Talk:Order of operations

There is some evidence to suggest that the order of operations exists as a way to reduce error if one were to mis-transcribe an equation or expression.

See http://mathandtext.blogspot.com/2005/05/order-of-operations.html.

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)

BODMAS' O

The O in bodmas is by no means universally held to be Orders. In fact, this is the first time that I have heard it to be specifically called this. Others that I have heard include:

Over (Similar to divide or brackets) Other (would include exponents) Of (Similar to divide) Order (Close to orders)

Essentially, the O is there really just to add a much needed vowel in the middle of a group of consonents. This is especially true when considering that it is really only useful for children - as an acronym wouldn't do for all of the mathematical operators (really).

When I was taught the Bomdas Rule (Ireland) 'O' meant "Of means multiply, and must be done as if inside brackets" for questions such as 2 + 1/2 of 4. Regards, MartinRe 10:39, 7 May 2006 (UTC)

Ref for above [1] MartinRe 10:41, 7 May 2006 (UTC)

I was also taught this (in England), and I believe it is wrong. Most people would evaluate "1/2 of 3 + 7" as 5, some as 8.5, and none as 7.16666 (as required by the "Of" version). Maproom (talk) 09:35, 4 April 2008 (UTC)

In the school text books of the school where I went in the North-east of England, about six years ago, it almost always said the O stood for 'of'.--Jcvamp 06:08, 14 February 2007 (UTC)

Wrong

Almost all descriptions of the order of operations are flatly wrong, including the one here. To give just one example, in 2 + 3 + 4*5 it is perfectly ok to add the 2 and 3 before you multiply the 4 and 5. In fact, the very phrase "order of operations" is misleading. What this article is really about is understood parentheses, since the operations of arithmetic are all binary operations. Small wonder that non-mathematicians have trouble understand algebra! Rick Norwood 14:18, 7 May 2006 (UTC)

Do you have a source? It looks like that fails the Google Test:

Results 1 - 10 of about 23 for "understood parentheses".

Results 1 - 10 of about 631,000 for "order of operations". capitalist 02:29, 8 May 2006 (UTC)

Yes, as I said, almost all. In this case 631,000 to 23. All professional mathematicians know better, but few spend any time fighting the battle to improve elementary education. It doesn't do any good, and annoys the education majors. Rick Norwood 19:26, 8 May 2006 (UTC)

Well if the goal is to improve communication between the 23 and the 631,000, wouldn't it seem easier to convince the smaller group to change their terminology instead of trying to get the rest of the planet to go along with the 23? That would be the quickest way to get everyone on the same sheet of music. Or are there factual issues beyond just the terminology? In other words, if the 631,000 build computers using the information in this article, will they not work as well as computers built by the 23? If that's the case then the 631,000 must change in favor of the 23. If it's just terminology then I think the reverse is true. capitalist 04:24, 9 May 2006 (UTC)

All computers and most calculators use the correct heirarchy. The one in this article is the one usually taught, and in a sense, it works, but nobody who understands mathematics would insist, in the problem 1 + 2 + 3*4, that you must do the 3*4 before the 1 + 2. Clearly, you get the same answer either way. Rick Norwood 21:18, 10 May 2006 (UTC)

The article doesn't claim that you won't get the same answer. In the first line it states that O.O.O. is a notational convention, not mathematical fact. It sounds like just a terminology issue. Given the preponderance of the term "order of operations" over the term "understood parantheses" my guess would be that the latter term is disappearing from the language. An interesting case of linguistic evolution I suppose, but at any rate I wouldn't recommend trying to stem the tide through a change in the title of the article. capitalist 03:24, 11 May 2006 (UTC)

I, too, would oppose a title change. I do think the rule can be stated precisely. Just what more a precise statement would take is something I've been working on. Rick Norwood 17:27, 11 May 2006 (UTC)

Question

Anyone knows what would be the result of the equation y = 2x-x^2+6 when x = -0.5? Would it be 4.75 or 5.25? Is the negate applied to the x^2 first or the order 2 applied to x first, then negate?

xieliwei 16:39, 25 June 2006 (UTC)

Substitution, in an ambiguous case, must be carried out within parentheses. Thus, replacing x with -0.5 yields 2(-0.5) - (-0.5)^2 + 6 = -1 - (0.25) +6 = 4.75. Rick Norwood 14:01, 25 June 2006 (UTC)

I have a question also, when did mathamatics start using the Order of Operations? Was it something that came with the computer age or did the likes of Sir Isaac Newton or Pythagoras use it?--SerialCoyote 19:29, 17 November 2006 (UTC)

Obviously it was not as recent as the computer age. Just look at math journals and books of the 19th century or journals and books that appeared in the first half of the 20th century. How could they have done without such things?? It seems weird to think they could have. Michael Hardy (talk) 17:04, 5 July 2008 (UTC)

That is an excellent question, and a subject worth researching. This article would benefit greatly from such information. Rick Norwood 13:33, 4 December 2006 (UTC)

Mathematics started using the Order of Operations immediately when multiplication was understood as a concept. It is not a convention that is true because it is agreed upon. It is a theorem (although not usually stated as such) that falls out from the definition of multiplication as repeated addition and the axioms of arithmetic (in particular, the distributive law and the multiplicative identity). Ditto for exponents because they are defined as repeated multiplication.Guildwyn (talk) 14:46, 5 July 2008 (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)

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 [2]. 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 gald to see that these important facts are now in the article. Rick Norwood 14:02, 3 July 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)

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)

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)