25 '16 at 18:30
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There is a perfectly good reason to write $(5)(2)$ rather than $5\cdot2$, which is that the dot can easily be confused with a decimal point. I believe Russians use a comma as a decimal point, so it makes sense that Russians would not see any danger of error in writing $5\cdot2$.

Also, scientists and engineers typically think of numbers as having units (unlike most mathematicians, who seem to think of units as part of the definition of the number), and therefore consider it scientifically illiterate to write numbers without units. So suppose we're going to multiply 5 m/s by 2 s. Writing it as $5\ \text{m}/\text{s}\cdot2\ \text{s}$ is ugly and hard to read, so we write it as $(5\ \text{m}/\text{s})(2\ \text{s})$. I find, e.g., $2\pi r=2\pi(1.2\ \text{cm})$, much more readable than $2\pi r=2\pi\cdot1.2\ \text{cm}$. Even for unitless numbers, an expression like $2\cdot7.8\cdot0.134$ looks like a hot mess to me compared with $2(7.8)(0.134)$

The disease that I've been seeing a lot within the last 5 years or so is the use of parentheses in totally unnecessary ways like this:

$(2)(x)$

$\sin(x)$

$((2)(x))/((y)(z))$

I suspect this is actually because of the increasing use of software on computers, phones, and calculators that has fancier symbolic math capabilities than most calculators used to have. Math and science students are typically entering their homework answers into computer software systems such as Mastering Physics or MyMathLab.

Many such systems require parentheses for function evaluation.

Computers don't have common sense or understanding of context or meaning, so they can't interpret expressions like $\sin\omega t$ the way any engineer would (as $\sin(\omega t)$, not $(\sin\omega)t$).

Also, students will have had the experience of having the software interpret their input according to the normal rules of precedence of operations, just as a human would, but the result isn't what the student expected because the student doesn't understand the order of operations. For example, the student intends $a/(b+c)$ but types $a/b+c$. Instead of learning from this experience that they need to get more fluent with the order of operations, they just decide that it's safer to use lots and lots of parentheses.

Another thing that's going on now with systems like Mastering Physics or MyMathLab is that many lazy teachers take them as excuses to avoid ever reading students' work and writing comments on it with a red pen. Therefore when students make stylistic mistakes such as $(2)(x)$ or $\sin(x)$, they never get any feedback from a human telling them that their style is wrong.

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    $\begingroup$ "Instead of learning from this experience that they need to get more fluent with the order of operations, they just decide that it's safer to use lots and lots of parentheses." is an excellent insight. $\endgroup$ – Chris Cunningham Oct 29 '18 at 14:40
  • $\begingroup$ I don't agree with "an expression like $2\cdot7.8\cdot0.134$ looks like a hot mess to me compared with $2(7.8)(0.134)$" at all. In fact, "$2(7.8)(0.134)$" is a hot mess, while the former expression is as clean as it gets -- exactly because it doesn't have anything unnecessary in it. To counter your other point, countries with the decimal point can use asterisks for multiplication such as "$2*5$" -- I do it all the time when teaching here in the US. Other than that, thank you for your answer! Especially for the great insight regarding influence of software and calculators. $\endgroup$ – zipirovich Oct 30 '18 at 4:00

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