The Museum of HP Calculators

HP Forum Archive 21

 PEMDASMessage #1 Posted by Les Koller on 24 July 2013, 8:09 p.m. This has probably been discussed here before, but I'd like some input. Without using ANY technology, just your brain and education, what is the value of 6÷2(1+2) What is your justification? Does your HP Calculator give the same answer? How about your TI? Thanks for humoring me here...I've never seen this one before.

 Re: PEMDASMessage #2 Posted by Kimberly Thompson on 24 July 2013, 8:23 p.m.,in response to message #1 by Les Koller Les Social media aside, the answer is in the title PEMDAS. Using the convention from my Engineering studies, I get ONE. Other conventions will possibly yield other results. SlideRule

 Re: PEMDASMessage #3 Posted by Les Koller on 24 July 2013, 9:42 p.m.,in response to message #2 by Kimberly Thompson 1 is exactly what I get too. My HP agrees. My TI disagrees and gives 9. If PEMDAS is followed EXPLICITLY would it not be 6 / 2*(1+2) = 6 / 2 * (3) = 9? Yes it would. So being educated as a mathematician and NOT an engineer, why did I arrive at 1? The whole point of the thing is, I thought this was all settled; I did not realize that physicists / engineers did it different, that implied multiplication and / vs ÷ were treated differently. Learn something new every day...hope it never ends!

 Re: PEMDASMessage #4 Posted by Thomas Klemm on 24 July 2013, 10:06 p.m.,in response to message #3 by Les Koller Quote: My HP agrees. My HP-48GX disagrees and gives 9 as well. Which calculator did you use? Cheers Thomas

 Re: PEMDASMessage #5 Posted by Les Koller on 26 July 2013, 10:38 p.m.,in response to message #4 by Thomas Klemm I used the HP 50g in equation writer and eval mode. Entering it in strictly post fix mode gave me 9 too.

 Re: PEMDASMessage #6 Posted by Gerson W. Barbosa on 26 July 2013, 11:57 p.m.,in response to message #5 by Les Koller Quite different things:

 Re: PEMDASMessage #7 Posted by Kimberly Thompson on 24 July 2013, 10:32 p.m.,in response to message #3 by Les Koller Les I learned my MATH from mathematicians, while attaining an Engineer Degree. I'm very happy with their tutoring. SlideRule Edited: 25 July 2013, 7:45 a.m.

 Re: PEMDASMessage #8 Posted by William L. Drylie on 28 July 2013, 3:41 a.m.,in response to message #3 by Les Koller Hi Les; Physicists and Mathematicians do not do it differently. The answer is = 1. Any competent Algebra text with a review of the basic axioms of Arithmetic will tell you to treat the numerator and denominator as if they were enclosed in parentheses. 6 is one expression and 2(1+2) is another expression. The 2(1+2) gets done first as the 1+2 is enclosed and the parentheses are removed by the distributive property. What is left then is 6/6=1. I showed this to a Physics Professor at Duke University who I am friends with and he said "surely you're joking"? I said "no" and he said "if you are going to make me tell you the answer to that, you will buy me a beer after work, and I may consider not having your degree revoked"!! (laughing) he said "1...you idiot" How shameful!! I actually treated him to a Bratwurst and an 8 dollar German beer at the Bavarian Brat House in Chapel Hill NC. after work and told him the story. The TI's, even with the N-Spire CX CAS, expect you to have a certain level of expertise with entering Algebraic expressions. If you enter (6)/(2*(1+2)) you will get the correct answer of 1. You can see it is also correct as two expressions and it satisfies TI's syntax for Algebraic expressions. My old 48SX shows the answer as 1, written as you stated the original. The hp does the work. Sincerely, Bill Drylie

 Re: PEMDASMessage #9 Posted by jep2276 on 28 July 2013, 11:23 a.m.,in response to message #8 by William L. Drylie Bill, Why do people think that 2(x+2) is actually 2*(x+2) and not understanding that 2 is the coefficient of (x+2) and therefore a factor that cannot be separated? That is exactly why the answer is 1. Anyway, I was told, in this discussion, that Quote:Well, I would say it's quite useless to discuss with you, because you seem to have no clue about mathematics. Using a(x-h)^2 for a*(x-h)^2 is your next nonsense, because a(...) is usually interpreted as "function of a" (as you did it on the LHS with f(x)). when I referred to the Standard form of the Quadratic as f(x) = a(x-h)^2 + k. Anyway, Programming convention does not dictate, nor override, proper Algebra and that seems to be the hang-up. John Edited: 28 July 2013, 11:54 a.m.

 Re: PEMDASMessage #10 Posted by Maximilian Hohmann on 28 July 2013, 1:12 p.m.,in response to message #9 by jep2276 Hello! Quote:Why do people think ... Because that's the way we (and obviously quite a lot of us all over the globe) were taught mathematics. Including "proper" algebra as well as programming computers. Nevertheless, this discussion has really been eye-opening for me as it showed me once again that even things I would have considered to be "universal" and "fundamental" can have two sides. Therefore the conclusion must be (which is taught to me in my current profession (flying the things that I helped designing in my previous life) in annual refresher courses): Never assume anything, always clarify/verify if the slightest doubt exists. Regards Max

 Re: PEMDASMessage #11 Posted by Thomas Klemm on 24 July 2013, 9:50 p.m.,in response to message #1 by Les Koller What makes 6/2*3 difficult while 5-2+3 isn't? Who thinks the latter should be 0? Cheers Thomas

 Re: PEMDASMessage #12 Posted by Les Koller on 26 July 2013, 10:39 p.m.,in response to message #11 by Thomas Klemm Original problem had 2(1+3), or 2(3) which is implicit multiplication. 2*3 is explicit. Apparently lots of applied mathematicians do the distribution first.

 Re: PEMDASMessage #13 Posted by jep2276 on 28 July 2013, 1:46 p.m.,in response to message #12 by Les Koller And, nobody would present the answer as 6/2(1+2), they would present the answer as a single value—as they are the one working their own independent problem. To calculate a value of 2(1+2) the Distributive Property is used and I think this is where most of the problems arise, as to apply the Distributive Property one uses multiplication so many interrupt 2(1+2) as 2*(1+2) and that would lead to ambiguity—when none existed. Bill said, Quote:… Any competent Algebra text with a review of the basic axioms of Arithmetic… And his statement did not say review PEMDAS or review some computer convention, it stated “… the basic axioms …” should be used. My big hang-up is that instead of teaching the basic axioms, we educate with tricks, e.g. for the inverse of a fraction, Copy-Change-Flip instead of teaching the basic axioms. So, when it comes to fully understanding the Distributive Property, no such trick is available thus many simply change it to a multiplication and apply their acronym of PEMDAS instead of the basic axioms resulting in the dislodging of the 2 from the (1+2). Another way to think of it is that the Distributive Property breaks a basic value into its factors, e.g. 2 and (1+2), so when applied to division, the numerator must be divided by all the factors of the denominator … the basic axioms of Arithmetic. John

 Re: PEMDASMessage #14 Posted by Don Shepherd on 24 July 2013, 10:01 p.m.,in response to message #1 by Les Koller I teach PEMDAS (aka order of operations) to middle-school kids, and the answer is 9. MD and AS in PEMDAS are misleading since they imply that you do all multiplies before divides and all adds before subtracts; but you do all multiplies AND divides left to right, then all adds AND subtracts left to right. I teach it like this: P E MD AS  Edited: 24 July 2013, 10:13 p.m.

 Re: PEMDASMessage #15 Posted by Mike Morrow on 24 July 2013, 11:59 p.m.,in response to message #14 by Don Shepherd Precisely! This topic came up in day one of introductory programming courses back in the days of FORTRAN-IV on the Univac 1108 at Georgia Tech, when I was a student there more than 40 years ago. Same too, on the Georgia Tech Burroughs B6500 in ALGOL-60. The natural and expected result is always 9. So also, in the more modern era, with the HP 17bii in ALG mode, and the HP 20S, and the HP 30b in ALGEBRAIC mode. Why is there any question about this ancient and unambiguous and universally-accepted convention now? Edited: 25 July 2013, 12:26 a.m.

 Re: PEMDASMessage #16 Posted by Pier Aiello on 25 July 2013, 4:15 a.m.,in response to message #15 by Mike Morrow Because i'm not the only guy with a "mechanical" learning (i mean: learn to do the following algorithm, without be conscious of the concept itself). I discover this in my early 20s :(. I'm so poor :( . 6:2(1+2) 6:2*3 3*3 9 Edited: 26 July 2013, 4:31 a.m.

 Re: PEMDASMessage #17 Posted by Don Shepherd on 25 July 2013, 6:59 a.m.,in response to message #15 by Mike Morrow While the 17bii in algebraic mode gets the correct answer to this particular problem, it fails for problems like 3+2x5 because it calculates intermediate results during problem entry, whereas calculators like the TI-83 get the correct results for these problems since they only deliver the final answer after entry is complete. My middle school students don't use calculators at all in my classes.

 Re: PEMDASMessage #18 Posted by Kimberly Thompson on 25 July 2013, 8:02 a.m.,in response to message #15 by Mike Morrow Mike My 17bii beeps at the implied multiply and will not process the equation until I press the * key, ie, I cannot enter the equation as writen on my 17bii. Interesting? SlideRule

 Re: PEMDASMessage #19 Posted by Les Koller on 26 July 2013, 10:36 p.m.,in response to message #14 by Don Shepherd I agree with you guys about the answer being 9 in strict PEMDAS order. However, when I did it in my head I got 1. I later learned that there were different conventions win the multiplication was implied (2(1+2)) vs explicit (2*(2+1)) which changes the order of PEMDAS. That got me to thinking whether the Distributive property should be included and codified or prioritized in PEMDAS. Apparently it is; however, the priority is different between pure and applied mathematics. I never knew that until 2 days ago.

 Re: PEMDASMessage #20 Posted by Maximilian Hohmann on 25 July 2013, 4:55 a.m.,in response to message #1 by Les Koller Good morning! I have no idea what PEMDAS might be (will probably find out by reading the answers already given :-) ) but the way mathematics is taught in german schools and universities leads to a result of 9. Regards Max NB: Both Ti Voyage 200 and HP50g (in algebraic mode) share my opinion Edited: 25 July 2013, 5:07 a.m. after one or more responses were posted

 Re: PEMDASMessage #21 Posted by Cristian Arezzini on 25 July 2013, 5:01 a.m.,in response to message #20 by Maximilian Hohmann That's what I thought too (by the way, I think that PEMDAS means: parentheses, then elevation, then multiplication and division, then addition and subtraction). But apparently, in some contexts, implied multiplication takes precedence over other multiplications/divisions. But I had never heard of this, even in my college years (aerospace engineering, in Italy).

 Re: PEMDASMessage #22 Posted by Les Koller on 26 July 2013, 10:42 p.m.,in response to message #21 by Cristian Arezzini English say Exponentiation instead of Elevation, but it is the exact same thing mathematically speaking.

 Re: PEMDASMessage #23 Posted by Bill (Smithville, NJ) on 25 July 2013, 8:00 a.m.,in response to message #20 by Maximilian Hohmann Max, Quote: I have no idea what PEMDAS might be Following link has a good description of PEMDAS: Bill Edited: 25 July 2013, 8:01 a.m.

 Re: PEMDASMessage #24 Posted by Walter B on 25 July 2013, 9:33 a.m.,in response to message #20 by Maximilian Hohmann Max, PEMDAS is the US-American acronym for "Klammer vor Hoch vor Punkt vor Strich". d:-)

 Re: PEMDASMessage #25 Posted by Chris Smith on 25 July 2013, 9:12 a.m.,in response to message #1 by Les Koller Some results: My brain (which did it automatically): 9 Other half's brain: 9 HP Prime emulator: 9 HP50g quoted alg expression: 9 C# translation: 9 C translation: 9 Python translation: 9 Logical consistency achieved!

 Re: PEMDASMessage #26 Posted by Gerson W. Barbosa on 25 July 2013, 9:52 a.m.,in response to message #25 by Chris Smith Excel gets this right, also.

 Re: PEMDASMessage #27 Posted by Chris Smith on 25 July 2013, 10:42 a.m.,in response to message #26 by Gerson W. Barbosa Never thought of Excel. Don't think I've started it up in years. Working in a software company, all we use it for is this: :)

 Re: PEMDASMessage #28 Posted by Walter B on 25 July 2013, 9:46 a.m.,in response to message #1 by Les Koller The tricky thing is the optical appearance of this equation suggests 2(1+2) being one term (due to the missing multiplication operator and the wide American division sign) - leading to evaluating this first. Once you insert the multiplication sign where it should be it becomes obvious. 6/2*(1+2) = 3*3 = 9. And good ol' RPN "inside out" method helps as well: 1 ENTER 2 + 2 * 6 x<>y / But optical appearance is tempting ... d:-)

 Re: PEMDASMessage #29 Posted by Gerson W. Barbosa on 25 July 2013, 10:24 a.m.,in response to message #28 by Walter B Quote: (due to the missing multiplication operator and the wide American division sign) ÷ has always looked universal to me (that's what I see on this fine German calculator and on many others). Also, its first appearance in a book titled "Teutsche Algebra" (1659) doesn't quite make it an American division symbol :-)

 Re: PEMDASMessage #30 Posted by Walter B on 25 July 2013, 11:15 a.m.,in response to message #29 by Gerson W. Barbosa Depends. What pupils learn in school here is : for division. This is the reason for the general rule "Punkt vor Strich" (i.e. "dots shall precede lines" since : and • shall be evaluated before + and - ). Later on (when it comes to fractions) they learn the horizontal fraction bar. Later on (when it comes to programming languages) they may learn the / for divisions. I've not seen anybody here writing a ÷ as division operator - it's a symbol being more complex than necessary and (looking at it from far) it may be confused with +. Well, but with the advent of the electronic calculating machines and their keyboards all the bad habits like printing × for •, ÷ for :, and . for , flooded good ol' Europe and we learned swimming. d;-/

 Re: PEMDASMessage #31 Posted by Chris Smith on 25 July 2013, 12:30 p.m.,in response to message #30 by Walter B We don't usually see the typical division sign in the UK past the first couple of years of mathematics (6-8 year olds) other than on calculator buttons. I've always considered the dots as placeholders for numbers. Things are usually written x over y (on paper) or x / y (on a computer) after that and that is it. I rather prefer : if written as a binary operator though. It makes more sense. There's a great discussion of the history of the division symbol in "Mathematics: From the Birth of Numbers" by Jan Gullberg: http://www.goodreads.com/book/show/383087.Mathematics

 Re: PEMDASMessage #32 Posted by Kimberly Thompson on 25 July 2013, 7:39 p.m.,in response to message #28 by Walter B Walter I like - are we thinking {6 / z} where z=2x & x= (2+1)? SlideRule

 Re: PEMDASMessage #33 Posted by Walter B on 26 July 2013, 1:15 a.m.,in response to message #32 by Kimberly Thompson Kimberly, Quote: I like - are we thinking {6 / z} where z=2x & x= (2+1)? I don't know whether we think this way but I found me falling into that trap and guessed others might do as well. d;-)

 Re: PEMDASMessage #34 Posted by jep2276 on 26 July 2013, 10:37 a.m.,in response to message #28 by Walter B You do know that your RPN evaluates to the correct answer of 1.

 Re: PEMDASMessage #35 Posted by Walter B on 26 July 2013, 11:30 a.m.,in response to message #34 by jep2276 Rats! Fell twice into the same trap :-( Should read 1 ENTER 2 + 6 ENTER 2 / * of course! d:-/

 Re: PEMDASMessage #36 Posted by jep2276 on 26 July 2013, 11:54 a.m.,in response to message #35 by Walter B I think we all can agree, with RPN instructions there is no ambiguity ... I like your first set!

 Re: PEMDASMessage #37 Posted by Walter B on 26 July 2013, 12:29 p.m.,in response to message #36 by jep2276 Even better - please look at my earlier post below. d:-)

 Re: PEMDASMessage #38 Posted by jep2276 on 25 July 2013, 11:51 p.m.,in response to message #1 by Les Koller 6÷2(1+2)=1 The implicit coefficients and operands are forever bound to the parentheses. This is called the Distributive Property. 6÷6=1 6÷(2+4)=1 6÷2(1+2)=1 An implicit multiplication is not evaluated like an explicit, so 6÷2x(1+2)=9. <---Explicit If the implicit is incorrectly changed to an explicit then 6÷6=1 6÷(6)=1 6÷1(6)=1 6÷1x(6)=36

 Re: PEMDASMessage #39 Posted by Maximilian Hohmann on 26 July 2013, 4:01 a.m.,in response to message #38 by jep2276 Good morning! Quote:The implicit coefficients and operands are forever bound to the parentheses. This is called the Distributive Property. Interesting. This would mean, that the expression 2(1+2) is o be read as (2 x (1 + 2)). First time in 45 years since I was first introduced to mathematics that I hear about that. I just read the articles in Wikipedia (the American version) about "Ditributive Property" and "Multiplication". And the bit on "Implicit multiplication" in the manual of a Ti-84 (was the first result that google found, a Ti-84 is still missing from my collection). No mention anywhere of this "forever bounding". Let's hope some consensus on this is agreed upon _before_ the first manned mission to Mars will be planned by an international team! Regards Max NB: I just entered the expression (with the implicit multiplication) into Wolfram Alpha and Google. Both evaluate it to 9.

 Re: PEMDASMessage #40 Posted by Pier Aiello on 26 July 2013, 4:41 a.m.,in response to message #38 by jep2276 Umm, i won't sign that convention. Too much ambiguous. You need an extra check to see if there is an explicit multiplication mark next to parentheses. for example n/(5+3)2(18+4) = n / [(5+3)*2*(18+4)] The first is ugly and not clear, i don't like it. Clarification: i mean, each convention, even the craziest, can be followed. But if i don't like it, i won't follow it.

 Re: PEMDASMessage #41 Posted by fhub on 26 July 2013, 4:48 a.m.,in response to message #40 by Pier Aiello Quote: Umm, i won't sign that convention. Too much ambiguous. Well, that's just a 'convention' by a few physicists who are too lazy to write the necessary parentheses, so nothing that we (mathematicians) should worry about. ;-) Franz

 Re: PEMDASMessage #42 Posted by Maximilian Hohmann on 26 July 2013, 4:48 a.m.,in response to message #40 by Pier Aiello Quote:Clarification: i mean, each convention, even the craziest, can be followed. But if i don't like it, i won't follow it. No! If a convention exists, you _must_ follow it, whether you like it or not. Unless you alone build your own spaceship in your back garden.

 Re: PEMDASMessage #43 Posted by Pier Aiello on 26 July 2013, 1:43 p.m.,in response to message #42 by Maximilian Hohmann Argh what i read! No, really, but this is another story (really OT). I don't know if OT discussion are welcome here, so if you want to talk about it suggest another discussion place.

 Re: PEMDASMessage #44 Posted by Kimberly Thompson on 26 July 2013, 2:05 p.m.,in response to message #43 by Pier Aiello How about a NEW post. You got my attention.

 Re: PEMDASMessage #45 Posted by Pier Aiello on 26 July 2013, 2:37 p.m.,in response to message #44 by Kimberly Thompson I use HPmuseums' forum as an high quality technical forum (as other forums for other things) so i don't want to go OT (and this discussion is already long). So choose: a new topic or a different place to discuss (Reddit? Quora? Facebook? twitte...no twitter no; Emails? Comp.hp48 newsgroup? ) Because following a convention (like implicit or explicit multiplication sign) is generalized problem (for all conventions). For example, no one assure you that i write (bad) English sentences that are related to the subject "when you must follow conventions" with the right semantics. These statement above can refer to pasta and pizza without problems.

 Re: PEMDASMessage #46 Posted by Pete Wilson on 31 July 2013, 7:41 p.m.,in response to message #42 by Maximilian Hohmann Salvage One ftw!

 Re: PEMDASMessage #47 Posted by fhub on 26 July 2013, 5:08 a.m.,in response to message #38 by jep2276 Quote: 6÷6=1 6÷(2+4)=1 6÷2(1+2)=1 6÷6=1 6÷(6)=1 6÷1(6)=1 6÷1x(6)=36 Sorry, but your 'transformations' are pure nonsense! With such crazy 'rules' you could in fact 'prove' everything. That's not mathematics but simply 'Voodoo'. ;-) Franz Edited: 26 July 2013, 5:10 a.m.

 Re: PEMDASMessage #48 Posted by jep2276 on 26 July 2013, 8:40 a.m.,in response to message #47 by fhub Sorry, but your 'transformations' are pure nonsense! With such crazy 'rules' you could in fact 'prove' everything. That's not mathematics but simply 'Voodoo'. ;-) Franz THANK YOU. It is not my 'Voodoo'. It is the 'Voodoo' of the ones that keep separating factors! And the ones that have 45 years of math that do not understand that an implied multiplication is exactly that--a factor. Ok, one more time 6÷(2x-2), if I factor (2x-2) I get 2(x-1). Now if we take 45 years experience and PEMDAS incorrecly applied and not understand implicit vs. explicit, I get 2*(x-1) and put that back into the original equation, then I get the wrong answer of 6÷2*(x-1), or 3*(x-1). Now lets test it when x=4. Original equation 6÷(2x-2) = 6÷(2*4-2) = 6÷(8-2) = 6÷6 = 1 Factor wrong 3*(x-1) = 3*(4-1) = 3*3 = 9. BTY. Do not let the PEMDAS brains fry on this one, Sin (pi/2) ........Which letter do I use, is it the P, the E .... help. Edited: 26 July 2013, 8:57 a.m.

 Re: PEMDASMessage #49 Posted by Maximilian Hohmann on 26 July 2013, 8:59 a.m.,in response to message #48 by jep2276 Quote:... if I factor (2x-2) I get 2(x-1) ... Sorry, no, you get (2(x-1)) instead. You need to preserve the original parentheses that were there for a reason. Factoring introduces a _new_ set of parentheses. "2x-2" always becomes 2(x-1) whatever the context is, or 2*(x-2) or whatever multiplication sign you like best.

 Re: PEMDASMessage #50 Posted by jep2276 on 26 July 2013, 9:25 a.m.,in response to message #49 by Maximilian Hohmann Max, You are wrong. The problem is you keep trying to make 6÷6 = 6÷3*2 = 4. The Standard form of the Quadratic is not f(x) = (a*(x-h)^2 + k) It is: f(x) = a(x-h)^2 + k Edited: 26 July 2013, 9:31 a.m.

 Re: PEMDASMessage #51 Posted by fhub on 26 July 2013, 9:34 a.m.,in response to message #50 by jep2276 Quote: The Standard form of the Quadratic is not f(x) = (a*(x-h)^2 + k It is: f(x) = a(x-h)^2 + k Well, I would say it's quite useless to discuss with you, because you seem to have no clue about mathematics. Using a(x-h)^2 for a*(x-h)^2 is your next nonsense, because a(...) is usually interpreted as "function of a" (as you did it on the LHS with f(x)). So I'll stop here, it's a waste of time ... BTW, with your 'Voodoo-math' I can even prove that 8=18 ;-) 12/2*3 = 12/3*2 (because you'll certainly confirm that * is commutative). Now LHS=12/2*3=6*3=18 and RHS=12/3*2=4*2=8, and thus 18=8 q.e.d. :-) Franz Edited: 26 July 2013, 9:36 a.m.

 Re: PEMDASMessage #52 Posted by jep2276 on 26 July 2013, 9:40 a.m.,in response to message #51 by fhub You said "Sorry, no, you get (2(x-1)) instead. You need to preserve the original parentheses that were there for a reason. Factoring introduces a _new_ set of parentheses. "2x-2" always becomes 2(x-1) whatever the context is, or 2*(x-2) or whatever multiplication sign you like best. " So it is you that implied the Quadratic is f(x) = (a*(x-h)^2 + k ), not me. Good you can correct yourself.

 Re: PEMDASMessage #53 Posted by fhub on 26 July 2013, 9:45 a.m.,in response to message #52 by jep2276 Quote: Good you can correct yourself. And bad that you can't even reply correctly! It was not my answer that you quoted ... Franz

 Re: PEMDASMessage #54 Posted by jep2276 on 26 July 2013, 10:43 a.m.,in response to message #53 by fhub Sorry, you are correct that I need to learn how to use the response button correctly. I will try harder. Thank you. But, my math is correct.

 Re: PEMDASMessage #55 Posted by fhub on 26 July 2013, 9:21 a.m.,in response to message #48 by jep2276 Quote: ... and not understand implicit vs. explicit ... In mathematics there's only ONE multiplication, no matter if you write it * or . or x or no character at all. Mathematically there's no difference between 'explicit' and 'implicit' multiplication, these are just 2 names for whether you write or omit a multiplication symbol. Franz

 Re: PEMDASMessage #56 Posted by jep2276 on 26 July 2013, 9:30 a.m.,in response to message #55 by fhub Yes there is The implicit cannot be changed to a explicit so that the two factors get separated and not divided properly. 6÷2x is not 6÷2*x--> or 3*x

 Re: PEMDASMessage #57 Posted by Kimberly Thompson on 26 July 2013, 9:36 a.m.,in response to message #56 by jep2276 JEP Thank you. In some advanced MATH SOFTWARE, implicit multiply is directly expreseed and maintained by a SPACE. Just google implicit math for such a reference. SlideRule

 Re: PEMDASMessage #58 Posted by fhub on 26 July 2013, 9:42 a.m.,in response to message #57 by Kimberly Thompson Quote: JEP Thank you. In some advanced MATH SOFTWARE, implicit multiply is directly expreseed and maintained by a SPACE. Just google implicit math for such a reference. SlideRule Can you please tell me which 'advanced' (?) math software would evaluate 6/2 3 (with the space between 2 and 3) as 1 (instead of the correct value 9)??? Franz

 Re: PEMDASMessage #59 Posted by kimberly thompson on 26 July 2013, 10:42 a.m.,in response to message #58 by fhub Franz I'm making NO claim about the evaluation of any advanced software other than a recognition of implicit multiplication; hence the reference to 'Googling'. This url http://community.wolframalpha.com/viewtopic.php?f=32&t=78280 contains Here's a tip for people new to Maple or to 2-D input: always use a space for implied multiplication. 2-D math input in Maple allows for implicit multiplication, which is writing a multiplication operation without an explicit multiplication operator. and this url https://epsstore.ti.com/OA_HTML/csksxvm.jsp?nSetId=103994 the following The TI-89 family, TI-92 family, and the Voyage 200 recognize implied multiplication... I am in considerable agrreement with this url http://community.wolframalpha.com/viewtopic.php?f=32&t=78280 and this text Re: Evaluation of 6÷2(1+2) by WolframAlphaTeam » Thu Feb 21, 2013 4:04 pm Try avoiding ambiguous notation in queries. Instead of 6÷2(1+2) use (6÷2)*(1+2) or 6÷(2*(1+2)) For either of these, there is no confusion as to what this means. To see how W|A interprets the order of operations, see: http://www.wolframalpha.com/input/?i=order+of+operations In my years of number crunching for the military, NO ONE writes equations in this form '6 / 2(1+2)'. We use that funny little bar ___ in ALL expressions with a numerator/denominator. I wanted to answer the question AS QUERIED and make NO claim beyond that. I will say, I have NEVER had a calculation submitted for final approval/inclusion rejected or altered for algebra errors. I find the intensity of the conversation illuminating & appreciate the sincerity of the respective authors without umbrage! SlideRule

 Re: PEMDASMessage #60 Posted by Maximilian Hohmann on 26 July 2013, 10:56 a.m.,in response to message #59 by kimberly thompson Quote:The TI-89 family, TI-92 family, and the Voyage 200 recognize implied multiplication... Yes, they certainly do. But as I already wrote in my first reply, the Ti Voyage 200 treats that implicit multiplication exactly as any other multiplication and returns 9. Just as Wolfram Alpha does, whatever some users may write in Wolfram Alpha forums. Edited: 26 July 2013, 11:20 a.m.

 Re: PEMDASMessage #61 Posted by jep2276 on 26 July 2013, 11:04 a.m.,in response to message #59 by kimberly thompson Yes you are correct, who would write anything like '6 / 2(1+2)' as a stand alone calculation. I only see it in when evaluating an equation like 6 / 2(x+2). So, when x=1 it becomes '6 / 2(1+2)'. If in that evaluation it is changed to '6 / 2*(1+2)' then the answer will be incorrect. I can reasonably infer that one would only be evaluating '6 / 2(1+2)' when substituting a value into an equation. The the answer is 1.

 Re: PEMDASMessage #62 Posted by Kimberly Thompson on 26 July 2013, 11:19 a.m.,in response to message #61 by jep2276 AMEN! (see post #2) SlideRule

 Re: PEMDASMessage #63 Posted by jep2276 on 26 July 2013, 9:45 a.m.,in response to message #57 by Kimberly Thompson Thanks for the info. BTW, how hard would it be to solve complex problems if all multiplications would need to be recorded with a separate symbol and plus all the extra parenthesis that would follow?

 Re: PEMDASMessage #64 Posted by jep2276 on 26 July 2013, 9:51 a.m.,in response to message #63 by jep2276 I missed your question about the simple math. You are correct that it does not require advanced math. But, if the basics are incorrectly taught at a young age, then it is just one more bad habit to correct later. This post is a simple example of the problems that will occur. Just teach the kids that when you divide by a number, always divide by all the factors. And, the answer is not 9 Edited: 26 July 2013, 9:58 a.m.

 Re: PEMDASMessage #65 Posted by Walter B on 26 July 2013, 10:51 a.m.,in response to message #64 by jep2276 To settle the case, let's have a look to the world's collected sciolism (i.e. Wikipedia). Let's reduce the demands further and look to only some quarter of the world's sciolism (i.e. English Wikipedia). Look here: http://en.wikipedia.org/wiki/Order_of_operations#Exceptions_to_the_standard and find that implied multiplication was apparently a side track followed by a few people. For sake of unambiguity it was abandoned meanwhile. At least that's what I read out there. d:-) Edited: 26 July 2013, 11:19 a.m.

 Re: PEMDASMessage #66 Posted by John B. Smitherman on 26 July 2013, 2:33 p.m.,in response to message #65 by Walter B What? No standard from DIN or ISO on this? ;-) Regards, John

 Re: PEMDASMessage #67 Posted by Les Koller on 27 July 2013, 8:48 p.m.,in response to message #65 by Walter B Walter, I have read somewhere in wiki that some conventions consider the slash or fraction bar symbol ( / ) differently from the old time, rarely-used division symbol ( ÷ ). After trying both in alpha wolfram, with both implied and explicit multiplication (that's 4 different combos) and getting 9 every time, I am still puzzled why I got 1 (being trained mathematically)...but I realize that 9 is correct, and i'll just be more careful from now on.

 Re: PEMDASMessage #68 Posted by Ron Ross on 26 July 2013, 9:46 a.m.,in response to message #56 by jep2276 I see that everyone is against you. But that doesn't make you wrong (including math software and calculators). However I am of the same opinion as you. Implied takes precedence. Very similar to Ti's (and others) missuse of the neg sign in front of raised powers (even hanging out in front). Today's current interpretation is that you need to use parentheses around the value. I have issues with that too. But that isn't as serious as this. I am glad you have made an excellent argument.

 Re: PEMDASMessage #69 Posted by Manolo Sobrino on 26 July 2013, 10:51 a.m.,in response to message #47 by fhub No, it's not Voodoo, yet it is quite interesting. The matter here is the precedence of implied multiplication vs that of explicit multiplication, and the answer depends on conventions. The shared ones: (1) Explicit multiplication and division have the same precedence. (2) When dealing with a series of operators with the same precedence (and in absence of parentheses) you have to evaluate them from left to right. Now, there is a third convention to be considered and there are two camps: (3a) Implied multiplication precedes explicit multiplication and thus division. Then the answer is 1, and the table written by jep2276 is correct. (3b) Implied multiplication and explicit multiplication have the same precedence. Then the answer is 9. Note that there is no right way of choosing a convention. Such expressions are mathematically ambiguous, and if you want to be sure you have to introduce parentheses. If you were writing that down, the order of the calculations would be fixed immediately according to your intention. Some calculators/programs follow (3a): All Casio calculators that I know of. A few TI calculators (they changed that at some point, see https://epsstore.ti.com/OA_HTML/csksxvm.jsp?nSetId=103110). Apparently Mathcad too: http://blogs.ptc.com/2011/11/10/math-basics-order-of-operations-precedence-and-tricky-brainteasers/. I can't find a reference and I don't remember about this but Wikipedia claims that Wolfram Alpha switched to (3b) in 2013. An old guide from the AMS includes this: "We linearize simple formulas, using the rule that multiplication indicated by juxtaposition is carried out before division" http://www.ams.org/authors/guide-reviewers.html. (I had to check the Internet Archive, versions of this page after 2002 have removed this paragraph). Although (3b) seems more reasonable, (3a) has a point. If I had to write 1/(2*Pi), according to (3a) that would be simply 1/2Pi. According to (3b) that would be 1/(2Pi). That's the way lots of Physics books are written BTW... So my first answer was 1. Edited: 26 July 2013, 11:11 a.m.

 Re: PEMDASMessage #70 Posted by jep2276 on 26 July 2013, 11:25 a.m.,in response to message #69 by Manolo Sobrino Like stated before: I can reasonably infer that one would only be evaluating '6 / 2(1+2)' when substituting a value into an equation. So if the equation 6÷(2x+4) was written 6÷2(x+2) and x=1, then (3b) would yield a different answer than the original equation.

 Re: PEMDASMessage #71 Posted by Kimberly Thompson on 26 July 2013, 11:32 a.m.,in response to message #70 by jep2276 AMEN! see post #23 SlideRule

 Re: PEMDASMessage #72 Posted by Don Shepherd on 26 July 2013, 3:32 p.m.,in response to message #70 by jep2276 Quote: I can reasonably infer that one would only be evaluating '6 / 2(1+2)' when substituting a value into an equation. Why the restriction? This mathematical expression is perfectly valid just by itself. It evaluates to 9, following the standard rules of PEMDAS, where x is the same as () is the same as (raised dot).

 Re: PEMDASMessage #73 Posted by William L. Drylie on 28 July 2013, 3:08 a.m.,in response to message #72 by Don Shepherd Hi Don; It is a valid expression, but you are forgetting basic principles of Mathematics. Most importantly here, is that in dividing two expressions the numerator and the denominator are treated as if they were enclosed in parentheses. Begin with the innermost parentheses and work outward. The clue above is expressions. Since 6 (numerator) should be enclosed, 2*(1+2) is a complete expression by the distributive property(denominator). Perform multiplications first then divisions then additions and subtractions working from left to right, unless they are already enclosed as here. So, 6 is the first expression 2(1+2) the other. What I have imparted above is in the Mathematics text 'College Algebra by Michael Sullivan. Prentice Hall 6th edition page 7' (6)/2*(1+2)=1 is the way it should be written but if written 6/2*(1+2)is still = 1 by the basic axioms of arithmetic. Enter in the TI N-Sprire cx cas (6)/(2*(1+2))=1 that follows the rules of Arithmetic and syntax as two expressions. By the way, in the equation writer of my old hp 48 SX I wrote 6/2*(1+2) as the original in Les' post, entered it on the stack and hit the eval key and it returned the answer 1. If you have an old SX you can see for yourself. Sincerely Bill Drylie

 Re: PEMDASMessage #74 Posted by Don Shepherd on 28 July 2013, 9:48 a.m.,in response to message #73 by William L. Drylie Thanks William. I evaluated 6/2(1+2) on a few calculators I have. The TI-83 gave 9 as the result; the TI-Nspire CAS also gave 9, as did the HP-17bii. The Sharp EL-W516 WriteView gave 1, interestingly; when I changed the expression to 6/2x(1+2), it gave 9. I agree with the poster who said that the coefficient 2 is obviously meant to be distributed over the 1+2 rather than divided into 6. I also agree with what he said about factoring. I assumed that all calculators and computing devices implemented order of operations in the same way, and I found that not to be true. This has been an interesting and enlightening topic. I especially agree with the poster who suggested using parentheses to make it perfectly clear how the evaluation should be done.

 Re: PEMDASMessage #75 Posted by Gerson W. Barbosa on 28 July 2013, 10:47 a.m.,in response to message #74 by Don Shepherd Don, I think the problem arises from some people being more used to do most of their calculations with pencil and paper, therefore not being familiar with computer conventions. On a sheet of paper, using two lines per expression, I would do  6 ----------- = 1 2 x (1 + 2)  or  6 --- x (1 + 2) = 9 2  Alternatively, if I needed or wanted to use only one line per expression, I would write  / 6 / 2 x (1 + 2) = 1 (the longer dash to denote the division emcompasses the rest of the left side of the expression) /  or  6/2 x (1 + 2) = 9  However, on the limited 1-dimensional space of a text line an extra pair of parentheses have to be used in order to avoid ambiguity:  6/(2x(1 + 2)) = 1  or 6/2x(1 + 2) = 9  Regardless of the symbol used for division, no matter implicit or explict multiplication, the result of the original expression is only one: 9. Regards, Gerson.

 Re: PEMDASMessage #76 Posted by Don Shepherd on 28 July 2013, 11:07 a.m.,in response to message #75 by Gerson W. Barbosa Quote: the result of the original expression is only one: 9 That's what I thought too, but the Sharp's implementation of PEMDAS is different. See here

 Re: PEMDASMessage #77 Posted by htom trites jr on 28 July 2013, 11:27 a.m.,in response to message #76 by Don Shepherd Part of this confusion is that some seem to think that multiplications should be done before divisions. That is, in a / b * c / d it should be evaluated as (a / (b*c)) / d whereas I think it should evaluated as ((a / b) * c) / d

 Re: PEMDASMessage #78 Posted by Walter B on 28 July 2013, 12:43 p.m.,in response to message #77 by htom trites jr Quote: Part of this confusion is that some seem to think that multiplications should be done before divisions. May easily happen when taught "PEMDAS". No problem with "dots precede lines", however. A textbook example of what an awkward choice of arithmetic symbols may cause for generations - please compare the post above. d:-/

 Re: PEMDASMessage #79 Posted by Pier Aiello on 28 July 2013, 12:59 p.m.,in response to message #77 by htom trites jr Nice example for the "implicit multiplication". a/bc/d , WHOA WHOA WHAT IS THAT? Won't sign :P .

 Re: PEMDASMessage #80 Posted by Kimberly Thompson on 28 July 2013, 12:25 p.m.,in response to message #76 by Don Shepherd Don Simple curiosity; which SHARP Model is referenced? Thanks! SlideRule

 Re: PEMDASMessage #81 Posted by Don Shepherd on 28 July 2013, 1:26 p.m.,in response to message #80 by Kimberly Thompson That's the Sharp EL-W516 WriteView, a nice little inexpensive calculator that let's you store key sequences (like (-b+sqrt(b^2-4ac))/(2a)) )and evaluate them with a single keypress. Edited: 28 July 2013, 1:28 p.m.

 Re: PEMDASMessage #82 Posted by William L. Drylie on 29 July 2013, 5:03 p.m.,in response to message #75 by Gerson W. Barbosa Hi Gerson; In your second convention of removing the 2 from the denominator in this fraction and placing it under the 6 in the numerator, what justification are you using to do this? 6 is a simple numerator 2*(1+2) is an unsimplified denominator. No matter how you might try to cut this problem it is a fraction, nothing more, nothing less. Lets assume for a moment, that your answer of 9 is correct. If it is, we should be able to prove it with a simple axiom of Arithmetic. We will set it up as an equation, multiplying both sides by the denominator to arrive at the numerator thusly. 9=6/2*(1+2) the original problem. | 9*(2*(1+2))=6/2*(1+2)*2*(1+2)/1 the left side equals 54 when derived and the 2*(1+2) on the right side cancels out to leave 6. 54 does not equal 6.If you substitute 1 for nine on both sides of the equation you are left with 6 on both sides of the equation...the numerator of our problem. 1 is the correct answer. The division sign between numbers or expressions always implies parentheses in the numerator and denominator. I quote from the College Algebra text 'College Algebra, Michael Sullivan, 6th edition Prentice Hall' "it is understood that the division bar acts like parentheses; that is , 2+3/4+8=(2+3)/(4+8)" that is, the numerator and denominator are simplified on their own then divided. Sincerely, Bill Drylie

 Re: PEMDASMessage #83 Posted by fhub on 29 July 2013, 5:31 p.m.,in response to message #82 by William L. Drylie Quote: I quote from the College Algebra text 'College Algebra, Michael Sullivan, 6th edition Prentice Hall' "it is understood that the division bar acts like parentheses; that is , 2+3/4+8=(2+3)/(4+8)" Oh my god, now it's even getting worse - is this a forum for mathematical idiots??? Franz Edited: 29 July 2013, 5:31 p.m.

 Re: PEMDASMessage #84 Posted by Don Shepherd on 29 July 2013, 6:09 p.m.,in response to message #82 by William L. Drylie Bill, 2+3/4+8 evaluates to 10 3/4, following the standard rules of order of operations (division before addition). If you want it to equal 5/12, then you would need to use parentheses (2+3)/(4+8) to force the addition first. That's my take, anyhow. Franz, that language is not helpful.

 Re: PEMDASMessage #85 Posted by fhub on 29 July 2013, 6:33 p.m.,in response to message #84 by Don Shepherd Quote: Franz, that language is not helpful. Yes, I know that my reaction was not very friendly, but slowly I can't stand this ignorance here anymore. Although absolutely unexpected in a forum for high-level calculators, the math level of some (or even many) members here seems to be even below elementary school, and I'm afraid that reading such nonsense again and again might eventually be infective. ;-) Franz Edited: 29 July 2013, 6:42 p.m.

 Re: PEMDASMessage #86 Posted by Don Shepherd on 29 July 2013, 7:35 p.m.,in response to message #85 by fhub Franz, the posters in this thread who think that the answer to the OP's expression is 1, rather than 9 (as you and I and most others do), deserve to be heard and respected. There are calculators that evaluate 6/2(1+2) as 1; I mentioned a Sharp calculator in an earlier post. That surprised me, and I think we need to try to understand why there might be alternative explanations. The programmers at Sharp (and maybe more) obviously did it a different way, for whatever reason. I'd like to understand that better than I do now. I still will teach PEMDAS as I always have, and I know that there are some who think it is unnecessary to teach order of operations to school kids, but I believe that anything that makes kids actually THINK is a good thing, so I think teaching PEMDAS has value. I still think the answer is 9, but I also think it's good to keep an open mind about things.

 Re: PEMDASMessage #87 Posted by Pete Wilson on 2 Aug 2013, 2:01 p.m.,in response to message #86 by Don Shepherd I don't think Franz has a problem with thinking implied multiplication has a higher precedence than explicit operations (even if wrong) but the book quote that said 1+2/3+4 should be evaluated as if written (1+2)/(3+4) deserves a lot of astonishment and disbelief I think. And Prentice Hall should be ashamed of themselves.

 Re: PEMDASMessage #88 Posted by Thomas Klemm on 2 Aug 2013, 2:56 p.m.,in response to message #87 by Pete Wilson Quote: And Prentice Hall should be ashamed of themselves. It appears the book was misquoted as Gerson already has pointed out. Cheers Thomas

 Re: PEMDASMessage #89 Posted by Manolo Sobrino on 29 July 2013, 7:51 p.m.,in response to message #85 by fhub The "I just want the angle in degrees" thing in the complex exponentials was much worse. That was illiteracy, these are just different conventions. You should be able to switch among them and recognise the reasons for their use. It's really not that difficult, even the narrow-minded may eventually see the light (see, we all can patronise...) I highly recommend the paper '"Order of operations" and other oddities in school mathematics' from H. Wu quoted below. Mr H. Wu, of course, "gets" maths.

 (deleted post)Message #90 Posted by deleted on 29 July 2013, 8:07 p.m.,in response to message #89 by Manolo Sobrino This Message was deleted. This empty message preserves the threading when a post with followup(s) is deleted. If all followups have been removed, the original poster may delete this post again to make this placeholder disappear.

 Re: PEMDASMessage #91 Posted by Thomas Klemm on 29 July 2013, 9:31 p.m.,in response to message #90 by deleted Quote: would it take you long to figure how the author interrupted 1/2x? Probably not. But I'd advise him to use a notation that is not ambiguous. PS: I do not think that interrupted means what you think it means.

 Re: PEMDASMessage #92 Posted by William L. Drylie on 29 July 2013, 9:13 p.m.,in response to message #85 by fhub Hi Franz; If you think I am a Mathematical idiot, that's fine, you are entitled to think of me that way. I offered a simple proof to Gerson as to why the answer is one. I won't repeat myself, if you bothered to read it you would understand why I think that way, and typing the OP exactly as stated into my 48SX two different ways = 1, I don't think PEMDAS is useful for simplifying a fraction. At any rate, I have not posted to this forum for a long time as retirement has kept me going more than my old job of designing and repairing Steam turbines, and given the present atmosphere of this forum, probably will not post again. I really don't want to put any ones nose out of joint. Sincerely, William L. Drylie

 Re: PEMDASMessage #93 Posted by Ron Ross on 29 July 2013, 10:25 p.m.,in response to message #92 by William L. Drylie I have really enjoyed this exchange as it has taught me a lot about how people rely on PEMDAS as their bible, so to speak. I also believe the answer to be ONE! and am not liking the reasoning nor short tempered responses of other's who refuse to listen to the logic that JEP has provided for his argument. If the original post had 6/2*(1+2), I would be all on board with the answer 9. However, as JEP has stated and provided ample reasoning, implied multiplication in this instance would take precedence over PEMDAS. Of course PEMDAS supporters will also insist that the formula should be 6/(2(2+1)) to get 1 and by not providing that extra set of parenthesis, they rest on holy ground. They obviously NEVER use implied multiplication. Without that extra multiplication sign, THAT 2 is attached to the parenthesis FIRST! It could have also been (6/2)(1+2) and no one would argue that result either. How many of the PEMDAS crowd also believe -4^2= -16. I don't. I believe the -4 is implied to be attached to the number. Modern Ti calculators have changed math. Today's kids will insist that you need to show this as (-4)^2 to correctly compute this. Do any of you follow this in your own habits? How many of you have had to reason with a business calculator user who believes CHAIN operations are correct. He/she pulls out an older Hp 10B and proves him/herself correct. And with this lesson, I will surely avoid this ambiguity in any future formula's or papers that I will write. This has certainly opened my eyes to mathematical interpretation!!!

 Re: PEMDASMessage #94 Posted by Reth on 30 July 2013, 8:30 a.m.,in response to message #93 by Ron Ross Quote: Modern Ti calculators have changed math. Really? Thank God I was born before that great event. Otherwise I would be an idiot who can't figure out simple things as 6/2*(1+2) or 6/2(1+2) which are identical, or struggle to understand why -4 square can't be negative. I really think some guys here are not for real.

 Re: PEMDASMessage #95 Posted by Thomas Klemm on 29 July 2013, 10:40 p.m.,in response to message #92 by William L. Drylie Quote: probably will not post again Usually that's fhub's statement. Now I'm confused.

 Re: PEMDASMessage #96 Posted by Reth on 30 July 2013, 8:18 a.m.,in response to message #85 by fhub Franz, your language is just fine, even too mild to my taste. Nevertheless trolls and idiots will be always. Take it easy. Cheers,

 Re: PEMDASMessage #97 Posted by William L. Drylie on 29 July 2013, 8:22 p.m.,in response to message #84 by Don Shepherd Hi Don; That's my point. 5/12 is the answer. Convention dictates the division bar in a fraction implies parentheses, and I showed you the quote from the text. The proof I provided stands, it is valid. Sincerely Bill

 Re: PEMDASMessage #98 Posted by Gerson W. Barbosa on 29 July 2013, 9:53 p.m.,in response to message #97 by William L. Drylie Quote: Convention dictates the division bar in a fraction implies parentheses, and I showed you the quote from the text That's what your reference really looks like: (if someone has a better image, please share it) By what I can see, a more faithful quotation would be  "When we divide two expression, as in 2 + 3 4 + 8 it is understood that the division bar acts like parentheses; that is, 2 + 3 = (2 + 3)/(4 + 8) 4 + 8 "  In that case the author would be right and you would have made a mistake, I fear. Regards, Gerson. Edited: 29 July 2013, 9:57 p.m.

 Re: PEMDASMessage #99 Posted by Thomas Klemm on 2 Aug 2013, 2:44 p.m.,in response to message #98 by Gerson W. Barbosa From Chapter R It appears there are floating a lot of copies around. I didn't find the document that was cited. However since they all seem to agree I assume that in fact in was misquoted. Or then maybe William L. Drylie was not aware that there is a difference between a division bar and a division sign. Cheers Thomas

 Re: PEMDASMessage #100 Posted by Gerson W. Barbosa on 29 July 2013, 6:35 p.m.,in response to message #82 by William L. Drylie Hello William, Quote: In your second convention of removing the 2 from the denominator in this fraction and placing it under the 6 in the numerator, what justification are you using to do this? I just follow the order of operations universally accepted in Mathematics and Computer Science, a.k.a. PEMDAS: ------------------------------------------- highest precedence 1 | Parenthesis ----------+--------------------------- 2 | Exponents ----------+--------------------------- 3 | Multiplication & Division ----------+--------------------------- 4 | Addition & Subtraction lowest prececence ------------------------------------------- 6/2*(1+2) Here is the original expression, using symbols most compilers will understand; 6/2*(3) The expression between parenthesis is evaluated first 6/2*3 There remains / and *; 3*3 Since / and * have the same precedence, /, the leftmost operand, is evaluate first; 9 The remaining operand, *, is evaluated.  Quote: I quote from the College Algebra text 'College Algebra, Michael Sullivan, 6th edition Prentice Hall' "it is understood that the division bar acts like parentheses; that is , 2+3/4+8=(2+3)/(4+8)" that is, the numerator and denominator are simplified on their own then divided. Sometimes, on paper, I use a longer slash (not dash as I had said) to denote the division bar. But in this case my personal convention would give 9/4 as a result, not the 5/12 your reference states:  / 2 + 3/4 + 8 = 2 + 3/(4 + 8) = 2 + 3/12 = 9/4 /  That's the 2002 edition. I imagined it was much older! Best regards, Gerson

 Re: PEMDASMessage #101 Posted by Les Koller on 1 Aug 2013, 4:02 a.m.,in response to message #100 by Gerson W. Barbosa I NEVER let my algebra students use the / symbol for division. I make them write a division problem as a fraction, and explain why. They're 9th graders, and it's just easier for them.

 Re: PEMDASMessage #102 Posted by Gerson W. Barbosa on 2 Aug 2013, 12:26 a.m.,in response to message #101 by Les Koller Quote: I NEVER let my algebra students use the / symbol for division. I am glad I am not a student anymore and can use my own convention :-) Anyway, as I said elsewhere in this thread, I use the / symbol on paper only occasionally, when the expression has to be written in only one line (mostly due to lack of space). Regards, Gerson.

 Re: PEMDASMessage #103 Posted by Walter B on 2 Aug 2013, 1:11 a.m.,in response to message #102 by Gerson W. Barbosa Quote: I use the / symbol on paper only occasionally, when the expression has to be written in only one line (mostly due to lack of space). The : symbol may be a better (i.e. unambiguous) choice there. d:-)

 Re: PEMDASMessage #104 Posted by Thomas Klemm on 29 July 2013, 8:50 p.m.,in response to message #82 by William L. Drylie Quote: 2+3/4+8=(2+3)/(4+8) Then how do you write this expression in one line? $2+\frac{3}{4}+8$

 Re: PEMDASMessage #105 Posted by William L. Drylie on 29 July 2013, 9:41 p.m.,in response to message #104 by Thomas Klemm Hi Tom; You just did! I don't know, but I think there is confusion with what has been written in the original post. Lets take the fraction 5/12. How else could we write it? Well, we can expand the numbers in the numerator and the denominator into factors that represent the same fraction 2+3/4+8 1+4/3+9 it's all the same 5/12 just a different representation. In Arithmetic and Algebra it's a fraction, period no matter how it is represented. My point was, Algebra texts and the way I learned was the division bar calls for parentheses whether they are written or not so that is why (2+3)/(4+8)=5/12. The original post was 6/2*(1+2) still a fraction! The division bar implies parentheses around the (6) and around the (2*(1+2)) the numerator is just a factor representation of the number 6 2*1+2*2=6 then 6/6=1. I don't think this PEMDAS is useful in simplifying fractions. Read my simple proof in the thread to Gerson. You should be able to arrive at the numerator,(6) by multiplying both sides by the denominator. When the answer is 9, one side = 54 the other 6. With 1 as the answer both sides equal 6. Since some regard me as a Mathematical idiot, Don't bother with a reply or other question as I will no longer frequent this forum. Sincerely, William

 Re: PEMDASMessage #106 Posted by Thomas Klemm on 29 July 2013, 11:29 p.m.,in response to message #105 by William L. Drylie As I count 3 lines where you only count one let me rephrase it: how do you write that expression without the horizontal division bar? Meaning with just the operators +, -, * or /.

 Re: PEMDASMessage #107 Posted by Dave Shaffer (Arizona) on 26 July 2013, 4:42 p.m.,in response to message #69 by Manolo Sobrino Quote:and if you want to be sure you have to introduce parentheses This is what I agree with! Then, at the expense of a bit more ink and space, there will be NO ambiguities.

 Re: PEMDASMessage #108 Posted by Pier Aiello on 26 July 2013, 2:40 p.m.,in response to message #1 by Les Koller Another interesting debate (at least foor me) on the subject: Quora debate

 Re: PEMDASMessage #109 Posted by RogerBlake on 26 July 2013, 6:56 p.m.,in response to message #1 by Les Koller This is a fascinating discussion. After some thought I take the minority view (for this forum) that with 6/2(1+3), 2 is the coefficient of (1+3) making 2(1+3) a complete term to which the PEMDAS rule then applies. Thus getting 1 as the answer. And I think this is the usual convention for all written mathematics. I've seen some interesting posts on the web on this and one post (http://mathforum.org/library/drmath/view/57021.html) indicated that even the AMS support this view, but that was back in 2000 and I couldn't find a recent reference. The syntax of equations consisting of operators and terms, with terms having coefficients does not seem to have been considered here ? 1/2pi would not be regarded as meaning half-pi in written work, although a calculator would act differently. The usual processing by calculators seems to be a historic lack of capability to give a correct result, and over time has itself become a convention. JMHO. The same post refrred to above concluded: "Some people argue about arithmetic-operation precedence by referring to what this or that calculator or programming language does. However, I believe all such references are irrelevant; for what may be syntactically convenient for some computing device need not be convenient (or traditional) for human mathematical writing."

 Re: PEMDASMessage #110 Posted by Thomas Klemm on 27 July 2013, 4:42 a.m.,in response to message #109 by RogerBlake Quote: making 2(1+3) a complete term to which the PEMDAS rule then applies So 2(1+3)2 = 64 instead of 32?

 Re: PEMDASMessage #111 Posted by RogerBlake on 27 July 2013, 5:11 a.m.,in response to message #110 by Thomas Klemm "So 2(1+3)2 = 64 instead of 32? " No ! If you have 2x^2 the coeficient of x^2 is 2, so similarly in 2(1+3)^2 the coefficient of (1+3)^2 is 2, so you get the answer 32 you want. PEMDAS is a simplified rule for certain circumstances - such as programming or using calculators. It is only a useful but partial rule when considered in the context of all mathematical notation. It is unfortunate that it does not consider coefficients of terms ("implied multiplication") thus causing considerable arguments like this and much confusion. However I have learned that there are many conflicting notations is mathematics, and much is not consistant - such as f' is not a derived function in every use of the notation. Even something common such as "imaginary number" (which I take to mean a complex number lying on the imaginary axis) is not used consistantly by everyone. So all-in-all I think it would be extremely unfair to mark anyone stating either as 6/2(2+1)=1 or as =9 as being wrong ! A lesson in possible ambiguities inherent in mathematical notations and invaluable at an early age ! Also the context in which the expression is evaluated needs to be taken into account. If 6/2(1+2) was in a calculator manual for instance I would take the answer to be 9. But in a sequence of calculations in an elementary algebra book would probably take it to be 1. If in any doubt then use extra parenthesis for clarity. But blindly following PEMDAS is not always correct. There is no correct. Perhaps there should be - an ISO would be nice but AFAIK that doesn't exist yet, and even if it did not everyone would follow it. Just like dy/dx should have d in roman and x and y in italic (which is an ISO), many books have this all in one or the other - really meaning (d*y)/(d*x) if all italic if you're pedantic , but no-one actually complains ... not much anyway. Edited: 27 July 2013, 7:07 a.m.

 Re: PEMDASMessage #112 Posted by Don Shepherd on 26 July 2013, 8:02 p.m.,in response to message #1 by Les Koller Les, what scares me about some of the responses to this question, especially by engineers who build bridges I may drive over one day, is that some people do not seem to understand how the order of operations works. The expression, as written, evaluates to 9. If someone wanted it to yield the answer 1 (as some seem to do), then it would have to be rewritten as 6÷(2(1+2)). You have to be careful about how you construct mathematical expressions by knowing how PEMDAS really works, since computing machines also abide by those rules. Consider a 17b solver equation to calculate one root of a quadratic equation using the quadratic formula: x1 = (-B+SQRT(SQR(B)-4xAxC))/(2xA)  If you write it instead as: x1 = (-B+SQRT(SQR(B)-4xAxC))/2xA  (as I have erroneously done on occasion), you will get an incorrect result. I hope that every engineer in the world has this knowledge and practices it. My life depends on it!

 Re: PEMDASMessage #113 Posted by RogerBlake on 26 July 2013, 9:01 p.m.,in response to message #112 by Don Shepherd but you would write your example quadratic expression inline as x1 = (-B+/-SQRT(SQR(B)-4AC))/2A At least I think most people would, and would also know to add the extra brackets round the denominator when using a calculator. Edited: 26 July 2013, 9:31 p.m.

 Re: PEMDASMessage #114 Posted by jep2276 on 26 July 2013, 9:49 p.m.,in response to message #113 by RogerBlake That would require people to undertstand math ... Beginning Algebra teaches that both the original equation and the factored, or simplified, equations must yield the same results. So, y=6÷(2x+4) when x=1 y=6÷(2+4) = 1 factored, y=6÷2(x+2) when x=1, it must still equal 1 or the math is incorrect. y=6÷2(1+2)=1 or most every Algebra book in the world must be changed.

 Re: PEMDASMessage #115 Posted by Thomas Klemm on 27 July 2013, 6:09 a.m.,in response to message #114 by jep2276 Can we agree that there are situations where factoring needs parentheses like in (2x+4)2 = (2(x+2))2? I hope we can also agree that there are situations where we can avoid them: 5+(2x+4) = 5+(2(x+2)) = 5+2(x+2). That's because we agree on PEMDAS. What about your example? You say: sure! Implied multiplication implies parentheses as well. But of course only in case of division. So you have an additional rule and an exception to it. What do we gain? People not familiar with LaTeX can write 1/2x when in fact they mean: $\frac{1}{2x}$. And they can write 1/2 x which of course means $\frac{1}{2}x$. So there's another rule to it: don't use spaces between factors in implied multiplication. But since we're fine with 2 x2 there's an exception to that rule as well. IMHO implied multiplication can be used when both expressions with or without multiplication sign are the same. Otherwise like in the example given originally it should be avoided. It's just not worth the confusion. So I'd write 1/(2x) or 1/2*x and avoid any ambiguity. Cheers Thomas

 Re: PEMDASMessage #116 Posted by Pete Wilson on 2 Aug 2013, 2:15 p.m.,in response to message #114 by jep2276 The problem is your factorization implied the answer you desired. IOTW, y=6÷2(x+2) is incorrect factorization, it should be y=6÷(2(x+2)) in which case there is no ambiguity and the answer still matches.

 Re: PEMDASMessage #117 Posted by Don Shepherd on 26 July 2013, 9:52 p.m.,in response to message #113 by RogerBlake Quote: and would also know to add the extra brackets round the denominator when using a calculator Roger, I think the brackets around the denominator would be required, not just on a calculator solver equation like this, but on any computer system that processes mathematical expressions according to the rules of order of operations, which I believe means every computer system. I think this thread shows that almost all calculators, Excel, and most if not all programming languages evaluate this expression as 9. So if the brain says the result is 1 and every computer and calculator says the result is 9, we've got a problem. I don't know, maybe I'm too sensitive to this because I teach PEMDAS to kids and if a kid told me the answer to this expression is 1, I'd mark it as incorrect. That reminds me, school starts again in 1 month! Lord give me strength to get through another year. Don

 Re: PEMDASMessage #118 Posted by jep2276 on 26 July 2013, 10:05 p.m.,in response to message #117 by Don Shepherd So, when a student would factor (2x+4) to 2(x+2) you would mark that wrong? The 2 is a factor of the original equation and is not evaluated separately. Should we rewrite all the Algebra books so that (2x+4) gets factored to (2*(x+2))? See post #67

 Re: PEMDASMessage #119 Posted by Don Shepherd on 26 July 2013, 10:21 p.m.,in response to message #118 by jep2276 Quote: So, when a student would factor (2x+4) to 2(x+2) you would mark that wrong? Of course not. But if I said evaluate 6÷2(1+2) and the student answered 1, yeah, that's wrong, the answer is 9. Quote: Should we rewrite all the Algebra books so that (2x+4) gets factored to (2*(x+2))? No. Just be aware of how PEMDAS works. It's not going to change just because you don't think it is right.

 Re: PEMDASMessage #120 Posted by jep2276 on 26 July 2013, 10:30 p.m.,in response to message #119 by Don Shepherd No it is not. The answer of the original equation must match the answer of the factored equation. 6=2x, so x=3, or x=(1+2) 6÷2x=1, even if it is written 6÷2(1+2). Maybe we need to rewrite all the advanced math books that use such substitutions in the solving of equations. PEMDAS is an acronym to teach beginning math, IT IS NOT A RULE! Edited: 26 July 2013, 10:39 p.m.

 Re: PEMDASMessage #121 Posted by Walter B on 27 July 2013, 2:41 a.m.,in response to message #120 by jep2276 Regardless what you claim, please see above. I think Wolfram should be sufficient to outbalance even you. d:-)

 Re: PEMDASMessage #122 Posted by htom trites jr on 26 July 2013, 10:37 p.m.,in response to message #112 by Don Shepherd 9. Most computer languages (there are some notable exceptions) do not have implied multiplication. In either case, a op b (foo) should be evaluated as a op b * (foo), that is, ((a op b) * (foo)) and now the confusion starts. Normally, PEMDAS evaluates left to right. Some computer languages and programs do not promise to do this (worse, some promise and do not.) Indeed, one compiler I wrote explicitly did not. The language spec was that multiple (more than two) terms or factors were not evaluated left to right, right to left, or in any particular order (two were evaluated left first, then right, then combined.) If order was important, parentheses should be used, and would be obeyed. If the shuffle-terms options was selected, a+b+c had six possible orders of evaluation. One of a, b, or c was evaluated; one of the others was evaluated, the addition was evaluated, the last was evaluated, and then the last addition. (The second version added some shuffling where more terms or factors might be evaluated before operations were evaluated.) The compiler left notes in the code as to which order of evaluation was used, and comparisons of different results from the same data and different compilations revealed some unexpectedly sensitive floating point calculations. Even then, most of the programmers hated the shuffle-terms option.

 Re: PEMDASMessage #123 Posted by jep2276 on 26 July 2013, 10:57 p.m.,in response to message #122 by htom trites jr I just put 6+2*5 into a calculator and it gave me the answer of 40. Well that settles it, compilers and calculators cannot be wrong. See post #68 Edited: 26 July 2013, 10:59 p.m.

 Re: PEMDASMessage #124 Posted by Pier Aiello on 27 July 2013, 2:52 a.m.,in response to message #123 by jep2276 jep, you are fighting hard to impose your view to all of us. A bit tyrannic. You are using just a different convention, take it easy.

 Re: PEMDASMessage #125 Posted by John I. on 27 July 2013, 6:12 a.m.,in response to message #124 by Pier Aiello General comment on the discussion. At school in Australia in the 60s-70s I learnt it as BODMAS. Brackets, Operations (e.g. powers), Division, Multiplication, Addition, Subtraction. I see you can find this term on the web - e.g. http://www.mathsisfun.com/operation-order-bodmas.html A noted on that link, if you had to choose between the order of a divide and multiply, or an addition and subtraction, you worked left to right - i.e. do the leftmost of the two first. That was the convention. John.

 Re: PEMDASMessage #126 Posted by jep2276 on 28 July 2013, 12:22 p.m.,in response to message #124 by Pier Aiello yes, the correct one.

 Re: PEMDASMessage #127 Posted by Pier Aiello on 28 July 2013, 12:56 p.m.,in response to message #126 by jep2276 There is no doubt that your choices are correct for you :). Simply leave alone the others if they disagree with you ^_^ .

 Re: PEMDASMessage #128 Posted by jep2276 on 28 July 2013, 1:19 p.m.,in response to message #127 by Pier Aiello OK. But, if you read the threads when I tried to explain my view, it was: called Voodoo; implied I was stupid because someone else had 45 years ...; and told I had no concept of math when I listed the Standard Form of the Quadratic (as listed in 1,000s of books) properly. Please tell them to play nice as well. Just maybe I have a deep love for math! As to me, I will yield to your reasonable request. John

 Re: PEMDASMessage #129 Posted by Pier Aiello on 28 July 2013, 1:59 p.m.,in response to message #128 by jep2276 You are completely right; i mean: to hold your point of view (POV). As i said, implicit multiplication for me is ugly, but it is a convention (that i use sometimes on my sheets only with letters and not when i have numbers nor on published sheets where i use always "\cdot" or parentheses). I have just written "a/bc/d WHOA WHOA" :P . Given the right advice at document start, one can use whatever convention, even RPN (that is really really ugly on textbook imo). The problem is: for me seem that you want to force you POV as other want to force you when they said "i have 3 masters and 7 billion of years, i win!"; and that is not a constructive debate. Instead, everyone should explain why a convention is better than another (or when you can switch between them) for his POV. Many have done so, but force only one view is always wrong (and forcing that no one should force the others is wrong too. This statement is false :P ). Remember Lobačevskij when he tried to say "Hey guys, a different Geometry is possible!" , they replied "LOL". Edited: 28 July 2013, 2:02 p.m.

 (deleted post)Message #130 Posted by deleted on 28 July 2013, 2:10 p.m.,in response to message #129 by Pier Aiello This Message was deleted. This empty message preserves the threading when a post with followup(s) is deleted. If all followups have been removed, the original poster may delete this post again to make this placeholder disappear.

 Re: PEMDASMessage #131 Posted by Pier Aiello on 28 July 2013, 2:23 p.m.,in response to message #130 by deleted I assume that there is no problem on that, or, at least, the discussion will be really interesting because it requires good argumentations to be explained. even if... http://www.youtube.com/watch?v=G_gUE74YVos James is great. Edited: 28 July 2013, 2:24 p.m.

 (deleted post)Message #132 Posted by deleted on 28 July 2013, 2:52 p.m.,in response to message #131 by Pier Aiello This Message was deleted. This empty message preserves the threading when a post with followup(s) is deleted. If all followups have been removed, the original poster may delete this post again to make this placeholder disappear.

 Re: PEMDASMessage #133 Posted by Reth on 29 July 2013, 6:05 a.m.,in response to message #132 by deleted Mathematics's language is meant to be universal and unambiguous. And such it is. Whoever gets answer (to the OP) different to 9 speaks some weird dialect. In your case also known as a "troll". Edited: 29 July 2013, 6:07 a.m.

 Re: PEMDASMessage #134 Posted by Pier Aiello on 29 July 2013, 2:27 p.m.,in response to message #133 by Reth Argh, what i read (2nd)

 Re: PEMDASMessage #135 Posted by Gerson W. Barbosa on 29 July 2013, 4:03 p.m.,in response to message #133 by Reth Quote: Whoever gets answer (to the OP) different to 9 speaks some weird dialect.

 Re: PEMDASMessage #136 Posted by John B. Smitherman on 29 July 2013, 9:13 p.m.,in response to message #135 by Gerson W. Barbosa When you're holding a hammer everything looks like a nail. ;-) Regards, John

 (deleted post)Message #137 Posted by deleted on 29 July 2013, 6:38 p.m.,in response to message #133 by Reth This Message was deleted. This empty message preserves the threading when a post with followup(s) is deleted. If all followups have been removed, the original poster may delete this post again to make this placeholder disappear.

 Re: PEMDASMessage #138 Posted by Reth on 30 July 2013, 8:12 a.m.,in response to message #137 by deleted Do you want me to tell you what you can wipe out with these papers?

 Re: PEMDASMessage #139 Posted by Manolo Sobrino on 29 July 2013, 7:04 p.m.,in response to message #133 by Reth Well, every Casio calculator in the world "speaks some weird dialect" then. As they are the best sellers outside the USA it is convenient to be able to understand it. These are just conventions (I tried to make this point clear in other post), be civil and get over it. There is no truth or false in a convention, that is for theorems and possibly their proofs. Mathematics really are what's left after we strip the conventions away. I want to share this quote from R. P. Feynman: Quote: Every theoretical physicist who is any good knows six or seven different theoretical representations for exactly the same physics.

 Re: PEMDASMessage #140 Posted by Reth on 30 July 2013, 8:07 a.m.,in response to message #139 by Manolo Sobrino You must be joking. Carved in stone math rules do not depend on how some idiot have implemented them in a calculator.

 Re: PEMDASMessage #141 Posted by Manolo Sobrino on 30 July 2013, 11:33 a.m.,in response to message #140 by Reth You just don't get it. I'm sorry, but the only troll here is you. You don't provide any reasoning, just general rudeness. I wish there was an ignore button in this forum.

 Re: PEMDASMessage #142 Posted by Reth on 1 Aug 2013, 10:16 a.m.,in response to message #141 by Manolo Sobrino Quote: You just don't get it. I'm sorry, but the only troll here is you. You don't provide any reasoning, just general rudeness. I wish there was an ignore button in this forum. Reasoning? Go see at the end of this thread there is a picture showing how two casio crapculators give solution to identical problem. I'd gladly use ignore button for those who see reasoning in how math is implemented in any calculator. Some here miss badly basic math knowledge. Trolling can't compensate that.

 Re: PEMDASMessage #143 Posted by Manolo Sobrino on 1 Aug 2013, 8:11 p.m.,in response to message #142 by Reth We all know arithmetic here, thank you very much. I know I'm wasting my time, but I'll explain this just once more. The point of having a distinct implied multiplication operator with precedence over division in addition to the multiplication operator (*) with the same precedence allows us to suppress a lot of parentheses. Of course both are equivalent save for the precedence: you could even introduce twenty formally different operators with decreasing precedence for any operation so you would never have to write down parentheses. That's why there are conventions, clever conventions save time and effort so we can be more productive (I reach my Casios more often than my TIs because I take advantage of this). Before you dismiss somebody else's ideas you should try to understand them, more so when they've been there and you haven't: maybe you'll learn something. That might be called intelligence by some, maybe it's asking too much.

 Re: PEMDASMessage #144 Posted by Walter B on 2 Aug 2013, 12:06 a.m.,in response to message #143 by Manolo Sobrino "Irony is the sword of intelligentsia." d:-)

 Re: PEMDASMessage #145 Posted by Reth on 2 Aug 2013, 6:34 a.m.,in response to message #143 by Manolo Sobrino Quote: That's why there are conventions, clever conventions save time and effort so we can be more productive (I reach my Casios more often than my TIs because I take advantage of this). Which Casio? The one that gives the right answer or the other? Which TI? Not that I'm interesting in, if I were, I'd post in TI forums ;) Or in Casio forums ;) I've been using HP calcs all my professional life and have never had any problems. Now trolls come around to teach me "conventions" and other crap to tell me I've been wrong all that time. If TI or Casio were any good, they would have forums. But they don't. And don't tell me they do, cause otherwise we wouldn't be talking. Edited: 2 Aug 2013, 6:35 a.m.

 (deleted post)Message #146 Posted by deleted on 2 Aug 2013, 6:51 a.m.,in response to message #145 by Reth This Message was deleted. This empty message preserves the threading when a post with followup(s) is deleted. If all followups have been removed, the original poster may delete this post again to make this placeholder disappear.

 Re: PEMDASMessage #147 Posted by reth on 2 Aug 2013, 6:03 p.m.,in response to message #146 by deleted Quote: Learn math and forget what the calculators do! Which math? Your's? The one with 'coventions' that evaluates 6:2(1+2) as 1? No, thank you. I'll stick to the simple one.

 (deleted post)Message #148 Posted by deleted on 2 Aug 2013, 8:23 p.m.,in response to message #147 by reth This Message was deleted. This empty message preserves the threading when a post with followup(s) is deleted. If all followups have been removed, the original poster may delete this post again to make this placeholder disappear.

 Re: PEMDASMessage #149 Posted by Thomas Klemm on 2 Aug 2013, 8:37 p.m.,in response to message #148 by deleted Quote: That is exactly why the answer is 1. You seem to forget fast.

 (deleted post)Message #150 Posted by deleted on 2 Aug 2013, 9:03 p.m.,in response to message #149 by Thomas Klemm This Message was deleted. This empty message preserves the threading when a post with followup(s) is deleted. If all followups have been removed, the original poster may delete this post again to make this placeholder disappear.

 Re: PEMDASMessage #151 Posted by Thomas Klemm on 2 Aug 2013, 9:13 p.m.,in response to message #150 by deleted Message #1 Posted by Les Koller on 24 July 2013, 8:09 p.m. Quote: what is the value of 6÷2(1+2) Message #2 Posted by Kimberly Thompson on 24 July 2013, 8:23 p.m., in response to message #1 by Les Koller Quote: I get ONE. Message #3 Posted by Les Koller on 24 July 2013, 9:42 p.m., in response to message #2 by Kimberly Thompson Quote: 1 is exactly what I get too. Message #8 Posted by William L. Drylie on 28 July 2013, 3:41 a.m., in response to message #3 by Les Koller Quote: The answer is = 1. Message #9 Posted by jep2276 on 28 July 2013, 11:23 a.m., in response to message #8 by William L. Drylie Quote: That is exactly why the answer is 1.

 (deleted post)Message #152 Posted by deleted on 3 Aug 2013, 6:11 a.m.,in response to message #151 by Thomas Klemm This Message was deleted. This empty message preserves the threading when a post with followup(s) is deleted. If all followups have been removed, the original poster may delete this post again to make this placeholder disappear.

 Re: PEMDASMessage #153 Posted by Thomas Klemm on 3 Aug 2013, 6:28 a.m.,in response to message #152 by deleted *You're*

 (deleted post)Message #154 Posted by deleted on 29 July 2013, 9:42 p.m.,in response to message #133 by Reth This Message was deleted. This empty message preserves the threading when a post with followup(s) is deleted. If all followups have been removed, the original poster may delete this post again to make this placeholder disappear.

 Re: PEMDASMessage #155 Posted by Reth on 30 July 2013, 8:04 a.m.,in response to message #154 by deleted I don't know, you - "the troll" tell us.

 Re: PEMDASMessage #156 Posted by kimberly thompson on 27 July 2013, 9:38 a.m.,in response to message #122 by htom trites jr htom I LIKE paragraph three {Normally...obeyed} - more than ONE convention - thanks for the input! message #2 SlideRule Edited: 27 July 2013, 11:27 a.m.

 Re: PEMDASMessage #157 Posted by Kimberly Thompson on 27 July 2013, 9:58 a.m.,in response to message #112 by Don Shepherd Don Before I entered the Air Force, I was employed as an Engineer Technician by Figg & Mueller Engineers (Tallahassee FL) & worked exclusively on BRIDGES, including the Sunshine Skyway over Tampa Bay. At the time, it was the first center-cable-stayed bridge in North America. We used HP calculators with RPN only for all the Quantity Survey & Field Survey calculations: mine was a 41CX. I'm very proud of my contributions to this marvelous structure. After completing my Engineering degree, I was the assistant Engineer for the second largest bridge inventory/inspection program in the nation {11+ yrs}. .... to today. Yes, I understand engineering caculations are generally LIFE SAFETY calculations. You are CORRECT, undertanding the significance of the expression under evaluation is paramount! NO equivication! Thank you for reminding us of a meta-principle of our PROFESSION. SlideRule Edited: 27 July 2013, 10:04 a.m.

 Re: PEMDASMessage #158 Posted by Don Shepherd on 27 July 2013, 10:21 a.m.,in response to message #157 by Kimberly Thompson Thank you, Kimberly. I will gladly ride over any of your bridges anytime, not necessarily because you used HP calculators in their design, but because you have demonstrated that you are a professional. When I was a younger man, I developed and tested software for air traffic control systems, and I knew that the margin for error in my work was zero. It made a difference in how we did our jobs. Thanks again.

 Re: PEMDASMessage #159 Posted by Les Koller on 27 July 2013, 8:42 p.m.,in response to message #157 by Kimberly Thompson You were the first to respond to my query, and the answer was concise, succinct, and terse. Those are some great qualities for a great engineer, and I have no doubt you ARE one. I particularly liked hearing about your use of the 41CX in your work. What is your calculator of choice these days?

 Re: PEMDASMessage #160 Posted by Kimberly Thompson on 27 July 2013, 9:53 p.m.,in response to message #159 by Les Koller Hello Les I very recently completed an MBA (scl), so I've been 'playing' with Business Models, HP of course. I still have my 17bii, 20b 30b & 19bii(ii) calcs out, up & running. I recently repurposed a surplus 20b to a wp-34s. Very Interesting machine/software combo! I still use my original purchase HP-65 - (yes, I'm somewhat long in the tooth). Thanks for the interest & response. v/r SlideRule

 Re: PEMDASMessage #161 Posted by reth on 5 Aug 2013, 8:50 a.m.,in response to message #160 by Kimberly Thompson Quote: I still have my 17bii, 20b 30b & 19bii(ii) calcs LOL! And you really think you've got HP calculators? And you built bridges using them? What did you use the HP41 for? Adding two numbers? Quantity surveyors don't build bridges. Not to be confused with land surveyors like myself. Now I understand math ignorance better. Edited: 5 Aug 2013, 9:02 a.m.

 Re: PEMDASMessage #162 Posted by Bill (Smithville, NJ) on 28 July 2013, 4:16 p.m.,in response to message #1 by Les Koller Hi Les, Quote: This has probably been discussed here before While not discussed here before, it has been discussed extensively (same as here now) on the Yahoo Answers from 2 years ago: While most of the responses were 1 or 9, there was one person who said it was 7. Been very interesting following this. Bill

 Re: PEMDASMessage #163 Posted by Pier Aiello on 29 July 2013, 2:58 a.m.,in response to message #162 by Bill (Smithville, NJ) Even if the content is good, yahoo answer shows informations in a ugly (another ugly shout!) way. I prefer Quora http://pier-writings-on-quora.quora.com/Great-discussion-about-PEMDAS-6-2-1+3 or stack exchange http://math.stackexchange.com/questions/33215/what-is-48293

 Re: PEMDASMessage #164 Posted by Massimo Gnerucci (Italy) on 29 July 2013, 7:37 a.m.,in response to message #1 by Les Koller Interesting reading on Please Excuse My Dear Aunt Sally...

 Re: PEMDASMessage #165 Posted by Kimberly Thompson on 29 July 2013, 8:36 a.m.,in response to message #164 by Massimo Gnerucci (Italy) Massimo Greetings, great article - KUDOS! SlideRule

 Re: PEMDASMessage #166 Posted by Bill (Smithville, NJ) on 29 July 2013, 9:52 a.m.,in response to message #164 by Massimo Gnerucci (Italy) Massimo, Fantastic article. Thanks for pointing it out. One item in the article caught my attention: Quote: Students are thereby encouraged to memorize things they never learn to use, and their teachers are also dragged into the game, because they know that sure-fire points on the examinations can be achieved by this useless memorization. What these test items ultimately succeed in doing is to legitimize teaching- and learning-by-rote Reminded me of a professor I had in college. He said "I'll assume that each of you have the ability to either memorize the formulas or can find them in the book or your notes. Therefore, my tests will NOT be open-book or open notes, and you can NOT use a 'crib' sheet. I will hand out a 'crib' sheet for your use - it will have all the equations that will be required on the test. They will most likely NOT be in the form required to solve the problems, but the required form can be derived, if you use what has been taught you in this class. Anyone can memorize or look things up. I want to see if you can apply what has been taught." I had several courses with this professor and enjoyed them all. And he DID make you really think about how to apply the equations. Bill

 Re: PEMDASMessage #167 Posted by Les Koller on 4 Aug 2013, 1:15 a.m.,in response to message #166 by Bill (Smithville, NJ) Bill, I like that kind of professor also. I had a professor / adviser for my Bachelor's in Math in the 80's that was exactly the opposite. He would allow, even encourage, the use of any text we wanted...our class text, CRC Handbook, any of our own class notes...for every test. He told us that when we began using math in our occupation, we would have any text we felt necessary to buy, steal, or borrow, so why not start now? He went on to tell us that we better be very, VERY familiar with our texts, because, as you said, he was not going to give us nice clean problems where the equations or formulae would fit nicely and we would waste time looking for one.. Loved that man!

 Re: PEMDASMessage #168 Posted by Pier Aiello on 29 July 2013, 2:42 p.m.,in response to message #164 by Massimo Gnerucci (Italy) The article is great just for this: Quote: No convention is sacrosanct. Every convention is arti cial, and as such, should be kept only if it continues to serve a purpose Thanks for the article, or better: grazie mille ;) .

 Re: PEMDASMessage #169 Posted by Pier Aiello on 30 July 2013, 1:44 a.m.,in response to message #1 by Les Koller Anyway, some branches of this discussion remember me:Edited: 30 July 2013, 1:45 a.m.

 Re: PEMDASMessage #170 Posted by Les Koller on 30 July 2013, 2:34 a.m.,in response to message #169 by Pier Aiello I loved the Foundation Trilogy!

 Re: PEMDASMessage #171 Posted by Pier Aiello on 31 July 2013, 2:51 a.m.,in response to message #170 by Les Koller OT: For me foundation trilogy was only a Meh.

 Re: PEMDASMessage #172 Posted by Massimo Gnerucci (Italy) on 31 July 2013, 3:12 a.m.,in response to message #171 by Pier Aiello heretic!Go and re-read the Seven Books Of Foundation!:-)Greetings,Massimo

 Re: PEMDASMessage #173 Posted by Ethan Conner on 30 July 2013, 4:15 a.m.,in response to message #1 by Les Koller I've been watching this thread with a great deal of interest. This came across social media sometime last year and it was not surprising the range of answers that were given nor the fact that it degenerated into name calling. When it showed up here my initial feeling was that the thread would be short and sweet and everyone would get the same answer. Being no math expert and not taking sides i decided to try this on two different coworkers. While not in any way tipping them off or leading them on and approaching them separately just presented the problem and asked them to solve it as written in the OP. One of them is currently in college taking calculus and the other is a recently graduated engineer. They both gave the answer 1 and proceeded to explain to me about PEMDAS and write in another form why the answer was 1. Two people is not a large group to sample but it was the best i could do considering my work environment is fast paced and working nights makes finding more brainiacs difficult. The recently graduated engineer said math should be absolute but when you involve humans nothing is absolute. Oh well.

 Re: PEMDASMessage #174 Posted by Chris Osburn on 30 July 2013, 7:53 a.m.,in response to message #173 by Ethan Conner God invented math; humans invented typography. Who do you trust? :D

 Re: PEMDASMessage #175 Posted by Walter B on 30 July 2013, 9:47 a.m.,in response to message #174 by Chris Osburn Quote: Who do you trust? Sic! d:-D P.S.: I didn't want to open a new flame war, but apparently this language loses its last traces of grammar. Edited: 30 July 2013, 10:39 a.m.

 Re: PEMDASMessage #176 Posted by Chris Osburn on 31 July 2013, 7:53 a.m.,in response to message #175 by Walter B Quote: Sic! d:-D P.S.: I didn't want to open a new flame war, but apparently this language loses its last traces of grammar. J'Accuse!(itive case!) In my defense, I was paraphrasing someone else's grammar. :-)

 Re: PEMDASMessage #177 Posted by Gerson W. Barbosa on 31 July 2013, 8:47 a.m.,in response to message #176 by Chris Osburn Quote: J'Accuse!(itive case!) From Wikipedia: "The pronoun whom is a remnant of the dative case in English, descending from the Old English dative pronoun "hwam" (as opposed to the nominative "who", which descends from Old English "hwa")" As of me, I am glad there are only remnants of the Old English grammar cases in Modern English. I would never be able to learn this table by heart, I think:  Declension of the definite article se Singular Plural Masculine Femine Neuter All genders Nom. se seo þæt þa Acc. þone þa þæt þa Gen. þæs þære þæs þæra Dat. þæm þære þæm þæm Inst. þy þære þy  ( From The Oxford Companion to the English Language, page 724 ) Isn't it a blessing all of these have been simplified to "the" ?

 Re: PEMDASMessage #178 Posted by Gilles Carpentier on 30 July 2013, 6:28 p.m.,in response to message #1 by Les Koller I found this on the web : (c) Arthur in http://www.silicium.org/forum

 Re: PEMDASMessage #179 Posted by Thomas Klemm on 1 Aug 2013, 8:41 p.m.,in response to message #1 by Les Koller What about these expressions? $6\div2\;\frac{1}{3}$ Or: $\pi\div2\;\frac{7x-5}{x^3}$ Or even: $1\div\pi\;\int_{-\pi}^{\pi}f(x)\cos(nx)dx$ Now the same with ÷ replaced by /. $6/2\;\frac{1}{3}$ Or: $\pi/2\;\frac{7x-5}{x^3}$ Or even: $1/\pi\;\int_{-\pi}^{\pi}f(x)\cos(nx)dx$ I was curious at which point the implied multiplication is visually going to break apart. Is somebody insisting that the last expression should be read as: $\frac{1}{\pi\int_{-\pi}^{\pi}f(x)\cos(nx)dx}$ Kind regards Thomas

 Re: PEMDASMessage #180 Posted by Walter B on 2 Aug 2013, 12:13 a.m.,in response to message #179 by Thomas Klemm Quote: $6\div2\;\frac{1}{3}$ Anybody voting for implied multiplication here? Why not? Compare Quote: $\pi\div2\;\frac{7x-5}{x^3}$ for an arbitrary x and x = 2, for example. d:-)

 Re: PEMDASMessage #181 Posted by RogerBlake on 2 Aug 2013, 8:06 a.m.,in response to message #179 by Thomas Klemm You seem to be forgetting the use of typographical conventions. If your last expression was intended, then the slash would be the same height as the denominator (the integral expression) and all would be clear. As it is written 1/pi is clearly the coefficient of the integral term. Many of these supposed queries are in fact quite clear if following usual mathematical typographical conventions. These rules are just as important as other rules such as PEMDAS and sometimes override them. Some people seem to have no knowledge of these conventions and then accuse others of being mathematically ignorant which is slightly perverse to say the least.

 Re: PEMDASMessage #182 Posted by Thomas Klemm on 2 Aug 2013, 11:15 a.m.,in response to message #181 by RogerBlake Just to make it clear: I would avoid all variants using ÷ or /. So IMHO we should only use these two expressions: $\frac{1}{\pi}\int_{-\pi}^{\pi}f(x)\cos(ns)dx$ $\frac{1}{\pi\int_{-\pi}^{\pi}f(x)\cos(nx)dx}$ There's no reason to write these expression in an ambiguous way. We're not forced to use typewriters anymore. We can use LaTeX to write our expressions and don't have to bother about typographical conventions: \frac{1}{\pi}\int_{-\pi}^{\pi}f(x)\cos(ns)dx \frac{1}{\pi\int_{-\pi}^{\pi}f(x)\cos(ns)dx}  Presumably, the title of this thread was misleading from the outset. Because it was never about PEMDAS. It's about whether implied multiplication ties the factors stronger together than explicit multiplication. I must admit I was not aware of this convention. Still I can imagine why it was used. However I don't see why it should still be used today. Because obviously it may lead to ambiguity. And both google and WolframAlpha disagree. Of course you can still follow that convention even when the meaning of an expression depends so much on the width of a space. Just be aware that this may not always be understood the way you do. Cheers Thomas Quote: If your last expression was intended, then the slash would be the same height as the denominator (the integral expression) and all would be clear. Certainly that's not the way to make the expression clear without ambiguity.

 Re: PEMDASMessage #183 Posted by RogerBlake on 2 Aug 2013, 12:40 p.m.,in response to message #182 by Thomas Klemm I'd politely disagree, and one use for using / would be to avoid tiny unreadable fonts in complicated denominators ... P.S. this is definitely my last post on the subject :-) Edited: 2 Aug 2013, 12:55 p.m.

 Re: PEMDASMessage #184 Posted by Thomas Klemm on 2 Aug 2013, 1:01 p.m.,in response to message #183 by RogerBlake Quote: avoid tiny unreadable fonts Use [Ctrl] + [+]. But you do realize that the font-size is the same in either case? Edited: 2 Aug 2013, 1:05 p.m.

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