tag:blogger.com,1999:blog-7718462793516968883.post5868698143216044938..comments2020-09-20T00:47:09.385-04:00Comments on MadMath: Are Parentheses Multiplication?Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.comBlogger57125tag:blogger.com,1999:blog-7718462793516968883.post-86350309240538591782020-09-08T09:54:03.706-04:002020-09-08T09:54:03.706-04:00This comment has been removed by a blog administrator.Anonymoushttps://www.blogger.com/profile/00294680588502705618noreply@blogger.comtag:blogger.com,1999:blog-7718462793516968883.post-13603318463251132752018-12-22T22:37:56.888-05:002018-12-22T22:37:56.888-05:00Delta is right. In the facebook debate, the FX570M...Delta is right. In the facebook debate, the FX570MS calculator says 1 because it doesn't treat the slash as a division but as a fraction, like this: 6/(2(2+1))<br />That calculator is wrong.<br /><br />Juxtaposition is NOT part of a parenthesis.<br />6/2(2+1) or 6÷2x(2+1) = 6÷2x3 = 3x3 = 9<br /><br />Anonymoushttps://www.blogger.com/profile/04972887375254555533noreply@blogger.comtag:blogger.com,1999:blog-7718462793516968883.post-42785595237750952322018-12-08T21:38:18.079-05:002018-12-08T21:38:18.079-05:00Parentheses are a grouping symbol. So the expressi...Parentheses are a grouping symbol. So the expression two posts up is equivalent to (12)/(2(5-2)) = 2. Deltahttps://www.blogger.com/profile/00705402326320853684noreply@blogger.comtag:blogger.com,1999:blog-7718462793516968883.post-9707976202402570512018-12-07T10:45:10.397-05:002018-12-07T10:45:10.397-05:00Your explanations helped clarify a rationale when ...Your explanations helped clarify a rationale when deriving the trig identity: cos2θ=1-2sin^2θ<br /><br />When moving from <br /><br />cos2θ=(1 - sin^2θ) -sin^2θ<br /><br />I mistakingly attempted to distribute because of the presence of parentheses <br />MMhttps://www.blogger.com/profile/03858205409998243156noreply@blogger.comtag:blogger.com,1999:blog-7718462793516968883.post-85449993420714111012018-12-07T09:56:56.110-05:002018-12-07T09:56:56.110-05:00Delta, Thanks for your clarity of thought on the a...Delta, Thanks for your clarity of thought on the aforementioned. Please let me know how you would interpret the previous problem when written as:<br /><br /> 12<br />________<br /><br /> 2(5-2)<br /><br /><br />MMhttps://www.blogger.com/profile/03858205409998243156noreply@blogger.comtag:blogger.com,1999:blog-7718462793516968883.post-49425803952549181242018-06-11T21:57:59.511-04:002018-06-11T21:57:59.511-04:0018. E.g., see Google.18. E.g., see <a href="https://www.google.com/search?source=hp&ei=nCYfW4qBEoab5gKltp_oAw&q=12+%2F+2(5-2)+&oq=12+%2F+2(5-2)" rel="nofollow">Google</a>. Deltahttps://www.blogger.com/profile/00705402326320853684noreply@blogger.comtag:blogger.com,1999:blog-7718462793516968883.post-9191224113760375572018-06-04T04:17:29.626-04:002018-06-04T04:17:29.626-04:00So what would this be:
12 ➗ 2(5-2) = ?So what would this be:<br /><br />12 ➗ 2(5-2) = ?Anonymoushttps://www.blogger.com/profile/14622986191231739992noreply@blogger.comtag:blogger.com,1999:blog-7718462793516968883.post-45803354373692550512018-02-26T02:43:53.723-05:002018-02-26T02:43:53.723-05:00After reading all these comments I really enjoyed ...After reading all these comments I really enjoyed getting an idea how others approached this problem. I also approached and obtained the answer of 9 but also can see why it can be 1. I've always been taught math is absolute with 1 right answer. I can also say with the way it is written on paper it should be read like this "six divided by two times the sum of two plus one." In this wording the answer is 9. If the answer is to be 1 it would read "six divided by the quantity of two times the sum of two plus one." Since the original is leaving out a set of parentheses, it should be read like the first. Also since it leaves out a set of parentheses and is wrote the way it is on paper you can write it like this<br />6<br />/ (2+1)<br />2<br />This is easier to see each part in a more mathematical depiction. I'm just a college kid interested with math and figured I'd put a little of my own input. Let me know what ya think deltaAnonymoushttps://www.blogger.com/profile/14619001578738920805noreply@blogger.comtag:blogger.com,1999:blog-7718462793516968883.post-54094200309078602212018-01-10T11:49:00.907-05:002018-01-10T11:49:00.907-05:00This explanation is, unfortunately, nonsense. The ...This explanation is, unfortunately, nonsense. The exponent is not part of a "parenthetical term" (nor is that a real phrase). There is no "number inside... 1". <br /><br />Parentheses only direct operations INSIDE the parentheses to be done first. In this case:<br /><br />- There are no operations inside parentheses.<br />- Then, the exponent is computed.<br />- Then, the multiplying occurs.<br />- Then, the addition happens.<br /><br />Deltahttps://www.blogger.com/profile/00705402326320853684noreply@blogger.comtag:blogger.com,1999:blog-7718462793516968883.post-56812084934816346882018-01-10T11:42:15.834-05:002018-01-10T11:42:15.834-05:00P.S. 6/2(2+1) is not an equation; it is an express...P.S. 6/2(2+1) is not an equation; it is an expression. One cannot solve it; but one can simplify it.Deltahttps://www.blogger.com/profile/00705402326320853684noreply@blogger.comtag:blogger.com,1999:blog-7718462793516968883.post-29129767433077302642018-01-10T11:41:21.269-05:002018-01-10T11:41:21.269-05:00Parentheses direct that operations INSIDE the pare...Parentheses direct that operations INSIDE the parentheses be done first. They are not multiplication.<br /><br />While juxtaposition is a nice curt shorthand for multiplying, it does trick people into thinking that the juxtaposition is ~"glued tighter", i.e., i.e. has higher precedence than other operations. Whitehead's right that "By the aid of symbolism, we can make transitions in reasoning almost mechanically, by the eye" (<a href="http://www.madmath.com/2011/02/thoughts-and-cavalry.html" rel="nofollow">link</a>) but apparently this is not a perfect notation because it tricks people into perceiving rules that don't exist. Deltahttps://www.blogger.com/profile/00705402326320853684noreply@blogger.comtag:blogger.com,1999:blog-7718462793516968883.post-30092552955047133902018-01-10T11:35:54.999-05:002018-01-10T11:35:54.999-05:00I'm permitting this one comment but deleting s...I'm permitting this one comment but deleting several other (much longer ones) so as to not clutter the page. Here are a few top-level items.<br /><br />(1) Be careful to note that parentheses direct performing any operation INSIDE the parentheses first. <br /><br />(2) Parentheses can be removed after any multiplication step (not before).<br /><br />(3) Juxtaposition is used precisely because it is a short way of representing multiplication.Deltahttps://www.blogger.com/profile/00705402326320853684noreply@blogger.comtag:blogger.com,1999:blog-7718462793516968883.post-37287956735911392532017-12-22T02:44:06.059-05:002017-12-22T02:44:06.059-05:00Delta i'm not sure if you are still reading th...Delta i'm not sure if you are still reading this thread but if you are, I've been reading through all this and would love to know your thoughts on this as a possibility to why so many people get the answer 1 instead of 9.<br /><br />Apparently this is something that has been going on for a while and when I first saw this I was one of the people who got 1 instead of 9. As I've been reading and researching I've been trying to figure out why that is. I've come to the conclusion that i believe there are two factors to it. First is that I was taught that short cuts to writing equations gave the same meaning to different symbols. First learning that the obelus symbol can be written as the / symbol. so that the two are the same and interchangeable. then later learning that the / symbol can be written in place of the fractal bar. Now this is where I'm not sure if I was taught for just inferred then that the obelus symbol would work the same as the fractal bar. Such that everything to the left of the obelus would be separate from everything on the right. Certainly when i saw the equation that is how i wanted to intemperate it, but I cannot remember if i was miss informed as a student or just made that connection in my head.<br /><br />The other half for me comes from the idea of distribution in algebra, which i know isn't relevant to the original equation but bare with me a moment while I explain. The original equation 6/2(2+1) equals 9 according to the OoO (sorry I don't know how to type the obelus symbol on this computer), so we should be able to substitute X in for any number in this equation and solve for it. Meaning for example that 6/2(x+2) = 9 we should be able to solve x as 1. But to do that you need to resolve the 6/2 first, which isn't part of the parenthesis. To me I want to get rid of the parenthesis since its the P in PEDMAS, and to do that I want to distribute the 2 outside the parenthesis first because its the only way that makes sense to me. <br /><br />Once I got to this though, is where it really broke me of the idea that the equation as a whole should equal 1, because if I tried to distribute FIRST I could not balance the equation either way. If I made the equation equal to 9 or 1, either way x didn't come out right. I also think this may be a basis for why people want to make the implied multiplication by juxtaposition take precedence over the obelus symbol when readying left to right. people what to distribute the 2 in order to clear the parenthesis. <br /><br />I still think that the equation is confusing and would prefer to see more parenthesis to give it better definition, but at least now i believe i can understand it better, or at least understand what i was doing wrong myselfAnonymoushttps://www.blogger.com/profile/08886023069005259551noreply@blogger.comtag:blogger.com,1999:blog-7718462793516968883.post-30209029393492955222017-12-17T10:57:48.308-05:002017-12-17T10:57:48.308-05:00You simplified that wrong. If you were to simplify...You simplified that wrong. If you were to simplify d/a(b+c), it would look like this: d/ab+d/ac. As such, you would write this problem as (6/2*1)+(6/2*2)<br />(3*1)+(3*2)<br />3+6<br />9Anonymoushttps://www.blogger.com/profile/08539699116081246801noreply@blogger.comtag:blogger.com,1999:blog-7718462793516968883.post-40534054200844260012017-12-17T10:51:48.796-05:002017-12-17T10:51:48.796-05:00The reason that your simplification is wrong is be...The reason that your simplification is wrong is because of the order in which you simplified the problem.<br />6 / 2 (1 + 2) would be simplified completely different from 2 (1 + 2). It would become (6 / 2 * 1) + (6 / 2 * 2) or (3 * 1) + (3 * 2), or (3)+(6)=9.Anonymoushttps://www.blogger.com/profile/08539699116081246801noreply@blogger.comtag:blogger.com,1999:blog-7718462793516968883.post-52362920730955345702017-12-06T16:19:32.258-05:002017-12-06T16:19:32.258-05:00This blog has urged me to seek clarity regarding t...This blog has urged me to seek clarity regarding the understanding I have been having for so many decades. Daring to ask some elementary question or doubts. Please highlight wherever my understanding is incorrect.<br /><br />Why are equations written in following convention<br />a^2 + 2ab + b^2<br /><br />When they actually must be written as<br />a^2 + (2×a×b) + b^2<br /><br />Why are parenthesis omitted from 1st equation? Because omitting it does not challenge the order of operation?<br /><br />I read all arguments carefully on this page and have this persistent question about parenthesis as now its challenging my understanding.<br /><br />1. The order of operation calls for evaluation of parenthesis on priority basis, Are parenthesis to be evaluated (removed) "by performing the operation to obtain a value" or are they not to be removed? <br /><br />2. In other words is it acceptable "to keep parenthesis as is without performing the operation" as per order of precedence (where an operator is specified implicitly or explicitly in the problem statement) ?<br /><br />3. as per its order of operation, in a step of evaluation, Is it acceptable "to merely replace parenthesis with an operator" and "not actually perform that operation in that step"? (in the middle of evaluation merely Rewriting expression without performing operation).<br /><br />4. If juxtaposition is so irrelevant why is it used in the problem statement? What quality or quantity does it or can it represent in math when used with parenthesis?<br />Fedehttps://www.blogger.com/profile/07481445614615074601noreply@blogger.comtag:blogger.com,1999:blog-7718462793516968883.post-69661306249973183062017-11-06T23:46:35.227-05:002017-11-06T23:46:35.227-05:00My understanding of a strict order of operations i...My understanding of a strict order of operations is that they would be equal to 3n and a^2, respectively. Again, there is no agreed-upon rule that juxtaposition has higher precedence than other operations (and it would be weird if there were). <br /><br />However, we all agree that many people get confused by that, so it is best to avoid writing something like that, and certainly I never write anything that depends on people reading it properly. See also: <a href="https://matheducators.stackexchange.com/questions/13097/writing-fractions-correctly" rel="nofollow">https://matheducators.stackexchange.com/questions/13097/writing-fractions-correctly</a>Deltahttps://www.blogger.com/profile/00705402326320853684noreply@blogger.comtag:blogger.com,1999:blog-7718462793516968883.post-88754931751252518272017-10-25T19:06:00.859-04:002017-10-25T19:06:00.859-04:00I have a couple questions:
Given " 6/2n "...I have a couple questions:<br />Given " 6/2n ". Do read that as "six halves times n" or as "six over two n" ?<br />Also does a/1a = 1 or does it equal a^2 ?KG ProdDevhttps://www.blogger.com/profile/15986209311073863145noreply@blogger.comtag:blogger.com,1999:blog-7718462793516968883.post-10536797240650201402017-10-22T18:43:35.965-04:002017-10-22T18:43:35.965-04:00Thanks for the comment; I fundamentally agree with...Thanks for the comment; I fundamentally agree with that. It's well-put to talk about "juxtaposed multipliers", that is, to be clear that it's the juxtaposition that causes the multiplication -- and not something like (3) + 5. <br /><br />I also completely agree that people defending their position by resorting to the argument that basic arithmetic operations may have more than one interpretation of ordering is wildly insane (in fact, that's one of the few pieces of junk that I've actively deleted on this thread). Deltahttps://www.blogger.com/profile/00705402326320853684noreply@blogger.comtag:blogger.com,1999:blog-7718462793516968883.post-44291835920983022992017-10-21T08:34:43.148-04:002017-10-21T08:34:43.148-04:00I might be late to the party, so I will not be off...I might be late to the party, so I will not be offended if I am ignored or reprimanded, whatever. This is the only place I found that allowed me to join/comment. In the late 50's I was taught that simplifying parentheses involved solving everything inside the parentheses so that the final parentheses are then removed, which necessitates applying any juxtaposed multipliers out of necessity. Obviously the presence or absent of the final parentheses determines the answer to this now famous conundrum. Before going, I have to say that math might arrive at two different answers drives me crazy; this should not be. I enjoyed this thread immensely, thank you for making it available.Anonymoushttps://www.blogger.com/profile/10566968433304181906noreply@blogger.comtag:blogger.com,1999:blog-7718462793516968883.post-12844344764003845082017-09-19T19:09:32.099-04:002017-09-19T19:09:32.099-04:00Thanks for the thoughtful post. I wish I were awar...Thanks for the thoughtful post. I wish I were aware of some resource like OED/Webster's for math notation, but I don't think there is one. (Even for OED/Webster's, you quickly get into the "English is descriptive not prescriptive" argument which results in no one ever being wrong there, either.)<br /><br />I would guess that the ambiguity in precalculus texts isn't intentional, but more likely an accident of the authors themselves being unclear on what their starting point is (or possibly publisher pressure to reduce page count or formality). Even after a dozen years of teaching college algebra, I'm embarrassed about how unclear I am on what the starting assumptions are.<br /><br />The most helpful thing I've found this summer is to go back to my Abstract Algebra text (Hungerford) and work back through that again. My main observation is that almost all algebra/precalculus texts effectively start off presenting/assuming the (five) axioms for a field structure. I feel in a few ways that might not be the most self-evident starting point for those students, but that's what's almost universally done.<br /><br />Secondly I've been reading some material by Hung-Hsi Wu intended for formally training teachers in pre-algebra and algebra which I've found quite helpful.<br /><br />For links see my question on StackExchange Math Educators, <a href="https://matheducators.stackexchange.com/questions/12794/is-there-a-resource-that-formally-develops-the-topics-of-elementary-algebra" rel="nofollow">"Is there a resource that formally develops the topics of elementary algebra?"</a><br /><br />Deltahttps://www.blogger.com/profile/00705402326320853684noreply@blogger.comtag:blogger.com,1999:blog-7718462793516968883.post-21578573500419213992017-09-19T18:56:38.023-04:002017-09-19T18:56:38.023-04:00I disagree with the "invisible 1" argume...I disagree with the "invisible 1" argument. 5 - (2) does not inherently mean any multiplication, it is simply showing 5 minus the quantity (2).<br /><br />To argue that there are "invisible 1"'s around, one may as well also argue that there is a multiplication on the 5, to wit, 1×5. Or why not a whole bunch of them? 1×1×1×1×5. But that's not actually inherent or necessary in the given, written expression.<br /><br />But that isn't what the given, written expression actually means. Deltahttps://www.blogger.com/profile/00705402326320853684noreply@blogger.comtag:blogger.com,1999:blog-7718462793516968883.post-77934086435448519332017-09-19T01:07:19.967-04:002017-09-19T01:07:19.967-04:00Delta, I think you have the most consistent argume...Delta, I think you have the most consistent argument on the thread. That said, I think confusion comes from the notion that multiplication is implied merely through juxtaposition. The juxtaposition without parenthesis wouldn't work so they become intrinsic to the implication. That is not to say implication takes precedence. I'm coming to grips with relearning this myself. The problem with a lot of the school prealgebra textbooks is that there is a high amount of intentional ambiguity in presentation with the aim being to hallenge the students. It comes at the cost of misrepresenting how one would clearly and concisely write equations in an economical manner. As assignments near their final problems they come with a fair but present element of trickery the result being highly contrived equations.<br /><br />Anyway, Delta, you've convinced me that the answer to the problem 6 / 2(1+2) is 9 with what seems to be a grounded and unbiased reasoning, whereas those clinging to an answer of 1 (I'm still curious) are doing it with an unhealthy dose of habit. Are there any resources for math, akin to the English language's OED or Webster publications, that can help put this topic to bed once and for all for those having trouble accepting your authority on the matter? Sir Bacchanalia Trendy Liverhttps://www.blogger.com/profile/03469209530306897687noreply@blogger.comtag:blogger.com,1999:blog-7718462793516968883.post-89319093691505793562017-09-18T02:07:44.934-04:002017-09-18T02:07:44.934-04:00In your problem: "Evaluate 2+3(5)^2" The...In your problem: "Evaluate 2+3(5)^2" The exponent is part of the parenthetical term and must be done before the multiplication. By multiplying the exponent of the number inside(which is 1) the parenthesis with the exponent outside(which is 2) you get the new exponent of 2. Simplify: 2+3(5^2) = 2+3(25) = 2+75 = 77Bob Brusewitzhttps://www.blogger.com/profile/09487351863450714400noreply@blogger.comtag:blogger.com,1999:blog-7718462793516968883.post-31811011129921345672017-09-18T01:26:56.550-04:002017-09-18T01:26:56.550-04:00Actually there is a multiplication to get rid of t...Actually there is a multiplication to get rid of the parenthesis and it is by 1. So in your first equation you could write it as 5-1(2) but there is no need as that is implied. You could think of it as "five minus one group of 2". That is why the equation on facebook of 6÷2(2+1) resolves to 1. Again in written form this would be"Six divided by two groups of two plus one." If you wanted the answer of 9 then the equation should be 6/2*1(2+1) or just 6/2*(2+1). Bob Brusewitzhttps://www.blogger.com/profile/09487351863450714400noreply@blogger.com