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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.
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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
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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!
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Quote:
My HP agrees.
My HP48GX disagrees and gives 9 as well. Which calculator did you use?
Cheers
Thomas
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I used the HP 50g in equation writer and eval mode. Entering it in strictly post fix mode gave me 9 too.
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Quite different things:
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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.
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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 NSpire 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
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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(xh)^2 for a*(xh)^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(xh)^2 + k.
Anyway, Programming convention does not dictate, nor override, proper Algebra and that seems to be the hangup.
John
Edited: 28 July 2013, 11:54 a.m.
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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 eyeopening 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
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What makes 6/2*3 difficult while 52+3 isn't? Who thinks the latter should be 0?
Cheers
Thomas
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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.
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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 hangup is that instead of teaching the basic axioms, we educate with tricks, e.g. for the inverse of a fraction, CopyChangeFlip 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
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I teach PEMDAS (aka order of operations) to middleschool 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.
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Precisely!
This topic came up in day one of introductory programming courses back in the days of FORTRANIV 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 ALGOL60. 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 universallyaccepted convention now?
Edited: 25 July 2013, 12:26 a.m.
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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.
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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 TI83 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.
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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
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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.
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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
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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).
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English say Exponentiation instead of Elevation, but it is the exact same thing mathematically speaking.
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Max,
Quote:
I have no idea what PEMDAS might be
Following link has a good description of PEMDAS:
The Order of Operations: PEMDAS
Bill
Edited: 25 July 2013, 8:01 a.m.
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Max,
PEMDAS is the USAmerican acronym for "Klammer vor Hoch vor Punkt vor Strich".
d:)
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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!
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Excel gets this right, also.
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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:
http://wyorock.com/excelasadatabase.htm
:)
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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:)
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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 :)
http://en.wikipedia.org/wiki/Johann_Rahn
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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;/
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We don't usually see the typical division sign in the UK past the first couple of years of mathematics (68 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
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Walter
I like  are we thinking {6 / z} where z=2x & x= (2+1)?
SlideRule
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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;)
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You do know that your RPN evaluates to the correct answer of 1.
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Rats! Fell twice into the same trap :(
Should read 1 ENTER 2 + 6 ENTER 2 / * of course!
d:/
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I think we all can agree, with RPN instructions there is no ambiguity ...
I like your first set!
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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
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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 Ti84 (was the first result that google found, a Ti84 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.
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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.
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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
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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.
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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.
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How about a NEW post. You got my attention.
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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.
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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.
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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 thata factor.
Ok, one more time 6÷(2x2), if I factor (2x2) I get 2(x1). Now if we take 45 years experience and PEMDAS incorrecly applied and not understand implicit vs. explicit, I get 2*(x1) and put that back into the original equation, then I get the wrong answer of 6÷2*(x1), or 3*(x1).
Now lets test it when x=4.
Original equation 6÷(2x2) = 6÷(2*42) = 6÷(82) = 6÷6 = 1
Factor wrong 3*(x1) = 3*(41) = 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.
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Quote: ... if I factor (2x2) I get 2(x1) ...
Sorry, no, you get (2(x1)) instead. You need to preserve the original parentheses that were there for a reason. Factoring introduces a _new_ set of parentheses. "2x2" always becomes 2(x1) whatever the context is, or 2*(x2) or whatever multiplication sign you like best.
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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*(xh)^2 + k)
It is:
f(x) = a(xh)^2 + k
Edited: 26 July 2013, 9:31 a.m.
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Quote:
The Standard form of the Quadratic is not
f(x) = (a*(xh)^2 + k
It is:
f(x) = a(xh)^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(xh)^2 for a*(xh)^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 'Voodoomath' 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.
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You said "Sorry, no, you get (2(x1)) instead. You need to preserve the original parentheses that were there for a reason. Factoring introduces a _new_ set of parentheses. "2x2" always becomes 2(x1) whatever the context is, or 2*(x2) or whatever multiplication sign you like best. "
So it is you that implied the Quadratic is f(x) = (a*(xh)^2 + k
), not me. Good you can correct yourself.
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Quote:
Good you can correct yourself.
And bad that you can't even reply correctly!
It was not my answer that you quoted ...
Franz
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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.
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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
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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
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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
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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
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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 2D input: always use a space for implied multiplication. 2D 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 TI89 family, TI92 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 WA 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
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Quote: The TI89 family, TI92 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.
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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.
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AMEN!
(see post #2)
SlideRule
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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?
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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.
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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.
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What? No standard from DIN or ISO on this? ;)
Regards,
John
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Walter, I have read somewhere in wiki that some conventions consider the slash or fraction bar symbol ( / ) differently from the old time, rarelyused 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.
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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.
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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/mathbasicsorderofoperationsprecedenceandtrickybrainteasers/. 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/guidereviewers.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.
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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.
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AMEN!
see post #23
SlideRule
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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).
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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 NSprire 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
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Thanks William. I evaluated 6/2(1+2) on a few calculators I have. The TI83 gave 9 as the result; the TINspire CAS also gave 9, as did the HP17bii. The Sharp ELW516 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.
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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 1dimensional 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.
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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
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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
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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:/
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Nice example for the "implicit multiplication".
a/bc/d , WHOA WHOA WHAT IS THAT?
Won't sign :P .
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Don
Simple curiosity; which SHARP Model is referenced? Thanks!
SlideRule
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That's the Sharp ELW516 WriteView, a nice little inexpensive calculator that let's you store key sequences (like (b+sqrt(b^24ac))/(2a)) )and evaluate them with a single keypress.
Edited: 28 July 2013, 1:28 p.m.
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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
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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.
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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.
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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 highlevel 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.
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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.
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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.
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Quote:
And Prentice Hall should be ashamed of themselves.
It appears the book was misquoted as Gerson already has pointed out.
Cheers
Thomas
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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 narrowminded 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.
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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.
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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.
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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
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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!!!
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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.
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Quote:
probably will not post again
Usually that's fhub's statement. Now I'm confused.
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Franz, your language is just fine, even too mild to my taste.
Nevertheless trolls and idiots will be always. Take it easy.
Cheers,
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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
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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.
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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
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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
/
College Algebra, Michael Sullivan, 6th edition Prentice Hall
That's the 2002 edition. I imagined it was much older!
Best regards,
Gerson
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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.
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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.
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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:)
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Quote:
2+3/4+8=(2+3)/(4+8)
Then how do you write this expression in one line?
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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
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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 /.
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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.
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Another interesting debate (at least foor me) on the subject:
Quora debate
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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 halfpi 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 arithmeticoperation 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."
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Quote:
making 2(1+3) a complete term to which the PEMDAS rule then applies
So 2(1+3)^{2} = 64 instead of 32?
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"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 allinall 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 noone actually complains ... not much anyway.
Edited: 27 July 2013, 7:07 a.m.
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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!
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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.
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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.
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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:
. And they can write 1/2 x which of course means . So there's another rule to it: don't use spaces between factors in implied multiplication. But since we're fine with 2 x^{2} 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
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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.
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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
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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
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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.
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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.
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Regardless what you claim, please see above. I think Wolfram should be sufficient to outbalance even you.
d:)
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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 shuffleterms 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 shuffleterms option.
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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.
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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.
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General comment on the discussion.
At school in Australia in the 60s70s 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/operationorderbodmas.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.
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There is no doubt that your choices are correct for you :).
Simply leave alone the others if they disagree with you ^_^ .
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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
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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.
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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.
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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.
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Quote:
Whoever gets answer (to the OP) different to 9 speaks some weird dialect.
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When you're holding a hammer everything looks like a nail. ;)
Regards,
John
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Do you want me to tell you what you can wipe out with these papers?
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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.
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You must be joking. Carved in stone math rules do not depend on how some idiot have implemented them in a calculator.
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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.
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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.
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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.
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"Irony is the sword of intelligentsia."
d:)
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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.
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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.
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Quote:
That is exactly why the answer is 1.
You seem to forget fast.
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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.
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I don't know, you  "the troll" tell us.
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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.
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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 centercablestayed 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 metaprinciple of our PROFESSION.
SlideRule
Edited: 27 July 2013, 10:04 a.m.
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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.
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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?
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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 wp34s. Very Interesting machine/software combo! I still use my original purchase HP65  (yes, I'm somewhat long in the tooth). Thanks for the interest & response.
v/r
SlideRule
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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.
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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:
Answer this question 6÷2(1+2)= ?
While most of the responses were 1 or 9, there was one person who said it was 7.
Been very interesting following this.
Bill
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Interesting reading on Please Excuse My Dear Aunt Sally...
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Massimo
Greetings, great article  KUDOS!
SlideRule
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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 surefire points on the examinations can be achieved by this useless memorization. What these test items
ultimately succeed in doing is to legitimize teaching and learningbyrote
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 openbook 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
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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!
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The article is great just for this:
Quote:
No convention is sacrosanct. Every convention is articial, and as such, should
be kept only if it continues to serve a purpose
Thanks for the article, or better: grazie mille ;) .
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Anyway, some branches of this discussion remember me:
Asimov on liberation from the machine
Edited: 30 July 2013, 1:45 a.m.
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I loved the Foundation Trilogy!
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OT: For me foundation trilogy was only a Meh.
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heretic!
Go and reread the Seven Books Of Foundation!
:)
Greetings, Massimo
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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.
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God invented math; humans invented typography. Who do you trust? :D
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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.
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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. :)
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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")"
( https://en.wikipedia.org/wiki/Dative_case )
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" ?
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I found this on the web :
(c) Arthur in http://www.silicium.org/forum
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What about these expressions?
Or:
Or even:
Now the same with ÷ replaced by /.
Or:
Or even:
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:
Kind regards
Thomas
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Quote:
Anybody voting for implied multiplication here? Why not? Compare
Quote:
for an arbitrary x and x = 2, for example.
d:)
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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.
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Just to make it clear: I would avoid all variants using ÷ or /. So IMHO we should only use these two expressions:
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.
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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.
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Quote:
avoid tiny unreadable fonts
Use [Ctrl] + [+].
But you do realize that the fontsize is the same in either case?
Edited: 2 Aug 2013, 1:05 p.m.
