Talk:Order of operations/Archive 1

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Microsoft Excel

Would it be useful to mention that in Excel, arithmetic negation has a higher precedence than exponentiation? For example a formula such as -2^2 will return 4, as in (-2)^2, rather than -4, as in -(2^2). This is an endless source of pain and confusion because most other programs and languages interpret it as -(2^2). In particular, I've seen many people fall into this trap when trying to write a Gaussian function in Excel. For reference, see [1]. Itub 12:08, 19 February 2007 (UTC)

Unix's bc programming language, which uses a very C-like expression syntax plus exponentiation, does the same thing; I've noted this as well. Using binary subtract instead of unary negate avoids the trap (0-2^2 = -4), but I'm not sure it's worth mentioning that in this article. 65.57.245.11 03:16, 3 July 2007 (UTC)

I'm glad to see that these important facts are now in the article. Rick Norwood 14:02, 3 July 2007 (UTC)

Order of Unary Minus

I believe the 5th example (Examples section) to be incorrect. I'm loathe to go and change it since that could easily descend into arguments, so here's my understanding:

The article has $-3^{2}=[-3]^{2}=9$, but the OOO should actually be $-3^{2}=-[3^{2}]=-9$

I can't find a lot of authorative references on the internet, but the 2 I did find are:

• mathforum.org
• FindArticles.com

Intentional 01:59, 3 December 2006 (UTC)

You are correct. Rick Norwood 13:31, 4 December 2006 (UTC)

Not even the Math Forum writers agree 100% -- at least recommends using parentheses to remove ambiguity:

• another mathforum.org

In practice, the meaning comes from the context of the problem one is modeling. I don't know that the most common convention, $-3^{2}=-9$ can accurately be called "standard." Is there a standards body which publishes mathematical order of operations?

Gerardw 23:06, 11 July 2007 (UTC)

This standard comes from the way we write polynomials. Since $-x^{2}$ always means the opposite of the square of x, and for real x is never positive, we would wish to be able to "plug in" 3 for x and get the same answer: $-3^{2}=-9$ Rick Norwood 14:16, 21 September 2007 (UTC)

Fractional lines

A while ago I added "fractional lines" together with "roots". Now, Rick Norwood has removed it again, with the edit summary to say you do "fractional lines" first is confusing and misleading.

I don't get this. As I understand it, in ${\sqrt[{1+2}]{25+2}}\times 4$ and ${\frac {25+2}{1+2}}\times 4$, "1+2" and "25+2" are evaluated first, then the root respectively the division, and finally the multiplication. This is because roots and fractional lines come before multiplication in the hierarchy. In $25+2/1+2\times 4$, the division and the multiplication are done before the additions, because division and multiplication come before addition. So division written with a symbol like "/" and division written by fractional lines do not have the same place in the hierarchy. I find it confusing and misleading to omit fractional lines at this point.

It seems most people describing the hierarchy do omit fractional lines here; I just don't understand why. Can anyone explain this?--Niels Ø (noe) 18:58, 21 October 2007 (UTC)

To say "do" fraction lines first is confusing. A horizontal fraction line (but not a slanting fraction line) is both a symbol of division and a symbol of grouping. In 2 + 3*4 we "do" the multiplication first, meaning that we multiply first. But to "do" the fraction line first does not mean that we divide first. It means that we first treat the fraction line as a symbol of grouping, and only later treat it as an operation. This is too complicated to put in a chart.

I suspect that more lies have been told to students about the order of operations than about any other subject except American history. The order of operations is essential to progress in mathematics, but it is frequently misunderstood. I suspect misinformation about the order of operations is responsible for at least some of the failure in America to teach our children math. The mathematically able pick up on what the hierarchy "really" means by following examples, and discard the misinformation in the textbooks. Rick Norwood 21:29, 22 October 2007 (UTC)

OK, but why not omit roots from the chart too, then? ${\sqrt {}}5+1$ is - I believe - not an acceptable notation anyway, and in ${\sqrt {5}}+1$ or ${\sqrt {5+1}}$, the horizontal line is a symbol of grouping, much like the fractional line.--Niels Ø (noe) 06:43, 23 October 2007 (UTC)

Taking the nth root of m is a binary operation which can be indicated in several different ways. One way is m^(1/n). Root taking is done before multiplication or addition. 4*9^.5 = 12. Also note that the hierarchy was designed so an operation and its inverse are on the same level. Root taking is the inverse of exponentiation. 4^2^.5 = 4. Further, just as subtraction is a form of addition (addition of the opposite), and division is a form of multiplication (multiplication by the reciprocal), root taking is a form of exponentiation. The people who designed the hierarchy did not choose which operations to give precedence to randomly! There are many other patterns in this chart. Another symbol for root taking is the radical sign. It, like the horizontal line, is a symbol of grouping as well as an operation. The whole subject of symbols of grouping is an interesting one, but it is a different subject from order of operations. Rick Norwood 15:21, 23 October 2007 (UTC)

It may well be worth addressing 2D display formats like ${\frac {25+2}{1+2}}\times 4$ the rules for how you interpret them are somewhat different to 1D inline formats. Display formats are deserving of their own section. --Salix alba (talk) 17:55, 23 October 2007 (UTC)

4*9^.5 is not a root, it is a power, just as 4+(-3) is a sum, not a difference. The hierarchy is about how to interpret what is written in terms of operations to be performed (e.g. "add four and negative three"), not about other expressions having the same result (like the difference 4-3, "subtract three from four"). If roots are to be mentioned, it must refer to roots written as roots with a radix sign, but I still think it should be removed.

I think Salix has a point: The hierarchy is about interpreting math written in "1D" with "incomplete parenthesisation"; I don't know how best to express this. Interpreting "2D" features like fractional lines or radix signs, or interpreting parentheses, is in my experience as a teacher never a problem - but translating 2D expressions into 1D so that it can be entered into a computer or calculator often is, and so is removing parenthesis from expressions like 1-(2-3).--Niels Ø (noe) 11:41, 24 October 2007 (UTC)

First, please note that removing "roots" from the hierarchy is not an option, since the chart as given is standard in the literature and not ours to play around with. Second, 4^.5 is a root, because the definition of a unit fractional exponent is that it indicates a root. (It makes no sense to multiply 4 by itself half of a time.) The hierarchy only depends on the operation, not on the symbol used to indicate the operation. If I write using words instead of symbols "find two times the square root of 9" or "find the square root of 9 times 2", either way the answer is 6, because of the hierarchy.

Of course, we could all switch over to Polish prefix notation, but having taught classes using the old Hewlett Packard calculator, I don't think that would be a good idea. Rick Norwood 13:36, 24 October 2007 (UTC)

I'm not sure what you refer to by "the chart as given is standard in the literature"; it does not seem to be in the references or links cited in the article, and I have found several places that conform with the PEMDAS mnemonic - i.e., no mention of roots. I repeat, 4^.5 is not a root, and it is a different expression from 4^(1/2) (which, however, isn't a root either). Of course, they have the same value, but that is not the issue here. (Incidentally, CAS systems may consider them different as the expression with 1/2 may be evaluated exactly, where as the one with .5 may give a finite precision answer). Your argument about multiplying 4 by itself half of a time cannot be generalized to something like $4^{\pi }$, $4^{1/\pi }$, or $4^{0.1101001000100001...}$ with irrational exponents, anyway. I have to repeat: Order of operations is about how to interpret an expression as a sequence of computational steps. Expressions like $3\times 4^{0.5}$ are dealt with by the rule without the mention of roots, as exponents are evaluated before multiplication, so you have to come up with either several specific references where roots are mentioned in the hierarchy, or with an example using proper math notation (as opposed to various computer- or CAS-notations) where explicit mention of roots (as opposed to fractional lines) is needed to interpret the expression correctly.--Niels Ø (noe) 15:43, 24 October 2007 (UTC)

All mathematicians agree about the order of operations, which greatly aids international mathematical communication. As the article shows, some computer scientists disagree.

PEDMAS is used in grade school, and in every American grade school book I have examined it is stated incorrectly. For a good book on the subject, you need to go to another country, such as Finland or Singapore. The big problems with PEDMAS are first, it confuses symbols of grouping with operations, and second, it wrongly suggests that addition should precede subtraction.

In mathematics, an operation is a function, and two functions are equal if they give the same output for every input. A binary operation is a function whose input is an ordered pair. You are confusing the operation, that is to say the function, with the notation, which is traditional and arbitrary, and with the method by which the output is computed. The square root, the exponent 1/2, and the exponent .5 all indicate the same function, called the square root function, because in every case the output is, by definition, the non-negative number whose square is equal to the input. You say that the fact that they have the same value is not at issue. In mathematics, that is exactly the issue. Same value means same function.

You mention $4^{\pi }$. There is a lot of history, here. Originally, exponents were written as words, quadratum, cubum, and so on, and only whole number exponents were allowed. Square roots were indicated by the letter R. Thus the square root of 4 squared would be written R4quadratum, and the answer is the same whether the root is taken first or the exponent is taken first, thus roots and exponents are on the same level in the hierarchy.

Gradually, over the centuries, negative, fractional, irrational, and imaginary exponents were allowed, so that by the 18th Century, Euler could write e to the pi i equals minus one. $4^{\pi }$, is defined to mean the limit as n/m approaches pi of the mth root of 4 to the nth power, where m and n are relatively prime natural numbers. On the other hand $4^{\pi }$ can be computed using e to the power pi ln 4. The former is a definition, the later a theorem.

Each of these new kinds of exponents had to be defined, but in every case exponents were still given precedence over addition, subtraction, multiplication and division, and put on the same level with roots. The other three signs could be written either before or after a number, but a root sign must always precede the number it acts on, and an exponent sign must always follow the number it acts on. Thus in R4*9 you do R4, then *9, but in R4^9 you can do either operation first.

The bar over the root sign evolved from the expression RV, which stood for radix universalis, and meant to take the root of everything that followed. Thus RV4*9 meant R(4*9). In modern notation, ${\sqrt {}}4*9=18$, but ${\sqrt {4*9}}=6$. The bar is a symbol of grouping, not an operation.

Rick Norwood 14:15, 26 October 2007 (UTC)

Thanks for the history lesson, which I find very interesting. Perhaps we should have an article about that, or a history section in the present article. However, it does not change my opinion that order of operations, today, is about interpretation of notation, and ${\sqrt {4}}$ and $4^{0.5}$ are different notations. I agree that PIDMAS or whatever is not a good mnemonic, and I also agree that functions is an important concept here, but perhaps I do ont agree as to why. Function notation needs no hierarchy of operators to be unambiguous, so you could state the order-of-operations-thing as a set of rules for converting the strange way we write math into function notation (Łukasiewicz notation, if you like): $2+3\times 4+5\times 6^{2}$ = sum(2, product(3,4), product(5,power(6,2))).

Similarly, ${\sqrt {4}}={\sqrt[{2}]{4}}$ = squareroot(4) = root(4,2) = power(4,1/2) = power(4,0.5) = $4^{0.5}$. Here some equal signs represent translation between different notations for the same computation - some represent mathematical identity between different computations.

But, to return to the original questions about the radical sign and fractional lines, if a notation like ${\sqrt {}}5+2$ is considered acceptable - today, that is - I can see why one would need to include roots in the hierarchy. Otherwise, I can't. The notation ${\sqrt {blahblah}}$ is much like f(blah blah): You don't need a hierarchy to interpret it.--Niels Ø (noe) 16:11, 26 October 2007 (UTC)

The notation $sqrt{},$ was very common twenty years ago. Today, the use of calculators has just about wiped it out, because calculators usually treat the nth root of m not as a binary operation but rather as a family of functions "nthroot" acting on a single variable m, and calculators require function arguments to appear inside parentheses. Thus, if we only wanted to consider calculator mathematics, we could let exponents stand alone at the top of the chart. In years to come, the very idea of a "root" may wither away.

But that day is not yet. Also, I think the old chart is still useful as a mnemonic for many other rules. For example, in terms of the chart, each operation distributes over the two operations one line below, never distributes over the operations two lines below. Laws of logs and laws of exponents also follow patterns which the chart helps beginners learn. And there is someting pleasing about the fact that in the left hand column, each operation can be defined as repeated application of the operation below, at least for natural numbers, while the operations in the right hand column are inverses of the operations in the left hand column.

In any case, it does no harm to include roots. Rick Norwood 17:09, 26 October 2007 (UTC)

Parentheses

Should it be Parentheses then the rest? And should we show menomic devices such as Pemdas? —Preceding unsigned comment added by BrainiacMatt (talk • contribs) 19:44, 12 December 2007 (UTC)

All of the acronyms are misguided and, in my experience as a teacher, do more harm than good. The title of this article is, after all, order of operations, and parentheses are symbols of grouping, not operations. Confusing symbols of grouping with operations has, in my experience, done a lot of harm. Still, all Wikipedia can do is report what is, we can't fix it.

Rick Norwood (talk) 13:39, 13 December 2007 (UTC)

Calc.exe

No kidding, huh? Well, I apologize for removing that line without trying it myself. Just from memory, I could swear it wasn't right! Melchoir (talk) 20:57, 11 February 2008 (UTC)

"Proper use of parentheses..."

The notation for some operations implicitly groups operand expressions (e.g. expressions under a root or in an exponent). This grouping may need to be made explicit with parentheses when using alternative notations.

There's nothing else to say on the matter. In all other cases, parentheses are required in a plain-text expression iff they're required in the standard mathematical notation. (x) and x are equivalent everywhere else, too. sin(x+1) always needs parentheses. Et cetera. If I work in that brief explaination elsewhere, I think the whole section can go. --LuminaryJanitor (talk) 15:14, 4 May 2008 (UTC)

PEMDAS Deprecated

There's a well-intentioned addition of 15 Sep 2008 that puts at the beginning of the article the PEMDAS acronym, the "Please Execuse My Dear Aunt Sally" mnemonic, and an example. There are reasons something similar hasn't been provided. (1) PEMDAS, etc, are covered succinctly in the Acronyms section. The article has plenty of examples. (2) As the Acronyms section observes there are many other such memory devices. BOMDMAS, BOMDAS, BIDMAS, PEMDSA are among these. (3) The standard structure of a mathematics article would place this information later, not at the very top. (4) Also as observed in the current article, "Warning: Multiplication and division are of equal precedence, and addition and subtraction are of equal precedence. Using any of the above rules in the order addition first, subtraction afterward would give the wrong answer to 10 - 3 + 2...." PEMDAS and other simple acronyms can lead students to wrong results! Some presentations thus put it as PE(M or D)(A or S). Of course the parentheses kills memorability, and grin, introduces a little self-reference. I appreciate how PEMDAS helps students learn, but if they learn an untruth, it is really hard to unlearn that. That is why many Algebra books--McDougal Littel Algebra 1 (2008), Addison-Wesley Beginning Algebra (2007), Glencoe Mathematics Applications and Concepts (2006) and many more--do not mention PEMDAS. The harm PEMDAS does outweighs its temporary benefits.

In view of the parochial, misleading, and redundant qualities of this PEMDAS block, I recommend that the PEMDAS addition of 15 Sep 2008 be removed. --Gregoreo (talk) 15:53, 18 September 2008 (UTC)

Conflict with external source

In the article posted regarding the algebraic order of operations from PlanetMath, it is stated that multiplication is followed by division which is followed by addition, with subtraction implicitly following after. It then states that anything written higher has greater priority than those listed below; namely, this states that multiplication has higher precedence than division, and so on. If we are to state on the main article that multiplication and division have equal precedence, than we really should not be linking to an article that contradicts this. If this article is correct, then we should reevaluate our position on this. 141.224.33.11 (talk) 23:01, 6 November 2008 (UTC)

Lots of Irritating, Silly Acronyms

Where does the list in the sentence "Thus, we also have BEDMAS, BIDMAS, BIMDAS, BIODMAS, BODMAS, BOMDAS and BPODMAS." come from? What does using both I and O together mean? Does "BP" mean "Brackets then Powers" or "Brackets then Parentheses"?

If this is a list of acronyms that have been used traditionally, then we should have some references and descriptions on when these acronyms have been used. Otherwise, what use is the list? At the least, we should clarify to "For example, each of the following acronyms has been taught at some time and place:"

If on the other hand this is just a list of possible acronyms, why aren't other choices like BPOMDAS included?

Assuming this list isn't historical, I think just a couple extreme cases should be used to illustrate how you can really louse things up. For example include only BPODMAS. Or a sentence like "For example, some text books teach the acronym "BPODMAS".

And while I'm at it, how about a section called "Inconsistencies and Shortcomings of Order of Operations Systems" or somesuch to underscore the fact that the 'helpful' mnemonics possibly do more harm than good. Donimo (talk) 03:25, 23 May 2009 (UTC)

Current definition of order of operations omits unary operators

I'm a new contributor, please let me know if I'm violating any protocol despite reading the relevant guidelines.

The first sentence of the article states that the order of operations is involved when a number or expression is both preceded and followed by a binary operator; the sentence refers to algebra and computer programming. In computer programming the order of operations comes into play for unary as well as binary operators, and it's necessary to know the order of operations for all operators, unary included. A few examples in C are: *p++ (two unary operators), and *p.f and &s.f and !a < b (all with one unary and one binary operator).

Some languages (C, C++, Java are examples) have a ternary operator (the conditional operator); in such cases the order of operations must be defined for the unary, binary, and the ternary operator as well.

I would suggest an alternate wording to the sentence, but as a new contributor I thought I would begin by just bringing up the issue on the talk page, where I hope I am less likely to have my head bitten off if I'm out of line in some way.

Ybsgirg (talk) 06:41, 8 March 2009 (UTC)

I agree. The word "binary" can be omitted. Good point. I'll make the change.Rick Norwood (talk) 12:58, 8 March 2009 (UTC)

Under special cases, there is the following line: "in the case of a factorial in an expression, it is evaluated before exponents and roots" 1) Is there a source for this? (there obviously should be) 2) This is not in the main article on factorial 3) It doesn't even make sense: What is 5^2! according to the above rule, it would have to be 120^2Mortgagemeister (talk) 18:04, 31 July 2009 (UTC)

The latest addition to this special cases section, has the following statement: "This convention is prone to misreading except in the simplest cases, and so parentheses are recommended." This statement is clearly an opinion and sorta of sounds like original research. Mortgagemeister (talk) 14:21, 4 August 2009 (UTC)

More on latest addition to the special cases: "If exponentiation is indicated by stacked symbols, an author may treat physical height as a grouping convention, so that" $2^{1+3}$ would mean $2^{(1+3)}=2^{4}=16$ rather than $(2^{1})+3=5$ and $2^{3}!$ would mean $(2^{3})!=8!=40320$ while $2^{3!}=2^{6}=64$.

Not sure why the "if". In math, (as opposed to computers) exponentiation is usually indicated by "stacked symbols". Never heard it described like that, but okay. Why is this under a special case? It is the usual case. Moreover, who is this "author"? Why the restriction to the author? Can't anyone treat "physical height as a grouping convention"? Again - never heard it called this, but ok I suppose. And why "may treat"? "must" would be better. This whole thing really smacks of original research IMHO12.8.160.219 (talk) 22:14, 4 August 2009 (UTC)

More on latest addition: (not sure why the above didn't show my name and not sure how to fix it)

$2^{3}!$ would mean $(2^{3})!=8!=40320$ "would mean". As opposed to what? ?????? What else would it mean?????? Mortgagemeister (talk) 11:18, 5 August 2009 (UTC)

As written, no other meaning is possible. If written 2^3!, it would be ambiguous. Rick Norwood (talk) 13:53, 5 August 2009 (UTC)

Rick - I certainly agree with you re: $2^{3}!$ . I agree with you re: 2^3! only to the extent that anything with a caret just isn't "real" math. It is computer math and thus the question would be of course which computer and/or which compiler. Is that what you meant by ambiguous? 69.65.71.211 (talk) 21:38, 5 August 2009 (UTC) <==mortgagemeister (the 69.65.71.211 (talk) 21:38, 5 August 2009

(UTC) doesn't seem to work for me?????)

"Real math" is independent of notation. The superscript notation is traditional, but no more "real math" than the caret notation. Also, hierarchy of operations is not "real math" but is entirely conventional -- it "is" whatever people decide it is. Essentially everyone (except a few American grade school textbooks) agree that 3*4+5 = 17, not 27. There is no such general agreement about 2^3! nor is there ever likely to be. Rick Norwood (talk) 13:52, 6 August 2009 (UTC)

More metaphysical then I intended. All I meant was that the "superscript notation" is what is taught in almost every math classroom. If a student were to take no computer classes and not buy a computer he could still get an A in every class and never even "meet" the caret notation. Moreover, if we are going to get a bit metaphysical, the math is certainly notation dependent. The superscript notation is implicitly an unary operator. The caret is a binary operator. Perhaps that is why you consider it ambiguous. (If not why do you?)Mortgagemeister (talk) 15:38, 6 August 2009 (UTC)

Since we agree on the unambiguity of $2^{3}!$, I am going to edit that section out. Mortgagemeister (talk) 21:07, 6 August 2009 (UTC)

The difference between a binary and a unary operator is not a question of notation. Many notations can be used for either. And while factorial is unary, exponentiation is binary -- it has both a base and an exponent. And the caret notation is now common in e-mail and text messaging as well as on calculators and in computer programs. But I agree that $2^{3}!$ can go. Rick Norwood (talk) 12:55, 7 August 2009 (UTC)

But it seems (at least if I read the history correct) that you just put it $2^{3}!$ back in? And you took out the precedence rule for factorial vs exponent and left it what? unknown, unstated? Mortgagemeister (talk) 13:37, 7 August 2009 (UTC)

Your latest addition: "a factorial in an exponent applies to the exponent, while a factorial not in the exponent applies to the entire power." sounds like a forced rule, having nothing to do with precedence. Any "order" has to be more than just a bunch of rules, each of which dealing with only a single case. 69.65.71.211 (talk) 16:37, 7 August 2009 (UTC)

So Rick - your latest edit is a duplicate of what already exists (in the examples) of the first section. Moreover, it is a special case because it is an exception to the rule of equal precedence operators are evaluated left to right. If that is why you are putting it in the special case section, I certainly think you should make that clear. Mortgagemeister (talk) 21:57, 8 August 2009 (UTC)

One-half x

The paragraph just above the Examples section has the information backwards. I believe better wording would be the following:

Similarly, care must be exercised when using the slash ('/') symbol. The string of symbols "1/2x" is interpreted by the above conventions as 1/(2x). If what is meant is (1/2) × x, it should be written as (1/2)x. Again, the use of parentheses will clarify the meaning and should be used if there is any chance of misinterpretation.

Also, x/2 is not part of the misinterpretation. If there are no reasonable objections, I'll go ahead and make that change. JackOL31 (talk) 17:52, 5 December 2009 (UTC)

But if multiplication and division are of equal priority, and consecutive operations of equal priority are performed left to right, then the correct interpretation of "1/2x" is indeed (1/2)x. Also, the current statement contradicts the relevant claim in the "Calculators" section. I will reverse the statement unless someone has an objection. TrippingTroubadour (talk) 08:09, 27 February 2010 (UTC)

Minor Edits 25 March 2006

Changes inside the table only:

• capitalization more consistent
• removed & (ampersand) where is should be the word "and"
• made it clear that there is more than "comparision less-than"

Charles Gaudette 18:54, 25 March 2006 (UTC) there is so many things that people can do to save math —Preceding unsigned comment added by 65.209.37.196 (talk) 18:07, 1 March 2011 (UTC)

Confusion

Thought the following two sentences were possibly confusing:

"If PEMDAS is followed without remembering to do multiplication and division at the same time and done instead with multplication then division students will get wrong answers. For example: 6÷2(1+2)=9 not 1."

I.e. is 6/2(1+2)=9 a wrong answer by a student or the correct answer?

9 is correct, if you don't believe me copy and paste 6/2(1+2) into Google and hit enter —Preceding unsigned comment added by 129.82.99.144 (talk) 13:22, 29 April 2011 (UTC)

Made minor change to make meaning certain. —Preceding unsigned comment added by 92.28.212.122 (talk) 12:16, 29 April 2011 (UTC)

The answer to 6/2(1+2)=1, not 9. You are confusion multiplication with distribution. Distribution is NOT multiplication even if you multiple and it is not an operation because it does not change equality, it simply re-writes an equality in different ways.I will solve it two different ways to show. First, Lets Distribute the 2 into (1+2). 6/(2+4) Now Follow PEMDAS or PEDMAS. 6/(6)=1 Second way. 6/2(3) now, we DISTRIBUTE, NOT MULTIPLY the 2 into the 3. Although we multiply to make the work less, we are technically ADDING ((3)+(3))=(6). 6/(6) =1 The following question arises. Is 2(3) the same as 2*(3)? the answer is no, they are two different numbers because the operation splits quantities. —Preceding unsigned comment added by 50.46.18.174 (talk) 17:23, 29 April 2011 (UTC)

Sorry, but you're wrong. In order to distribute the 2 over the sum, using the rule multiplication distributes over addition, you have to multiply 2 times the sum before you divide the six by the two. That's wrong. The correct rule is that 6/2(1+2) means 6 * (1/2) * (1+2). Division is multiplication by the reciprocal. You seem to think distribution is different from multiplication. It's not. It is a property of multiplication. In 2(3) the operation is still multiplication, only the "multiplication" is "understood". In other words, concatination implies multiplication. Cut and paste 6/2*(1+2) into Google, as suggested above, if you're still in doubt. Rick Norwood (talk) 17:35, 29 April 2011 (UTC)

6/2*(1+2) is not the same expression as 6/2(1+2). A number, 6, divided by a number, 2, multiplied by a number,(1+2) is not the same as saying a number, 6, divided by a number, 2(1+2). Although they seem to be same because we learn that when you distribute a number in parenthesis you multiply that number. However, what you are failing to see is that 2(3) is a single number. Whereas 2*(3) are two different numbers separated by an operation. Distribution is not an operation (else it would be in the order of operations), and only operation separate numbers. —Preceding unsigned comment added by 50.46.18.174 (talk) 18:39, 29 April 2011 (UTC)

Looking a raft of high school algebra books, any California high school student that does not agree that 6/2(1+2)= 9 will be graded wrong. While I appreciate that there have been, and may still be, groups that give precedence to implicit multiplication over explicit multiplication, that is not the convention currently taught in high schools in California, at least. If a mathematical society wants to come up with some different rules, I suppose they can, but first they will have to unlearn what they were taught in high school, then live with the fact that the rest of the population is being taught rules that come up with 6/2(1+2)= 9. The links below jump directly to the pages spelling out the order of operations rules taught and tested for.

• "order%20of%20operations" Kaplan CSET: California Subject Examination for Teachers
• "order%20of%20operations" Roadmap to the California High School Exit Exam: Mathematics, 2nd Edition
• "order+of+operations" Passing the California Algebra 1 End of Course Test
• "order%20of%20operations" CliffsTestPrep California High School Exit Exam: Mathematics
• "order%20of%20operations" Beginning Algebra By Mustafa A. Munem, Carolyn West
• "order%20of%20operations" CAHSEE - Mathematics (REA): The Best Test Prep for the California High School Exit Examination in Mathematics
• "order%20of%20operations" Kaplan CAHSEE Mathematics: California High School Exit Exam Kaplan

Macchess (talk) 08:10, 3 May 2011 (UTC)

Unsigned: you do not get to make up the rules of mathematics as you go along. 2(3) is not a "single number", it is the product of two numbers. When two numbers are juxaposed, the juxaposition must mean something. What it means is a convention, but it is an important convention. In this case, what it means is multiplication, a decission arrived at, historically, in 17th century France, and accepted throughout Europe and later throughout the world.

Distribution is a law of mathematics, which can be taken as one of the field axioms, or proved as a consequence of the Piano postulates. Multiplication distributes over addition. In this case, however, the distributive law is not the problem. The question is whether to divide by the shortest complete symbol or string of symbols following the division sign, in this case the 2, or to divide by some longer string of symbols. The former choice was made. You can check this just by typing that string of symbols into any reasonably expensive calculator. Follow this link to see how Texas Instruments used to follow the opposite convention but changed its mind. http://epsstore.ti.com/OA_HTML/csksxvm.jsp?nSetId=103110.

But the most important rule is not a rule of mathematics but a rule of common sense. Don't write expressions like this.

Rick Norwood (talk) 12:13, 30 April 2011 (UTC)

I think that Wolfram Mathematica is quite an authority in case of mathematics, isn't it? ;) http://img96.imageshack.us/img96/3839/mathematica6221.png —Preceding unsigned comment added by 83.13.175.139 (talk) 22:23, 30 April 2011 (UTC)

Microsoft Excel and the unary minus

The article says that Microsoft Excel evaluates a unary minus at a higher order of precedence than exponentiation. This would mean, presumably, that it would interpret the polynomial −x^2 + x + 1 as equal to the polynomial x^2 + x + 1. Does anyone know if later versions of Excel have this same problem? Rick Norwood (talk) 12:18, 1 May 2011 (UTC)

I am using Excel 2003, which interprets −x^2 + x + 1 as equal to x^2 + x + 1. Surely later versions do the same, because they should not break existing formulas. Ceinturion (talk) 19:52, 3 May 2011 (UTC)

Not necessarily. The footnote to this section mentions this as a "problem" and offers a "fix". More recent versions may (or may not) provide that fix. Anybody have a version of Excel from the last seven years? Rick Norwood (talk) 12:29, 4 May 2011 (UTC)

Church's dot

What about the dot notation of Alonzo Church where left-grouping is assumed for the missing brackets and a dot stands for a missing left bracket whose mate is as far to the right as possible without changing the pairing of the existent brackets? It is regularly used for the λ-binder in the λ-calculus and occasionaly with non-assiciative algebraic operations. For example the logical axioms (A⇒(B⇒C)) and ((A⇒(B⇒C))⇒((A⇒B)⇒(A⇒C))) can be shortened as A⇒.B⇒C and (A⇒.B⇒C)⇒.A⇒B⇒.A⇒C Zinoviev (talk) 12:10, 2 November 2011 (UTC)

Trigonometry Traditions

I dislike the traditions of notation for trigonometry functions, but they exist. I'm looking at the 2004 Larson/Hostetler high school Trigonometry textbook. In defining simple harmonic motion, one reads: "d = a sin ωt" instead of "d = a sin( ωt )" . I've not seen an authoritative rule, but it seems to be, the first term after one of the six trigonometry function names (and their inverses) is implicitly bracketed as the function's input. Thus "tan 2πx + 3" is parsed as tan(2πx) + 3. Many books explicitly bracket a term that has a leading negative sign, as in sec(-2πx). There is of course the basic irritation that in a trigonometry context, "sin x" is not the product of four variables, but a function of one variable x. Comments? What other functions have this implicit bracketing? Is this tradition worth mention in the article? --Gregoreo (talk) 21:37, 11 August 2008 (UTC)

I just stumbled upon the issue of priorities between operations like composition of a function and exponentiation: one should read $\cos ^{2}(x)=\cos(\cos(x))$ but is very often understood as $(\cos(x))^{2}$ while it should be equivalent to $\cos(x)^{2}$. But the latter is sometimes interpreted as $\cos(x^{2})$ which is, in my opinion, plainly wrong... Are these caveats worth mentioning? Christian.Mercat (talk) 13:33, 10 November 2011 (UTC)

Printed mathematics textbook as a source (moved from top of page)

I'm dealing with some extremely "special" folks that don't believe -1^2 has a different meaning than (-1)^2. I see you need a citation for this result in the "gaps in the standard" section. This is addressed in every basic level mathematics textbook. Exactly what information do I need do I need to provide so that you'll be satisfied with the citation? I have many print books that clearly indicate -1^2 = -1, but I can't find anything online.209.124.221.148 (talk) 04:11, 28 September 2011 (UTC)

You may cite a published mathematics textbook if you want. The more authoritative the better. There is no need for it to be online. In maths, printed works may often be considered more reliable than web sites. Thincat (talk) 19:47, 27 November 2011 (UTC)

APL and Smalltalk wrong

APL and Smalltalk do have different precedence levels.

APL operators have higher precedence than functions and are left associative.

Smalltalk unary messages have higher precedence than >1-ary messages. — Preceding unsigned comment added by 24.130.147.208 (talk) 04:43, 15 July 2012 (UTC)

48÷2(12)

This problem has been meming its way around the internet lately. I checked out this article to see if it could shed any light on the question of whether the correct answer is 2 or 288. The real question is whether the expression could also be written as 48÷2×12, or whether its the equivalent of 48÷(2×12). We don't evaluate 48÷2x as 48÷2×x, which would be 24x, but it could be that this is only the case when using variables, and that when using numbers, multiplication is multiplication. There are some fairly smart and informed people here, so I thought I'd ask if anyone knows what the correct answer is. - Lisa (talk - contribs) 04:14, 10 April 2011 (UTC)

Reading 48÷2x is actually completely correct though. To answer your question directly though, the convention is to solve equal priority operators from left to right.97.94.226.5 (talk) 09:27, 10 April 2011 (UTC)

Thanks. But I think the question is, are 2(12) and 2×12 the same? Are they both the identical operator? Is 48/2x equal to 24x or 24/x? - Lisa (talk - contribs) 12:29, 10 April 2011 (UTC)

Yeah, I'm seeing it like this image here: http://apina.biz/39808.jpg

My Casio 9860GII and 991ES return 2, GNOME calculator results in 2. Sage and Maxima all fail to evaluate without an operator before the first (. Mathomatic returns 288.

2 in that case to me is clearly wrong.

Thoughts? --Lakkasuo (talk) 15:16, 10 April 2011 (UTC)

My thinking is that 2x differs from 2 × x. The notation 2x carries implicit grouping. Brackets and parentheses are evaluated first because they connote grouping, so a notation that groups implicitly should take precedence over one that does not. So the answer would have to be 2. - Lisa (talk - contribs) 16:39, 10 April 2011 (UTC)

From what I learned in grade school, I'm pretty sure that it's 288. The crux of the argument is as Lisa says - is 2(12) an equivalent statement to 2×12 (or rather - it should be the crux...some people fail to realize that multiplication and division are, more or less, the same operation). In my experience, a(b) means a×b, since when a statement is fully evaluated inside any sort of parenthesis, the parenthesis are simply removed and coefficients are multiplied having the normal multiplication priority. However, I believe that order of operations was not always taught this way, and so some may in fact have learned that 2(12) = (2(12)). Having not learned it this way, it makes little sense to me, but what can I say... Rill2503456 (talk) 01:00, 12 April 2011 (UTC)

I agree with Lisa. To me 2(12) implies grouping, and the question she raised here is if 2x12 and 2(12) are the same? I am not a math professional so most people here know more than I do. But from what I've read so far, although Order of Operations clearly says execute operators with equal priority from left to right, it never states if 2x12 and 2(12) have equal priority or not. I understand it's common to write 2x12 as 2(12) but are they exactly the same? Hoyun (talk) 15:28, 29 April 2011 (UTC)

In most situations 2(12) is shorthand for 2*12. Historically, it was invented to avoid 2×12 which could be mistaken for 2x12= 2*x*12 = 24x. Computer scientists needed a better multiplication sign and introduced * for that purpose, because it looked a little like ×, could not be mistaken for x, and was handy on the typewriter or cardpunch keyboard. As I mention below, this "left to right" rule is grade school, not used by professionals. Rick Norwood (talk) 15:37, 29 April 2011 (UTC)

Surely the correct interpretation is dependent on context? Considered in isolation it is unclear.

In the case of 100/2π, is it necessarily obvious which interpretation to take? Is '2π' taken as shorthand for a single constant or is it split to give (100/2)*π?

Similarly, 4i^2 can mean two entirely different things depending on whether the context indicates use of complex numbers.

Doctor Wibble (talk) 09:13, 13 April 2011 (UTC)

I didn't see anything in this article about working from left to right to calculate items in the same order of precedence. I think that's where a lot of this confusion is coming in--people don't realize they have to calculate the division part before the multiplication because it's the leftmost item. —Preceding unsigned comment added by 216.38.216.134 (talk) 22:34, 12 April 2011 (UTC)

In the United States, at least, there is one set of rules in grade school and a different set of rules for professionals. Grade school teachers tend to teach what they were taught in grade school, by teachers who taught what they were taught, and so on. There is absolutely no "rule" in mathematics about working from left to right. Real mathematicians, scientists, engineers and other professionals know not to write expressions that can be easily misunderstood, and that anything that can be misunderstood will be. Thus the horizontal fraction bar is preferable to the slanting fraction bar. Rick Norwood (talk) 12:25, 14 April 2011 (UTC)

That notation is ambiguous in mathematics, and so it would be fundamentally wrong just how it's written, a vinculum or grouping symbols would be used instead to clearly show the division, as it is it can't be solved in mathematics as it's badly constructed. In computing, the operation will be done left-to-right, and since we've been using computers for calculations for ages, that makes 'left-to-right' the de facto standard when ambiguity is found. The proper way to note 48/2(9+3) would be with the vinculum, or like this (48/2)*(9+3), putting back any implied operators and properly grouping the operations, and that's how a computer will solve the operation, so that's the way to go. 95.120.37.230 (talk) 02:44, 26 July 2012 (UTC)

No mention of "left to right"

"If PEMDAS is followed without remembering to do multiplication and division at the same time and done instead with multplication then division students will get wrong answers. For example: 6÷2(1+2)=9, not 1."

This is clearly a ridiculous portion, not because the answer is wrong, but because the justification is absurd. There is no way to do "multiplication and division at the same time" in this example.

Following PE(DM)(AS) as explained in this article the best you can get is the following:

6÷2(1+2)=6÷2*3

There is no way to perform the division of 6 by 2*3 "at the same time" as multiplication of 6÷2 and 3.

One possible convention you could adopt is, when all else is equal, perform operations from left-to-right, in which case you would come out with the answer 9. But nothing in this article mentions that left-to-right is a convention in BEMDAS and I can't find any sort of official opinion from any reputable organization regarding it. Another convention one could adopt is to view the multiplication implied by placing numbers adjacent to each other as a form of grouping, which would give an entirely different answer (one that many newer programs and calculators would give, although almost certainly not the majority). MarcelB612 (talk) 19:51, 30 April 2011 (UTC)

Good catch. At the same time is absurd. I'll fix it. Rick Norwood (talk) 20:52, 30 April 2011 (UTC)

I have added "left to right" to the main "order of operations" list. Reading this whole mulit-page article, there was still no mention how 10/5*2 should be evaluated. If we don't say left-to-right is a convention, one saying 10/5*2=10/10=1 would be right. — Preceding unsigned comment added by 137.138.122.18 (talk) 13:35, 26 July 2012 (UTC)

This has been discussed many times before. There is no left to right rule. Rather, /5 means *(1/5) (division is multiplication by the reciprocal) and 10 * 1/5 * 2 can be done in any order. Nobody in his right mind would do 13/17*17 from left to right. Rather, they would observe that the division by 17 and the multiplication by 17 cancel, leaving 13. In teaching this, It is better to teach students to use the vinculum rather than the solidus, that is the horizontal fraction line rather than the slanting fraction line, because the slanting fraction line often leads to this confusion. Nobody gets confused by ${\frac {10}{5}}$×2, Rick Norwood (talk) 15:04, 26 July 2012 (UTC)

Subtraction and division are binary operations

The article claims that "It is helpful to treat division as multiplication by the reciprocal (multiplicative inverse) and subtraction as addition of the opposite (additive inverse)", and proceeds to treat division and multiplication in that way. So, according to the article, $a-b$ represents $a+\mathrm {AdditiveInverse} (b)$. The article later says that "There exist differing conventions concerning the unary operator − (usually read "minus")", accepting unary expressions such as $-5$, to be interpreted as $\mathrm {AdditiveInverse} (5)$.

I claim that this attitude is not neutral and does not represent the point of view of all mathematically active people and machinery.

Subtraction is usually first introduced before negative numbers, as a (partially defined) binary operator in the set of natural numbers. In this context, $a-b$ can't be defined as $a+\mathrm {AdditiveInverse} (b)$, and in fact the result of $73-25$ can't be computed directly using the algorithm of addition.

After the introduction of negative numbers, subtraction of integer numbers can be defined, and the expression $-5$ can be interpreted as $0-5$, just as $+5=0+5$.

Analogous considerations affect the definition of division, which is usually first defined between (positive) integers and later generalised.

In rigorous treatments of mathematics, the set of natural numbers is usually not considered a subset of the set of integer numbers (which are more complicated objects, defined later). And the set of integers is not contained in the set of fractions. This care is also often taken by computers, which have different datatypes and operations for unsigned integers (natural numbers), signed integers and floating point numbers (fractions). For example, in C (programming language) you get $7/2==3$ and $7.0/2.0==3.5$.

If subtraction is considered a binary operation, it is usually given the same level of precedence that multiplication, and expressions involving both operations are evaluated from left to right, giving, for example, $5-4+3=(5-4)+3$.

(The left-to-right convention, also disambiguates multiple exponentiations, giving, for example, 5^4^3=(5^4)^3.) But see the article on operator associativity (which originally led me here), where the exponentiation is regarded as right-associative (evaluated from right to left). Left to right convention is also used in lambda calculus, which is where I started reading.

Conclusion: the point of view presented in the article isn't shared by computer programmers, nor rigorous mathematicians, nor schoolchildren. (Marcosaedro (talk) 08:05, 26 September 2012 (UTC))

Multiplication and division

I can't wrap my head around this paragraph:

Similarly, there can be ambiguity in the use of the slash ('/') symbol. The string of characters "1/2x" is interpreted by the above conventions as (1/2)x. The contrary interpretation should be written explicitly as 1/(2x). Again, the use of parentheses clarifies the meaning and avoids the possibility of misinterpretation.

It appears to be saying that $1/2x\,$ should be interpreted as ${\tfrac {1}{2}}x\,$ rather than ${\tfrac {1}{2x}}\,$. As far as I know, the correct interpretation (calculators notwithstanding) is ${\tfrac {1}{2x}}\,$. To check that I wasn't going crazy, I spent about 20 minutes flipping through various math and science books on my bookshelf and confirmed that, indeed, the convention seems to be that the solidus is evaluated last in these kinds of expressions. For example, in Griffiths' electrodynamics book we find expressions like $1/\mu _{0}\epsilon _{0}\,$ and $(2/\sigma ){\sqrt {\epsilon /\mu }}\!$, all clearly written with the convention that the solidus has lower precedence than multiplication. What gives? Am I missing something about this paragraph? Zueignung (talk) 00:56, 1 August 2012 (UTC)

The paragraph is right in that the expression is ambiguous, but I agree that the more common and more natural interpretation is $1/2x=1/(2x)$, whereas ${\tfrac {1}{2}}x$ is written as (1/2)x. I don’t known what kids learn in elementary school, it may well be that the convention they are taught doesn’t respect the practice in professional mathematics.—Emil J. 10:27, 1 August 2012 (UTC)

It depends a lot on the context. Consider, for example, $x^{2}+1/2x+1/3=0$. Most people would "see" the middle term here as (1/2)x because they are used to the form of a quadratic equation. On the other hand, in $1/2x=4$, many people would "see" 1/(2x) = 4, because they would expect (1/2)x in this case to be written x/2. The important point is that 1/2x is ambiguous. The only "rule" is that /expression is the same as (expression)^-1. When "expression" is not just a single symbol, it is safest to put it inside parentheses. Rick Norwood (talk) 13:49, 1 August 2012 (UTC)

Where are the references for these claims? Who says that ${\tfrac {1}{2x}}\,$ "should" be written as $1/(2x)\,$ with a solidus? It appears to contradict the convention used in published work. Zueignung (talk) 14:19, 1 August 2012 (UTC)

I never said that ${\tfrac {1}{2x}}\,$ "should" be written as $1/(2x)\,$. Of course it should not be so written, to avoid ambiguity. The question is, what to do when it is so written? Rick Norwood (talk) 15:46, 1 August 2012 (UTC)

The paragraph in question says: "The contrary interpretation should be written explicitly as 1/(2x)" (emphasis mine). Who is saying that it "should" be written this way? Why should I not remove this unsourced assertion, given that it appears contrary to established convention? Zueignung (talk) 16:20, 1 August 2012 (UTC)

The example was 1/2x. If you want this to be interpreted as 1 divided by the quantity 2x, you should use parentheses. But it would be better to avoid the solidus entirely. Rick Norwood (talk) 17:21, 1 August 2012 (UTC)

Please provide bibliographic references backing the assertion that 1 divided by the quantity 2x should have parentheses if written with a solidus. The assertion appears to be at odds with reality. Zueignung (talk) 19:57, 1 August 2012 (UTC)

That 1/2x is ambiguous and should be avoided is one of those things that is so obvious that I'm having trouble finding a reference that actually says so, just as I would have trouble finding a reference that says you shouldn't use the symbol "2" as the name of a variable. But saying it is obvious doesn't solve the problem, since Zueignung thinks it is obvious that there is no ambiguity, and provides one example from a serious math book where the solidus has lower precidence than multiplication, that is, where x/yz has a different meaning from x÷y*z. Can anyone help with a reference? Rick Norwood (talk) 16:02, 2 August 2012 (UTC)

Rick, you are constantly making a straw-man argument. The paragraph Zueignung is objecting to doesn’t just say that 1/2x is ambiguous. It says that 1/2x should be interpreted as (1/2)x, whereas 1/(2x) should be written with brackets. This is the main problem. I can’t speak for Zueignung, but I would be happy if the paragraph were trimmed to only state that 1/2x is ambiguous, without suggesting which of the two interpretations is right and which is wrong.—Emil J. 16:23, 2 August 2012 (UTC)

I agree that 1/2x is ambiguous. I disagree that the order of operations conventions as currently stated in the article would call for this expression to be interpreted as (1/2)x, and I disagree that if one intends to write ${\tfrac {1}{2x}}\!$ with a solidus that one "should" write $1/(2x)\,$. This is a value judgement that requires proper attribution. It is fine for the article to say something like "Mathematical typesetter John Smith (2003, p. 46) suggests that parentheses should be used to avoid ambiguity." I do not think it is appropriate to state—without attribution—that one should use parentheses in this situation, given that it does not seem to be borne out in practice. I agree with Emil J; I think it would be best to simply state that 1/2x is ambiguous and leave it at that. Zueignung (talk) 16:30, 2 August 2012 (UTC)

I can live with that, but I will also still search for an reliable reference. My reason for thinking that 1/2x "ought to mean" (1/2)x is as follows. The current rules for operations state that multiplication and division are on the same level, and that division is multiplication by the reciprocal. I've never seen a rule that says implied multiplication takes precidence over the solidus.

Would you agree that 1 ÷ 2 × x means (1/2)x, for the same reason and following the same rule that says 1 - 2 + x = -1 + x? (When my children were in grade school their textbook, written by an editor of The Mathematics Teacher, a journal in which I have published, said that 1 - 2 + x = 1 -(2+x) because My Dear Aunt Sally clearly states that addition takes precedence over subtraction. This article rightly rejects My Dear Aunt Sally. I only mention this because so many children are being taught so badly that I think the rules should be crystal clear.) So I ask, is 1 ÷ 2 × x equal to 1 ÷ 2x? Is it equal to 1/2×x? If the answer to either of these questions is no, then the precidence of an operation changes depending on which symbol is used to write that operation, which seems to cause needless confusion. And, of course, if we teach that 1/2x means 1/(2x), then we need to explain that calculators do operations differently from mathematicians, which also seems to cause needless confusion. I understand that Zueignung is expressing an honest opinion. But I teach this stuff all the time, both at the graduate level in abstract algebra and at the freshman level in our Math for Teachers course, and I take getting it right seriously. Rick Norwood (talk) 19:09, 2 August 2012 (UTC)

It may well be that some (or many) people are working with conventions that are ambiguous, that make operator precedence dependent on what symbol is being used, or that are at odds with calculator conventions. If that is indeed the case, these facts should be noted in the article. Sweeping these things under the rug and simply declaring what 1/2x "ought to mean" makes the article factually inaccurate. Zueignung (talk) 20:25, 2 August 2012 (UTC)

But the phrase you put in quotes, "ought to mean", is not in the article. It is in the discussion here, and was given to explain why I think the usual rules are good rules. What the article says is "The string of characters "1/2x" is interpreted by the above conventions as (1/2)x." And that is the case. If multiplication and division have no difference in precedence, then 1/2x means 1÷2×x =(1/2)x. That is just applying the rules as stated. The next sentence, which you have flagged as dubious, says "The contrary interpretation should be written explicitly as 1/(2x).[dubious – discuss]" But if the rules state that one interpretation obeys the rules as given, then it follows logically that the other interpretation does not obey the rules as given, and should be written differently. Is your objection the use of the solidus rather than the vinculum? Rick Norwood (talk) 14:57, 3 August 2012 (UTC)

Now the discussion is beginning to move in circles. This is my objection: 'I do not think it is appropriate to state—without attribution—that one should use parentheses in this situation, given that it does not seem to be borne out in practice.' Zueignung (talk) 15:10, 3 August 2012 (UTC)

I edited the paragraph to make explicit the facts that (a) if the solidus is interpreted as multiplication by the reciprocal, then 1/2x = (1/2)x, and (b) this is not the usual interpretation in published work. Zueignung (talk) 03:00, 5 August 2012 (UTC)

Evidently, physics has its own conventions. Rick Norwood (talk) 15:34, 6 August 2012 (UTC)

This difficulty has been met by a lot of people, and I think it should be more carefully addressed in the article. On one hand, computers usually evaluate from left to right, so that 8-4+3=(8-4)+3 (and analogously, $8\div 4*3=(8\div 4)*3$). On the other hand, the expression 8/4*3 (with the slash being usually taller than the surrounding symbols) is graphically similar to ${\frac {8}{4*3}}$, and is sometimes the chosen substitute when embedding in computer text a fraction of monomials. This convention makes sense when transcribing formulas from a physics textbook, because by giving the division operator a lower order of precedence than the multiplication operator, we can transcribe ${\frac {ab}{2cde}}$, as a*b/2c*d*e, which is more economic than (a*b)/(2c*d*e). The expression a*b/2/c/d/e doesn't resemble the original formula, and also leads to a bad strategy for computation (because multiplication is preferable over division). (Marcosaedro (talk) 09:22, 26 September 2012 (UTC))

People will do what they will do, but it seems to me like a really bad idea for physics and mathematics to have a different set of rules, especially when people use one rule part of the time and the other rule part of the time. Rick Norwood (talk) 12:04, 26 September 2012 (UTC)

Lexing Error?

There's a lexing error(whatever that is) on the page. Please fix it. — Preceding unsigned comment added by 98.15.248.218 (talk) 00:32, 2 October 2012 (UTC)

Request for a citation about what the TI-89 does.

I'm a little puzzled about a request for a citation for what the TI-89 does. If I push the buttons on the TI-89, that is what it does. But nobody is likely to say in a book that if you push those buttons that is what the TI-89 does. You can't really use the evidence of your own experience in Wikipedia, but I'm not sure how else to prove that the TI-89 does what it in fact does. Suggestions? Rick Norwood (talk) 22:26, 5 October 2012 (UTC)

I removed it, calling it a "trivially reproducible operation on the TI-89", I'm not sure how one would cite that, but anyone with a TI-89 can reproduce it without fail, so its pointless to ask for a citation. Zath42 (talk) 05:17, 15 October 2012 (UTC)

How do you write a^b^c?

I don't know how to get a^b^c to display in superscript form. Help? Rick Norwood (talk) 13:16, 7 August 2009 (UTC)

I was able to get this:

$(a^{b})$$^{c}$

Mortgagemeister (talk) 15:02, 7 August 2009 (UTC)

You misunderstand my question. I want to display a stack of powers without using parentheses. Rick Norwood (talk) 13:55, 8 August 2009 (UTC)

$a^{b^{c}}=a^{(b^{c})}\neq (a^{b})^{c}\,$

Mortgagemeister, you have wrongly placed the parentheses around a^b, and your method, using TWO sets of math tags, is uncouth. Michael Hardy (talk) 16:34, 8 August 2009 (UTC)

Mr. Hardy: I didn't wrongly place anything. I wasn't answering the question "What is the order of operations for a^b^c?" I thought Rick Norwood was trying to figure out the wiki syntax for presenting a^b^c in "stacked symbols". I was showing how far I had been able to get. As to the couthness of my method - well I'm from Brooklyn. I certainly congratulate you on your couth method. Mortgagemeister (talk) 21:51, 8 August 2009 (UTC)

And so I was. Thank you for your help. The use of set braces around the exponent was just what I needed, couth or no. Rick Norwood (talk) 11:52, 24 July 2010 (UTC)

I do think there is some current ambiguity in the Order of Operations as it currently stands, as the article only addresses hand written or typeset expressions. It does not address the current common usage of using the caret ^ or doubled asterisks ** to expression exponentiation. As a result, there is a perceived ambiguity to expressions such as a^b^c, and whether or not they should be computed as (a^b)^c or a^(b^c). In this case, I believe the most commonly used idea of operations of the same type being performed left to right is incorrect, as what most people visualize when seeing that expression would need to be read right to left. I cannot, however, find documentation for either way being the preferred method. 74.178.124.221 (talk) 22:19, 25 December 2012 (UTC)

Mnemonic used in NZ

"PEMA is one of the mnemonics taught in New Zealand.[citation needed]" Last time I checked, being a New Zealander and got taught it in primary 10-12 years ago, BEDMAS is what we use here. This is, of course, it has been changed in the last few years. I'll see what references I can find either way for this.

• I learnt BEDMAS too, albeit before NCEA. Neftaly (talk) 00:03, 9 January 2013 (UTC)

• I learnt BEDMAS and BODMAS (during NCEA changeover). Don't know where this PEMA crap came from... — Preceding unsigned comment added by 122.248.130.1 (talk) 02:58, 9 January 2013 (UTC)

recent edit

It is not enough to know how to use fractions to write about them. To write about them requires understanding and care. I've reverted the following addition to the article "The "fraction bar" or viniculum has the same role as the parenthesis. It instructs you to treat the quantity above the numerator as if it were enclosed in a parenthesis, and to treat the quantity below the numerator as if it were enclosed in yet another parenthesis." There is no "quantity above the numerator". In any case, the article already explains that the vinculum is used as a symbol of grouping. Rick Norwood (talk) 15:40, 10 January 2013 (UTC)