# Math problem **URL:** \ **Category:** Factual Questions\ **Created:** [April 28, 2011, 9:54pm UTC](https://boards.straightdope.com/t/math-problem/580084 "2011-04-28T21:54:00Z")\ **Posts on this page:** 20\ **Page:** 4
**Author:** ![Pasta](https://avatars.discourse-cdn.com/v4/letter/p/ecccb3/32.png) [@Pasta](https://boards.straightdope.com/u/Pasta)\ **Post date:** [April 29, 2011, 4:47pm UTC](https://boards.straightdope.com/t/math-problem/580084/61 "2011-04-29T16:47:50Z")
> [@CaveMike](#): > > Do you or anyone else have a cite that there is an accepted convention that multiplication should be evaluated before division? I see a lot of anecdotal evidence, but I don’t see a cite (maybe I missed it). > > Wikipedia supports the notion that the multiplication and division are to be evaluated left-to-right: [http://en.wikipedia.org/wiki/Order\_of\_operations](http://en.wikipedia.org/wiki/Order_of_operations) Please see your very own link. The fourth sentence of the introduction, when juxtaposition is mentioned (in passing), gives a citation that describes some history and includes many of the arguments mentioned in this thread. A more general response, though, is that that Wiki page does not address all operations that exist, and it doesn’t treat juxtaposition in particular. It’s true that juxtaposition implies “multiplication” (usually), but multiplication isn’t the thing that gets a precedence – the symbol for multiplication does. And, juxtaposition itself (as a notation) carries different, and sometimes ambiguous, precedence rules, which is why you don’t find 1/xy written in formal settings. Again: that page (and everyone’s PEDMAS 6th-grade math class) is assuming a very narrow space of possible operations. Juxtaposition, factorials, vector products, derivatives, integrals, differentials (“dx”), … all fall outside the scope. Juxtaposition in particular is ambiguous in certain circumstances because of how math is actually written (with pen and paper) and the limitations of transferring that written math into typed text. My next post will hopefully illuminate that, but in the meantime, please check out the reference in your link. ---
**Author:** ![Hypnagogic\_Jerk](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/hypnagogic_jerk/32/4252_2.png) [@Hypnagogic\_Jerk](https://boards.straightdope.com/u/Hypnagogic_Jerk)\ **Post date:** [April 29, 2011, 5:23pm UTC](https://boards.straightdope.com/t/math-problem/580084/62 "2011-04-29T17:23:47Z")
> [@Indistinguishable](#): > > (In a way, what’s going on in this thread is a clash of descriptivism and prescriptivism… some of us are advocating recognition of the realities of how mathematicians and mathematical tools read and write expressions, while others are advocating strict adherence to a particular set of passed-down rules regardless of whether these rules are universally adhered to in actual practice) Exactly. It’s really a language question, not a “math problem”, despite the title. By the way, thanks for your posts in this thread, **Indistinguishable**. ( **Great Antibob** and **Pasta** also made good posts.) Upon reading the OP, I did the computation and got 1, but found it rather ambiguous. Then I read some of the posts claiming it was unambiguously 9, and got confused, especially given that I am a mathematician, so one assumes I should be able to do such things. (Although, as I tell my students, there are three kinds of mathematicians: those who know how to count, and those who don’t.) It all made sense when you and **Antibob** pointed out the use of the juxtaposition operator, which I just hadn’t thought about. And it just goes to show that this is actually a linguistics question: when you speak a language fluently, you may be hard-pressed to figure out explicitely what its little conventions and rules are. You can’t see them because they’re just too natural to you. > [@SeaDragonTattoo](#): > > The fact that math seems to be open to interpretation and two different answers could each possibly be right based on how you argue the equation, is making my brain leak out of my ears. I wish I could otherwise participate in the argument, I gotta go find some gray matter… :eek: It’s not the math that’s open to interpretation, it’s the language. We’re in front of a sentence written in an odd way, and we’re trying to understand what it says. If you don’t speak the language fluently, no amount of gray matter is going to help you here, because the point is that _for fluent speakers_ this sentence sounds weird and ambiguous. ---
**Author:** ![Pasta](https://avatars.discourse-cdn.com/v4/letter/p/ecccb3/32.png) [@Pasta](https://boards.straightdope.com/u/Pasta)\ **Post date:** [April 29, 2011, 5:24pm UTC](https://boards.straightdope.com/t/math-problem/580084/63 "2011-04-29T17:24:03Z")
For this post, I’ll use the scratchwork shown in this image: [Scratchwork](http://www.flickr.com/photos/53143438@N04/5669657905/in/photostream) **Part 1** When you write a fraction by hand, the leftmost approach is easy. Sometimes, for space or clarity reasons, you might use the middle version. In either case, if you want to migrate to typed text, you are forced to use the rightmost version. And, as Google Calculator, Wolfram Alpha, the first reference in the Wiki OoO article, and many posters in this thread all suggest, this is common, if potentially ambiguous. Thus, it is prudent to avoid it, but that doesn’t remove the ambiguity. **Part 2** Is the left hand expression equal to 1.3333… or 4.3333…? You might argue that it must be 1.3333…, but in a recipe (and when kids first learn fractions), 4.3333… is the more likely correct interpretation. If a recipe writer intended 4/3, they would likely have (by convention) written “4/3” rather than “4” juxtaposed with “1/3”, which may be read as addition instead of multiplication in that context. **Part 3** No one would mistake 5/4 as equal to 1/45. Why not? Well, there is an unwritten convention that “juxtaposition implies multiplication” doesn’t apply to two numerals. Two _numbers \* are okay (2_e\*), a number and a variable are okay (2x), and two variables are okay (xy), but if two numerals appear next to each other, it means something else entirely. Thus, 1/45 does not equal 5/4. This last example may sound silly, but it (along with the others) should show that you take a lot of grammatical conventions for granted that aren’t written down anywhere. This is a genuine language we are talking about, and languages are at times redundant, at times ambiguous, and nearly always nuanced. ---
**Author:** ![TriPolar](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/tripolar/32/3008_2.png) [@TriPolar](https://boards.straightdope.com/u/TriPolar)\ **Post date:** [April 29, 2011, 7:07pm UTC](https://boards.straightdope.com/t/math-problem/580084/64 "2011-04-29T19:07:03Z")
> [@Indistinguishable](#): > > Last edited by Indistinguishable; Today at 12:14 PM. Reason: Also, I am a fan of TriPolar’s last post, as well as their “There is no controlling legal authority” remark… Thank you. I’m honored. ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [April 29, 2011, 7:14pm UTC](https://boards.straightdope.com/t/math-problem/580084/65 "2011-04-29T19:14:40Z")
Quoth **md2000** : > [@](#): > > The fact that all programming languages also obey the rule of evaluating (add, subtract) from left to right and (multiply. divide) from left to right merely means those languages were constructed to follow what was established convention for mathematics. As has been mentioned several times already in this thread, not all programming environments use the same rules. ---
**Author:** ![etv78](https://avatars.discourse-cdn.com/v4/letter/e/d78d45/32.png) [@etv78](https://boards.straightdope.com/u/etv78)\ **Post date:** [April 29, 2011, 8:05pm UTC](https://boards.straightdope.com/t/math-problem/580084/66 "2011-04-29T20:05:54Z")
showed it to friend, FINALLY saw where 9 as an answer comes from. (Still think 1 is correct, BTW) ---
**Author:** ![needscoffee](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/needscoffee/32/1076_2.png) [@needscoffee](https://boards.straightdope.com/u/needscoffee)\ **Post date:** [April 29, 2011, 8:19pm UTC](https://boards.straightdope.com/t/math-problem/580084/67 "2011-04-29T20:19:26Z")
> [@etv78](#): > > showed it to friend, FINALLY saw where 9 as an answer comes from. (Still think 1 is correct, BTW) Do you understand that multiplication doesn’t come BEFORE division; they are considered to be the same thing. (In other words, 6 ÷ 2 is the same as 6 x the reciprocal of 2, which is ½. 6 ÷ 2 is the same as 6 x ½. They both equal 3.)Therefore, you do the equation by **adding the parentheses terms first, then do the problem from left to right**. 6 ÷ 2(1+2)=? 6÷2 x 3=? Now do 6÷2 before you do x3 because you must do it from left to right. 3 x 3=? 3 x 3=9 Math equations are written so that we do problems from left to right, so that we’re all on the same page. You can’t just decide to do it the other way because it looks nice to you. Remember that multiplication = division, and addition = subtraction. ---
**Author:** ![Snarky\_Kong](https://avatars.discourse-cdn.com/v4/letter/s/a183cd/32.png) [@Snarky\_Kong](https://boards.straightdope.com/u/Snarky_Kong)\ **Post date:** [April 29, 2011, 8:21pm UTC](https://boards.straightdope.com/t/math-problem/580084/68 "2011-04-29T20:21:30Z")
> [@needscoffee](#): > > Do you understand that multiplication doesn’t come BEFORE division; they are considered to be the same thing. (In other words, 6 ÷ 2 is the same as 6 x the reciprocal of 2, which is ½. 6 ÷ 2 is the same as 6 x ½. They both equal 3.)Therefore, you do the equation by **adding the parentheses terms first, then do the problem from left to right**. > 6 ÷ 2(1+2)=? > 6÷2 x 3=? Now do 6÷2 before you do x3 because you must do it from left to right. > 3 x 3=? > 3 x 3=9 > > Math equations are written so that we do problems from left to right, so that we’re all on the same page. You can’t just decide to do it the other way because it looks nice to you. Remember that multiplication = division, and addition = subtraction. Have you read any of the discussion on ambiguity in this thread? ---
**Author:** ![etv78](https://avatars.discourse-cdn.com/v4/letter/e/d78d45/32.png) [@etv78](https://boards.straightdope.com/u/etv78)\ **Post date:** [April 29, 2011, 8:23pm UTC](https://boards.straightdope.com/t/math-problem/580084/69 "2011-04-29T20:23:31Z")
> [@needscoffee](#): > > Do you understand that multiplication doesn’t come BEFORE division; they are considered to be the same thing. (In other words, 6 ÷ 2 is the same as 6 x the reciprocal of 2, which is ½. 6 ÷ 2 is the same as 6 x ½. They both equal 3.)Therefore, you do the equation by **adding the parentheses terms first, then do the problem from left to right**. > 6 ÷ 2(1+2)=? > 6÷2 x 3=? Now do 6÷2 before you do x3 because you must do it from left to right. > 3 x 3=? > 3 x 3=9 > > Math equations are written so that we do problems from left to right, so that we’re all on the same page. You can’t just decide to do it the other way because it looks nice to you. Remember that multiplication = division, and addition = subtraction. I was ALWAYS taught (and THIS is the source of the debate in the thread) PEMDAS, which in case would get rid of the parenthesis. General question to everyone: If PEMDAS is wrong (and many of you apparently think it is) WHY was it taught? Someone up-thread mentioned a newer version of PEMDAS. I thought math was static, why the need to throw away a perfectly valid method? ---
**Author:** ![Jragon](https://avatars.discourse-cdn.com/v4/letter/j/e19b73/32.png) [@Jragon](https://boards.straightdope.com/u/Jragon)\ **Post date:** [April 29, 2011, 8:27pm UTC](https://boards.straightdope.com/t/math-problem/580084/70 "2011-04-29T20:27:18Z")
> [@etv78](#): > > I was ALWAYS taught (and THIS is the source of the debate in the thread) PEMDAS, which in case would get rid of the parenthesis. > > General question to everyone: If PEMDAS is wrong (and many of you apparently think it is) WHY was it taught? Someone up-thread mentioned a newer version of PEMDAS. I thought math was static, why the need to throw away a perfectly valid method? PEMDAS itself isn’t wrong, it’s just taught poorly. It’s not the process that’s bad, it’s the fact that the mnemonic leads to misconceptions about said process really easily. The other thing people in this thread are arguing is that PEMDAS isn’t COMPLETE, because there exist (in a descriptive sense), certain scenarios in syntax like “implicit operators” that popularly take a higher precedence. ---
**Author:** ![etv78](https://avatars.discourse-cdn.com/v4/letter/e/d78d45/32.png) [@etv78](https://boards.straightdope.com/u/etv78)\ **Post date:** [April 29, 2011, 8:31pm UTC](https://boards.straightdope.com/t/math-problem/580084/71 "2011-04-29T20:31:03Z")
> [@brickbacon](#): > > Just an FYI: Many kids today learn [GEMS](http://www.acronymfinder.com/Grouping-Exponents-Multiplication-and-division-Subtraction-and-addition-%28Order-Of-Operations-Pneumonic%29-%28GEMS%29.html) instead of PEMDAS. GEMS is **G** rouping symbols, **E** xponents, **M** ultiplication/division, **S** ubtraction/addition. I believe this change was to address the concerns that came up in this thread. This certainly seems simpler, though I suspect 20 years from now the same discussion will be had on this board, with GEMS students defending how they were taught. ---
**Author:** ![sich\_hinaufwinden](https://avatars.discourse-cdn.com/v4/letter/s/c5a1d2/32.png) [@sich\_hinaufwinden](https://boards.straightdope.com/u/sich_hinaufwinden)\ **Post date:** [April 29, 2011, 8:42pm UTC](https://boards.straightdope.com/t/math-problem/580084/72 "2011-04-29T20:42:19Z")
> [@etv78](#): > > showed it to friend, FINALLY saw where 9 as an answer comes from. (Still think 1 is correct, BTW) If you, or anyone else, still thinks there is a ‘‘correct’’ answer to the equation as written, I believe you have missed the point, IMHO. Only the person writing the original equation knows what they intended with such an ambiguous format. Short of input from that person, the equation is open to some interpretation, regardless of what the mathematical prescriptivists would have you believe. However, if this was a real world problem there would likely be units involved that would indicate how the numbers should be crunched. But then there would be nothing to talk about and this thread would have no reason to exist. ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [April 29, 2011, 8:43pm UTC](https://boards.straightdope.com/t/math-problem/580084/73 "2011-04-29T20:43:16Z")
> [@etv78](#): > > General question to everyone: If PEMDAS is wrong (and many of you apparently think it is) WHY was it taught? Someone up-thread mentioned a newer version of PEMDAS. I thought math was static, why the need to throw away a perfectly valid method? I agree with everything **Jragon** says in their reply above, but also please take to heart, as has repeatedly been noted throughout this thread, PEMDAS isn’t math. PEMDAS is just a notational convention. “Order of operations” isn’t part of the concepts of addition, multiplication, etc. It’s just part of the conventions we choose for how to write these down (and the only reason we have to establish conventions to disambiguate “order of operations” is because we happened to settle on this silly notation where we put binary operators inbetween their inputs; in a different world, where people wrote “+ 3 4” instead, or such things, none of this “order of operations” business ever would have come up). PEMDAS, properly interpreted, hasn’t changed (though it may go by the mnemonic GEMS now, the concept is the same). But it certainly could change! Saying “How can a perfectly valid method like PEMDAS change? Isn’t math static?” is like saying “I used to be taught to write ‘I et IV est V’, and now I’m taught to write ‘1 + 4 = 5’. How could it have changed? Isn’t math static?” ---
**Author:** ![sich\_hinaufwinden](https://avatars.discourse-cdn.com/v4/letter/s/c5a1d2/32.png) [@sich\_hinaufwinden](https://boards.straightdope.com/u/sich_hinaufwinden)\ **Post date:** [April 29, 2011, 8:49pm UTC](https://boards.straightdope.com/t/math-problem/580084/74 "2011-04-29T20:49:02Z")
> [@brickbacon](#): > > Just an FYI: Many kids today learn [GEMS](http://www.acronymfinder.com/Grouping-Exponents-Multiplication-and-division-Subtraction-and-addition-%28Order-Of-Operations-Pneumonic%29-%28GEMS%29.html) instead of PEMDAS. LOL - From that link I see that it means ‘‘Grouping Exponents Multiplication and division Subtraction and addition (Order Of Operations Pneumonic)’’. ---
**Author:** ![Nuveena](https://avatars.discourse-cdn.com/v4/letter/n/a587f6/32.png) [@Nuveena](https://boards.straightdope.com/u/Nuveena)\ **Post date:** [April 29, 2011, 8:57pm UTC](https://boards.straightdope.com/t/math-problem/580084/75 "2011-04-29T20:57:40Z")
> [@etv78](#): > > I was ALWAYS taught (and THIS is the source of the debate in the thread) PEMDAS, which in case would get rid of the parenthesis. > > General question to everyone: If PEMDAS is wrong (and many of you apparently think it is) WHY was it taught? Someone up-thread mentioned a newer version of PEMDAS. I thought math was static, why the need to throw away a perfectly valid method? PEDMAS isn’t wrong. It’s just easily misunderstood. It means (and was always supposed to mean) Parentheses Exponents Multiplication and Division (left to right) Addition and Subtraction (left to right) It doesn’t mean to do all multiplication before division and all addition before subtraction. The source of debate is whether putting two terms smack dab next to each other is a higher order of multiplication than using a \* between them (i.e. does 1/xy mean 1/(x\*y) or (1/x)\*y?). And the only answer to that is to ask the person who wrote the problem in the first place. ---
**Author:** ![Chessic\_Sense](https://avatars.discourse-cdn.com/v4/letter/c/7c8e57/32.png) [@Chessic\_Sense](https://boards.straightdope.com/u/Chessic_Sense)\ **Post date:** [April 29, 2011, 9:26pm UTC](https://boards.straightdope.com/t/math-problem/580084/76 "2011-04-29T21:26:44Z")
> [@needscoffee](#): > > Therefore, you do the equation by **adding the parentheses terms first, then do the problem from left to right**. > 6 ÷ 2(1+2)=? > 6÷2 x 3=? Now do 6÷2 before you do x3 because you must do it from left to right. > 3 x 3=? > 3 x 3=9 > > Math equations are written so that we do problems from left to right You keep insisting on this, and yet you’ve been told several times in this thread that it just ain’t so. You seem to think that 6/2(3) is the same thing as 6/2\*3. It’s been repeatedly stated that it’s not. Implicit multiplication may or may not take precedence over explicit multiplication/division…why do you continue to be obtuse and stomp around with your fingers in your ears, swearing that that’s never the case and acting like your convention was delivered by the hand of God? Look, it’s pretty simple: Does the 3 go in the numerator or the denominator? IT’S AMBIGUOUS - NO ONE KNOWS FOR SURE! I don’t know why this thread is continuing… ---
**Author:** ![Thudlow\_Boink](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/thudlow_boink/32/320_2.png) [@Thudlow\_Boink](https://boards.straightdope.com/u/Thudlow_Boink)\ **Post date:** [April 29, 2011, 9:34pm UTC](https://boards.straightdope.com/t/math-problem/580084/77 "2011-04-29T21:34:18Z")
> [@sich\_hinaufwinden](#): > > If you, or anyone else, still thinks there is a ‘‘correct’’ answer to the equation as written, I believe you have missed the point, IMHO. Only the person writing the original equation knows what they intended with such an ambiguous format. Short of input from that person, the equation is open to some interpretation, regardless of what the mathematical prescriptivists would have you believe. As long as we’re clearing up confusion, can we please not refer to it as an equation? It’s not an **equation** ; it’s an **expression**! From [here](http://www.squarecirclez.com/blog/the-equal-sign-more-trouble-than-its-worth/4986): > [@](#): > > [This](http://www.squarecirclez.com/blog/wp-content/uploads/2010/08/evaluate-the-equation.jpg) is a screen shot from a video that extols the virtues of interactive white boards for math teaching. > > The instructor’s question is “Evaluate the equation”. Can you really “evaluate an equation”? > > You can **solve** an equation for some variable, or you can state that expressions are equal in an equation, but you can’t evaluate an equation. > > We need to be careful how we express mathematics questions since this is where misconceptions can arise. ---
**Author:** ![etv78](https://avatars.discourse-cdn.com/v4/letter/e/d78d45/32.png) [@etv78](https://boards.straightdope.com/u/etv78)\ **Post date:** [April 29, 2011, 9:38pm UTC](https://boards.straightdope.com/t/math-problem/580084/78 "2011-04-29T21:38:35Z")
> [@Nuveena](#): > > PEDMAS isn’t wrong. It’s just easily misunderstood. It means (and was always supposed to mean) > Parentheses > Exponents > Multiplication and Division **(left to right)** > Addition and Subtraction **(left to right)** > > It doesn’t mean to do all multiplication before division and all addition before subtraction. The source of debate is whether putting two terms smack dab next to each other is a higher order of multiplication than using a \* between them (i.e. does 1/xy mean 1/(x\*y) or (1/x)\*y?). And the only answer to that is to ask the person who wrote the problem in the first place. :smack: This shows how long it’s been since I’ve been in grade school! ---
**Author:** ![etv78](https://avatars.discourse-cdn.com/v4/letter/e/d78d45/32.png) [@etv78](https://boards.straightdope.com/u/etv78)\ **Post date:** [April 29, 2011, 9:39pm UTC](https://boards.straightdope.com/t/math-problem/580084/79 "2011-04-29T21:39:47Z")
> [@Indistinguishable](#): > > I agree with everything **Jragon** says in their reply above, but also please take to heart, as has repeatedly been noted throughout this thread, PEMDAS isn’t math. PEMDAS is just a notational convention. “Order of operations” isn’t part of the concepts of addition, multiplication, etc. It’s just part of the conventions we choose for how to write these down (and the only reason we have to establish conventions to disambiguate “order of operations” is because we happened to settle on this silly notation where we put binary operators inbetween their inputs; in a different world, where people wrote “+ 3 4” instead, or such things, none of this “order of operations” business ever would have come up). > > PEMDAS, properly interpreted, hasn’t changed (though it may go by the mnemonic GEMS now, the concept is the same). But it certainly could change! Saying “How can a perfectly valid method like PEMDAS change? Isn’t math static?” is like saying “I used to be taught to write **‘I et IV est V’, and now I’m taught to write ‘1 + 4 = 5’. How could it have changed? Isn’t math static?** ” This a change in SPOKEN/WRITTEN language, NOT MATH!!! :rolleyes: ---
**Author:** ![Irishman](https://avatars.discourse-cdn.com/v4/letter/i/b487fb/32.png) [@Irishman](https://boards.straightdope.com/u/Irishman)\ **Post date:** [April 29, 2011, 9:52pm UTC](https://boards.straightdope.com/t/math-problem/580084/80 "2011-04-29T21:52:32Z")
> [@md2000](#): > > The biggest problem in this whole thread is that on paper or from math textbooks (where most of us learned to do math) equations are written properly. It is only the lack of total text editor functions that forces us to put what should be a multiple line division into a linear format. This is a big part of the problem with what several people are writing. A/BC carries the visual impression it means A BC This is a limitation of text editors, and conventions for simplified representation. Part is not using ÷ , but most is the grouping AB instead of using an explicit multiplication sign. > [@](#): > > There is NO RULE that says implied operations take precedence over explicit operations. It’s not about written rules, it’s about expectations. A lot of this is visual placement. The visual representation carries connotations to people based upon what they learned, and also there are conventions about how to use those visual placement elements that are understood rather than written down. This example violates those unstated conventions, which is why it is ambiguous. It’s all about the non-stated assumptions. How many arguments have you encountered that boil down to unstated assumptions affecting how each party interprets the meaning of statements? This is exactly the same thing. > [@etv78](#): > > General question to everyone: If PEMDAS is wrong (and many of you apparently think it is) WHY was it taught? Someone up-thread mentioned a newer version of PEMDAS. I thought math was static, why the need to throw away a perfectly valid method? It’s wrong only in the sense that it is misunderstood, often by the teachers. It should be PE(MD)(AS). But we don’t put parenthesis in acronyms. If the point of the mnemonic is to help the user remember something specific, then the mnemonic needs to be clear in what it represents. PEMDAS apparently is taken to mean something it isn’t intended - that M preceeds D and that A preceeds S. And if math is static, why did Newton and Leibnitz have to invent calculus? [Previous page](https://boards.straightdope.com/t/math-problem/580084.md?page=3) [Next page](https://boards.straightdope.com/t/math-problem/580084.md?page=5)