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Bedlam with BODMAS and BEDMAS

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brianvds
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« on: May 19, 2014, 19:30:32 PM »

Suppose you have this calculation:

One third of 3^2 + 1

Depending on the order in which you do the operations, you could get either 2 or 4 as answer. My intuition tells me you should get 4, i.e. if we combine the BODMAS and BEDMAS rules, we should get BEODMAS, not BOEDMAS.

Any comments from our fine mathematical minds here?

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Mefiante
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« Reply #1 on: May 19, 2014, 19:39:42 PM »

As it stands, the result is 10/3.  That’s because the wordy part of the question enjoys the lowest priority, and the question is not (32)/3+1 = 4, but instead (32+1)/3 = (9+1)/3 = 10/3, or 3.333…

I don’t see how you can get 2.  Even 9 is more likely: (32+1)/3 = 27/3 = 9.

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brianvds
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« Reply #2 on: May 19, 2014, 20:30:38 PM »

And if I rephrase:

1/3 of 3^2 +1   ?

What do we get now?

In short, BODMAS and BEDMAS seem to be in potential conflict. How do we resolve the conflict? What gets precedence, "of" or the exponent? Is it BEODMAS or BOEDMAS? Or neither, and if so, what then?


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Rigil Kent
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« Reply #3 on: May 19, 2014, 20:51:54 PM »

What gets precedence, "of" or the exponent?
The exponent. "Of" is the same as "multiply with what follows".

1/3 of 3^2 +1  = ⅓ x (32 +1) = 10/3

r.
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Mefiante
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« Reply #4 on: May 19, 2014, 20:54:33 PM »

The mathematical/arithmetical formulations always take precedence over the worded parts.  That’s the way I’ve both seen it treated and treated it myself.  Thus, “1/3 of 3^2 +1” is still “1/3×(3^2 +1)”, as before.

Any mathematician worth their salt would in any case carefully avoid such potential ambiguities.

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Rigil Kent
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« Reply #5 on: May 19, 2014, 21:15:04 PM »

I don’t see how you can get 2.

Maybe thus

1/3 of 3^2 +1  = (⅓(3))2 +1 = 2

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Mefiante
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« Reply #6 on: May 19, 2014, 21:22:57 PM »

Yes, that’ll get you 2 as an answer.  With a real stretch, I can get the answers 2/3 and 4/3.  These results hinge on a different interpretation of the “^” operator, one with which C programmers will be familiar.

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The Vulcan
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« Reply #7 on: May 19, 2014, 22:05:09 PM »



I like
1/3 x 3^2 + 1 = 4

1/3 of 3^2 +1   ?

Well this is just me as an ordinary layman, I like this answer best, because when I read 1/3x3^2 +1 I don't see why I must assume that the operation (3^2+1) would be meant to be in brackets the way the question was put, but I'm certainly no math guru, just to me that makes the most sense, but if you would've asked

What is one third of the sum of 3^2 and 1 I would then think to use brackets
« Last Edit: May 19, 2014, 22:25:57 PM by The Vulcan » Logged
brianvds
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« Reply #8 on: May 20, 2014, 04:54:06 AM »



I like
1/3 x 3^2 + 1 = 4

1/3 of 3^2 +1   ?

Well this is just me as an ordinary layman, I like this answer best, because when I read 1/3x3^2 +1 I don't see why I must assume that the operation (3^2+1) would be meant to be in brackets the way the question was put, but I'm certainly no math guru, just to me that makes the most sense, but if you would've asked

What is one third of the sum of 3^2 and 1 I would then think to use brackets

And this is indeed what I was taught in grade 8 (then standard 6) when I first learned it - that "of" takes precedence. Hence BODMAS, not BDMASO. Apparently they taught me wrong, but that BODMAS rule is still taught in schools all over the world. Under that rule, the calculation 1/2 of 4 + 2 would yield 4, not 3.

Except in some schools they teach BEDMAS, and pretend that "of" does not exist as operator.

Question is now what the heck do I teach a grade 7 class? Seems to me BEDMAS is probably better, and then to word problems in such a way that ambiguity will not arise when "of" makes its appearance? When at all in any doubt, always use brackets?
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Rigil Kent
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« Reply #9 on: May 20, 2014, 06:23:54 AM »

When at all in any doubt, always use brackets?
Precisely. You can never use too many brackets. (It's by far the most useful thing (short of the author himself pointing out what he meant originally (when he first wrote the text (whether at his home or office (whilst slightly drunk)))) for clarifying the meaning of an expression.)

R.
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Mefiante
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« Reply #10 on: May 20, 2014, 07:07:33 AM »

I must ask:  Did this question appear in a maths test?  And at what level?  If it did, shame on the teacher who set up the test.  The question is meant to trick pupils, not test any understanding of BODMAS.  That teacher no doubt meant that (1/3)×(3²)+1 = (1/3)×9+1 = 3+1 = 4 is the correct answer.  It is this sort of question that validates the complaint about pupils being trained to pass exams, not understand the subject matter.

If one were to encounter such an expression in a book or a journal paper, that would be a very rare occurrence, and the context would make clear what is meant, for example, “… and so the final answer becomes one-third of the previous expression 3²+1, or 10/3.”

As a side note, (and I think I may have pointed this out in another thread) it is beyond me why BODMAS has been made so overly complicated.  There is no reason to distinguish between multiplication and division, or between addition and subtraction.  Division is just multiplication by the divisor’s inverse, and subtraction is just addition by the addend’s negative.  Thus, BEMA or BOMA would concisely capture all of the precedence rules of arithmetical operations.

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Rigil Kent
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« Reply #11 on: May 20, 2014, 10:16:33 AM »

Thus, BEMA or BOMA would concisely capture all of the precedence rules of arithmetical operations.
With roots presumably lumped in with exponents, since nx = x1/n ?
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Mefiante
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« Reply #12 on: May 20, 2014, 10:41:57 AM »

Yup, just as division is multiplication in drag and subtraction is addition in a clown suit, extracting roots is exponentiation in a tuxedo.

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brianvds
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« Reply #13 on: May 20, 2014, 15:10:35 PM »

I still don't see why the rule is BODMAS, or BOMA, if "of" does not in fact take precedence...

But I agree that it overly complicates the issue. On the other hand, if I teach the kids BEMA, and then they go to another school, they may end up confused.
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Brian
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« Reply #14 on: May 20, 2014, 15:36:06 PM »

Had this quiz on Facebook the other day: 4x4-4x4+4-4x4=? 20 in my book despite the lack of brackets...many said 320.
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Mefiante
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« Reply #15 on: May 20, 2014, 15:55:36 PM »

It’s the introduction of the “of” construct in mathematical/arithmetical expressions that is the problem.  It is completely artificial and exists only in the strangely contorted OCD minds of postmodern maths teachers and educational authorities whose megalomania vastly exceeds their numeracy.  In the substantially more real world inhabited by mathematicians and mathematically literate professionals, such forms are NEVER used.  (Is that sufficient emphasis?)  And they are NEVER used precisely because they are ambiguous, and ambiguity goes with maths the way cheese goes with barbed wire.

4x4-4x4+4-4x4=?
–12.  (4×4 – 4×4 + 4 – 4×4 = 16 – 16 + 4 – 16 = 0 + 4 – 16 = –12.)

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Brian
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« Reply #16 on: May 20, 2014, 16:19:23 PM »

sorry my bad: the original quiz was: 4x4+4x4+4-4x4=?? 20 or 320 according to some. I was taught that multiplying and dividing take predence over +/-....in other words the order is key.
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cr1t
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« Reply #17 on: May 20, 2014, 16:24:09 PM »

sorry my bad: the original quiz was: 4x4+4x4+4-4x4=?? 20 or 320 according to some. I was taught that multiplying and dividing take predence over +/-....in other words the order is key.

Yes that is what I was thought as well , how the hell do they get to 320?
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Mefiante
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« Reply #18 on: May 20, 2014, 17:13:55 PM »

4x4+4x4+4-4x4 = ??
20.  (4×4 + 4×4 + 4 – 4×4 = 16 + 16 + 4 – 16 = 20.)

In naïve left-to-right reading order, the result obtained is wrong:
4×4 + 4×4 + 4 – 4×4
= 16 + 4×4 + 4 – 4×4
= 20×4 + 4 – 4×4
= 80 + 4 – 4×4
= 84 – 4×4
= 80×4
= 320.

'Luthon64
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brianvds
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« Reply #19 on: May 20, 2014, 17:25:42 PM »

It’s the introduction of the “of” construct in mathematical/arithmetical expressions that is the problem.  It is completely artificial and exists only in the strangely contorted OCD minds of postmodern maths teachers and educational authorities whose megalomania vastly exceeds their numeracy.  In the substantially more real world inhabited by mathematicians and mathematically literate professionals, such forms are NEVER used.  (Is that sufficient emphasis?)  And they are NEVER used precisely because they are ambiguous, and ambiguity goes with maths the way cheese goes with barbed wire.


Yes, it is kind of difficult to imagine many real world problems where the "of" construct would be used, except perhaps in things like "only 12% of the 300 math educators could do elementary multiplication." :-)

Now I am lucky enough to work for an independent school, and we do not absolutely have to do whatever the education department wants, but we nevertheless try to remain fairly close to the official syllabus because the students will eventually have to pass a government exam. Also, as I have pointed out before, if one of our students go to another school, or we get students from other schools, it can create difficulties if the two schools followed too radically different syllabuses.

Hmm, reading around on the web, I now see a new twist in the tale. When I was in school, they told us the O in BODMAS stands for "of". But according to some pages, such as this one:

http://thehalltruth.com/tag/bodmas/

it actually stands for orders - also known as exponents. I.e. BODMAS and BEDMAS are not two conflicting rules. They are the same thing. I.e. my grade 8 teacher was just an uneducated moron. As if the fact that he was also a sadistic psychopath wasn't bad enough.

One more data point in my theory that we need teachers who are educated in the subjects they teach, rather than in education, and that closing down the teachers' colleges might well have been the one rational thing the ANC did in its quest for better education. :-)

PS: I also get -12 on that thing with the long string of multiplied, subtracted and added 4s... :-)
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Mefiante
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« Reply #20 on: May 20, 2014, 17:33:04 PM »

Aha, difficulty solved!  I wasn’t aware that you thought the “O” in BODMAS stood for “Of” instead of “Orders” (which is the archaic form of “exponents”, although still used today when describing polynomials “of order n” where n is the highest power that occurs in the polynomial).  If “Of” is to be read as a multiplication operator (as logic dictates it must) then it would be pointless to include it alongside “D” and “M” because all three are at the same precedence level.

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brianvds
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« Reply #21 on: May 20, 2014, 18:02:32 PM »

Aha, difficulty solved!  I wasn’t aware that you thought the “O” in BODMAS stood for “Of” instead of “Orders” (which is the archaic form of “exponents”, although still used today when describing polynomials “of order n” where n is the highest power that occurs in the polynomial).  If “Of” is to be read as a multiplication operator (as logic dictates it must) then it would be pointless to include it alongside “D” and “M” because all three are at the same precedence level.

Indeed. Of course, after grade 8, we never, ever again saw problems with the "of" construct, so I cheerfully forgot about it, and order of operations was just never an issue again, until I got confronted by it once again in primary school materials! But it now seems I was quite simply fed completely false information by a high school teacher, way back in the early 14th century. Whatever else I teach my own students, I really must get this into their heads: never, ever, blindly believe a teacher. :-)

I wonder whether it was only my own high school though, or whether that crap was taught to an entire generation of students. I should ask around some teens of my acquaintance and find out whether they perhaps STILL teach it that way...

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Rigil Kent
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« Reply #22 on: May 20, 2014, 18:44:26 PM »

It looks like public opinion remains divided as to what the "O" in BODMAS stands for. https://answers.yahoo.com/question/index?qid=20090413141638AA0xR8p. So at some point the idea of "of" must have snuck in.

I recall my teacher applying the Brackets, Addition, Division, Angular Substitution and Simplification approach. Tongue

r.
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brianvds
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« Reply #23 on: May 21, 2014, 04:21:33 AM »

It looks like public opinion remains divided as to what the "O" in BODMAS stands for. https://answers.yahoo.com/question/index?qid=20090413141638AA0xR8p. So at some point the idea of "of" must have snuck in.

Hmm, yes, lots of people apparently think the O stands for "of", so one must presume that at some point, this is what schools taught. One wonders how that happened. There may be a whole interesting book there. At least I got the right information eventually, even though my hair is graying, and despite my schooling.

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Brian
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« Reply #24 on: May 21, 2014, 07:51:28 AM »

Quote
At least I got the right information eventually, even though my hair is graying, and despite my schooling
      Yes sometimes I wonder that we actually survived. I had a German teacher who was quite adamant that the whole manned space satellites thing in the early 60's was a scam, but then again I had a brilliant Afrikaans and Math teacher (my math was abysmal)in Affies, Pretoria.
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