Mathematics and precedence rules

[pendell]

Recently I saw the following on facebook:

6-1x0+2/2=?

My solution uses the precedence rule I was taught in school -- My Dear Aunt Sally (MDAS) -- multiply first (1x0), then divide (2/2), then add (0+1) then subtract (6-1) = 5.

However, I noticed some other people were using a very different precedence rule -- BODMAS (open brackets, division, multiplication, addition, subtraction). It reverses the order of division and subtraction from what I was taught.

Who uses BODMAS? Where is it taught? Is there any likelihood for a mistake if, say, you're calculating a tax result using MDAS when the original author intended BODMAS? Are there any other precedence rules I should know of?

ETA: I'm not convinced my answer is right. There may have been a mistake. Different people have come up with 0, 1, 5, and 7. I'm not sure whether they are correctly applying alternate precedence rules, whether they made a mistake, or whether *I* made a mistake.

Respectfully,

Brian P.

[Siosilvar]

PEMDAS and BODMAS are the same thing.

1) Do things inside parentheses, or brackets, if you're not American - in either case, they're these things: ().
2) Exponents/Orders/Powers/Indices
3) Multiplication AND Division from left to right
4) Addition AND Subtraction from left to right

6-1x0+2/2=?

6-0+2/2

6-0+1

6+1

7

The AND in those last two is the important part. Division is just* multiplication by the reciprocal, so it has the same priority. And subtraction is just* addition of the negative, so you do it at the same time as well.

/thread

*well, for the reals at least.

[GnomeFighter]

To answer your question, BODMAS or MDAS should not make any difference. The basic reasoning is multiply/divide THEN add/subtract.

For example:

2x2/6:

2x2=4 4/6=0.66666...
2/6=0.3333... 2x0.3333...=0.6666...

To answer the question on facebook,one answer is 7, the other is "didn't your maths teacher teach you anything? That is very badly written". Those silly examples annoy me as lots of people get to show how "clever" they are by knowing the basic rules, but non of them point out that the problem is that the original question is badly written.

[pendell]

Noted.

I have reviewed the wiki article on precedence here and have been reminded of the following:

These mnemonics may be misleading when written this way, especially if the user is not aware that multiplication and division are of equal precedence, as are addition and subtraction. Using any of the above rules in the order "addition first, subtraction afterward" would also give the wrong answer.

The correct answer is 9 (and not 5, which we get when we add 3 and 2 first to get 5,and then subtract it from 10 to get the final answer of 5), which is best understood by thinking of the problem as the sum of positive ten, negative three, and positive two.

So, given this new information, I do multiplication and division at the same time to get

6-0+1,

which gives me 7. So I agree with the two of you.


If you can see the original post , you'll see answers very from 0, 1, 5, and 7. And there are more than 100,000 answers.

I think it's a valuable observation on how poorly written and imprecise documents can lead to crazy misunderstandings even in something as simple and precise as arithmetic. If the original author had written
6 - ((1*0) + (2/2))

there would have no debate or discussion. But poor communication means that people of good will can apply average intelligence to a simple problem and still get the wrong answer because they don't remember every jot and tittle of the precedence rules.

It's useful for all species of argument. It's not that your interlocutor is stupid or willfully ignorant, it's that it's hard for humans to communicate even simple ideas.

Respectfully,

Brian P.

[Rawhide]

1) Do things inside parentheses, or brackets, if you're not American - in either case, they're these things: ().
2) Exponents/Orders/Powers/Indices
3) Multiplication AND Division from left to right
4) Addition AND Subtraction from left to right

This, pretty much. Multiplication and division have the same priority, as do addition and subtraction.

6-1*0+2/2=
6-0+1=
6+1=
7

Google will give you the same answer, as will Wolfram Alpha.

Copy and paste that into Windows Calculator on Standard (which does everything in the order it is entered), and you will get 1, but copy and paste it into Windows Calculator on Scientific, and you'll also get 7.

There's nothing wrong with the way it is structured. Some brackets could make it less confusing at first glance, but they aren't needed if the order is followed correctly.

[Yora]

The reason we use bracets is because there is no single true way to solve it.
I'd say the one way to do it wrong, is to not use bracets when one should have.

The only rule I know is points before lines (in German schools, we use · and ÷ for * and /.) And Intuition would say adding first, dividing second.

I would calculate the example as 7. But if I had to calculate something like that, I'd probably go to the person who wrote that and demand to get it with bacets.

[Brother Oni]

The question's been pretty much answered, but I was taught BODMAS at school in the UK (and I end up with 7 as well).

[TheFallenOne]

It's not imprecise. Any brackets (correctly) placed in that equation can be removed because they're trivial. If anyone gives an answer except 7 it's not the fault of the equation for being imprecise, poorly written or ambiguous, but entirely the failing of the person giving the answer. It's math at its most basic level.

Incidentally, you placed the brackets wrong.

6 - ((1*0) + (2/2))

This would give 5.

[AsteriskAmp]

I think it's a valuable observation on how poorly written and imprecise documents can lead to crazy misunderstandings even in something as simple and precise as arithmetic. If the original author had written
6 - ((1*0) + (2/2))

there would have no debate or discussion. But poor communication means that people of good will can apply average intelligence to a simple problem and still get the wrong answer because they don't remember every jot and tittle of the precedence rules.

It's useful for all species of argument. It's not that your interlocutor is stupid or willfully ignorant, it's that it's hard for humans to communicate even simple ideas.

Respectfully,

Brian P.

It's not badly written. 6-1*0+2/2 is perfectly valid and unequivocal.

Would writing it the other way make it easier to solve for those who don't really understand operation order?
Yes. The convenience gain in the regular form is minimal but that doesn't mean it's wrong or imprecise.

The precedence rule is not some insanely complex formula, or rules to memorize. It is simply remembering what each operation means and going from left to right (the way everyone read on the western hemisphere). You go from most "complex" to least.

Parenthesis->Exponentiation->Multiplication->Addition

Radicals are fractional exponents, division is multiplication by the reciprocal, addition is the sum of the additive inverse of the number.

[pendell]

Yes. it should be ... let's see ...

(6 - (1 * 0)) + (2/2).

which resolvs to
(6 - 0 ) + (2/2)

which gives us

6+1 = 7.

...

I wonder if there is a web site which does drills so I can brush up on this?

It's not imprecise. Any brackets (correctly) placed in that equation can be removed because they're trivial. If anyone gives an answer except 7 it's not the fault of the equation for being imprecise, poorly written or ambiguous, but entirely the failing of the person giving the answer.

I must respectfully disagree. When 100,000 people respond to this question and only a fraction of them get it right, this is an indication that the equation is poorly written. It may be unambiguous to a person who has a proper understanding of precedence rules, but it is obvious that this is a fraction of the general population. So if I was writing this for a general audience, I would include brackets and parenthesis to ensure there would be less chance of misinterpretation.

Respectfully,

Brian P.

[AsteriskAmp]

Yes. it should be ... let's see ...

(6 - (1 * 0)) + (2/2).

which resolvs to
(6 - 0 ) + (2/2)

which gives us

6+1 = 7.

...

I wonder if there is a web site which does drills so I can brush up on this?

Respectfully,

Brian P.

Kahn Academy is great for all math related learning.

I must respectfully disagree. When 100,000 people respond to this question and only a fraction of them get it right, this is an indication that the equation is poorly written. It may be unambiguous to a person who has a proper understanding of precedence rules, but it is obvious that this is a fraction of the general population. So if I was writing this for a general audience, I would include brackets and parenthesis to ensure there would be less chance of misinterpretation.

Respectfully,

Brian P.

No, it means many of them have deficient understanding of math, either due to a failure to properly understand the concepts involved or due to wilful ignorance; not that it is badly written. The operation is correct, those answering are wrong, in this case interpretation is unique, people are just plain wrong.

While in literature one could ask for a simpler vocabulary (at the cost of losing the poeticalness and layers of interpretation) in mathematics asking to make it easier to solve is indulging in wilful ignorance; it is not the writer's fault but the audience for not knowing something which is taught in elementary school.

[Rawhide]

Question to those thinking it's imprecise.

Would ((((1+2)+3)+4)+5)= be better than 1+2+3+4+5=?

Both of them result in the same answer, and the operations are performed in the same order (which is the important bit), one just tells you how to do it by brackets, while the other relies on the order (and I think we'd agree that the first is just silly).

6-1*0+2/2= is perfectly valid and will always result in the same answer if you follow the correct order. There's no point during the equation where, by the order of operations, you could do it one way or another.

Personally, I'd probably include the brackets, but they aren't required.

[TheFallenOne]

This 'proper understanding of precedence rules' is something one should have by 5th grade. You can't solve an equation with more than one operator without it(!)

It is shocking so many people seem to get it wrong, but that doesn't mean we should add brackets left and right as training wheels. Rather, think about where things went wrong. I don't think you're flat-out stupid, but something in your math education or memory thereof went terribly wrong. I'd advise you to take a math book that starts at zero and go through that. You need to get the bare basics down or anything involving numbers will be troublesome.

[pendell]

Question to those thinking it's imprecise.

Would ((((1+2)+3)+4)+5)= be better than 1+2+3+4+5=?

No, because all operations are at the same precedence level. 1+2+3+4+5 is clear and unambiguous without the need for brackets.

No, it means many of them have deficient understanding of math, either due to a failure to properly understand the concepts involved or due to wilful ignorance

More likely because those reading do not apply precedence operations on a daily or even a weekly basis. Don't confuse lack of practice because it's not necessary with willful ignorance.

Speaking as a sometime author, I contend that it is the responsibility of the author to ensure that his meaning is clearly and correctly understood by the audience. If he is writing over their heads, it is his responsibility, not theirs, to lower the exchange until they understand what he's saying.

Insulting the audience is also a poor way to win their loyalty, I believe.

That is why I believe the writing of equations is an art form. It is possible, by not using brackets, to allow normal people to misunderstand an equation. It is also possible, by over-use of brackets, to obscure what is normally obvious.

I am a software person and I deal with general business types all the time. We frequently have mis-communications about requirements. Nothing quite on this level but bad enough. And I will say from personal experience that "the other person is too stupid to understand what I'm saying" is not a valid answer. I have a responsibility on my part to ensure that my audience understands exactly what I am writing , to the extent it depends on me, and that means writing to their level. Even if it means using what some would consider training wheels. Because when talking with normal business types about software , it is all over their head, outside their area of expertise. Since they don't understand my AOE and I do, the burden is on me to make this intelligible to them, not vice versa.



Respectfully,

Brian P.

[Yora]

It's not badly written. 6-1*0+2/2 is perfectly valid and unequivocal.

Would writing it the other way make it easier to solve for those who don't really understand operation order?

It's not about understanding, but about the existing of several conflicting operation orders.

[Rawhide]

Precedence order exists for a reason, so that brackets aren't required. While my example uses only addition, it highlights how crazy things get if you can't rely on the precedence order.

6-1*0+2/2= is clear and unambiguous without the need for brackets.

[WalkingTarget]

It's not imprecise. Any brackets (correctly) placed in that equation can be removed because they're trivial.
[snip]
Incidentally, you placed the brackets wrong.
This would give 5.

Well, it's not that any brackets/parentheses are trivial; it's that whatever brackets/parentheses you add makes it a different equation. There is (at least one) set of them that results in the same equation as is present without them, but it's not that other placements are "wrong" unless you know what answer you want at the end.

Whoops - misread this as a general statement about them, not as a statement about "that equation" in particular.

[nedz]

It's what's known as a parsing problem.

For instance a 'pocket' calculator would assume the following bracketing

(((6-1)x0)+2)/2 which equals 1

rather than (the correct)

6-(1x0)+(2/2) which equals 7

Out of interest what are the proportions of the different answers on facebook ?

[Rawhide]

Well, it's not that any brackets/parentheses are trivial; it's that whatever brackets/parentheses you add makes it a different equation. There is (at least one) set of them that results in the same equation as is present without them, but it's not that other placements are "wrong" unless you know what answer you want at the end.

There is one set of brackets which will not only result in the same answer, but the same order of operations. i.e. It will not change the order you perform the operations in.

This is the bit that matters - that the order remains the same.

[pendell]

Precedence order exists for a reason, so that brackets aren't required. While my example uses only addition, it highlights how crazy things get if you can't rely on the precedence order.

6-1*0+2/2= is clear and unambiguous without the nead for brackets.

With respect, Rawhide, when 30,000+ out of 100,000 read that and come out with the wrong answer, I contend that is not , in fact, the case.

When thousands of people can copy-paste that equation into one of the most popular calculator programs and get the wrong answer , I contend it requires revision.

My idea of "clear and unambiguous" is that when 100,000 people read the equation, 99,000 come out with the right answer. 99,000 people of average intelligence and education.

If i write

2+2 = ?

How many wrong answers am I going to get from 100,000 average people?

I don't know about the other people here, but if I write something and a significant part of my audience misunderstands what I have written, I'm not going to think "what a stupid and uneducated audience." I'm going to think "I need to make this more clear".

This is a subject of note to me as some software I have written is intended to be used by people with a high school education, no previous computer experience, and do not speak English as a first language. I have to revise all kinds of things because what I have written, though correct, is not intended to be read by college graduates , and I can't expect minimum wage employees to take college courses simply to use my software.

Out of interest what are the proportions of the different answers on facebook ?
_______

Regrettably, no one seems to have tabulated and I myself do not have time to do so by hand. Does anyone have a suggestion for a software solution to do this?

Respectfully,

Brian P.

[factotum]

I have to agree with Brian here--when so many people are misunderstanding what you've written, it's better to assume that what you've written isn't clear than to assume everyone reading is an ignoramus.

[AsteriskAmp]

More likely because those reading do not apply precedence operations on a daily or even a weekly basis. Don't confuse lack of practice because it's not necessary with willful ignorance.

It is still a trivial procedure taught throughout basic education. There is no need to practice it, it is simple logic.

Speaking as a sometime author, I contend that it is the responsibility of the author to ensure that his meaning is clearly and correctly understood by the audience. If he is writing over their heads, it is his responsibility, not theirs, to lower the exchange until they understand what he's saying.

The meaning is clear, there is no ambiguity, or possibility for misunderstanding. Getting it wrong is not the author's fault, it's the readers fault for not knowing something taught throughout basic education, education pre-required for any career path.

Insulting the audience is also a poor way to win their loyalty, I believe.

It's not insulting the audience so much as a minimum expectation. It's not area of expertise related, this is something taught along with basic reading comprehension skills.

That is why I believe the writing of equations is an art form. It is possible, by not using brackets, to allow normal people to misunderstand an equation. It is also possible, by over-use of brackets, to obscure what is normally obvious.

It is not. It is a simple procedure which can be embellished or not depending on the writer, but it is a mathematical procedure and unequivocal.

I am a software person and I deal with general business types all the time. We frequently have mis-communications about requirements. Nothing quite on this level but bad enough. And I will say from personal experience that "the other person is too stupid to understand what I'm saying" is not a valid answer. I have a responsibility on my part to ensure that my audience says exactly what I am writing , to the extent it depends on me, and that means writing to their level. Even if it means using what some would consider training wheels. Because when talking with normal business types about software , it is all over their head, outside their area of expertise. Since they don't understand my AOE and I do, the burden is on me to make this intelligible to them, not vice versa.



Respectfully,

Brian P.

Order of Operations is a basic mathematical skill, it's not a Lebesgue Integral or programming in Python. It's something everyone actually is taught at elementary school, and belongs to no AOE, it's a necessary universal knowledge. If you got to university or college you had to know this.

It's not about understanding, but about the existing of several conflicting operation orders.

There are no conflicting orders of operations, just wrongly done mnemonics in place of a correct understanding. The order or system is one, it's the mnemonics which are wrong or the understanding of them which producers errors.

[tensai_oni]

I'm sorry, but if people cannot get such a simple equation right, this is not the fault of mathematics themselves - they do not have to be bogged down by unnecessary brackets, rather people need to get better in maths as a whole.

If anything, this is a social experiment that shows a person on average does not know mathematics very well. And we exist in a society that allows, even encourages that. If you fail a basic history or biology question, like saying World War II started in year 1549 or that dogs are fish, you'll get laughed at (and rightfully so). But with harmful notions such as "math sure is haaard" or "you don't need to know math, it's useless since I have a calculator" being so prevalent, especially in media (it doesn't help that writers tend to have studied humanities and thus didn't have much contact or liked mathematics very much), and with the associated stereotype of someone math-savvy being an uptight dork with no fashion sense or humor; the society creates a situation where for a normal, peer-pressurized individual it becomes TRENDY and DESIRED to be math-illiterate.

And that has to change. Also, I have to write shorter sentences.

By the way, substraction and addiction exist on the same level because they're the same thing. 6-2 is the same as 6+(-2). Same for multiplication and division - 4 /2 is the same as 4 x 0.5.

[TheFallenOne]

Speaking as a sometime author, I contend that it is the responsibility of the author to ensure that his meaning is clearly and correctly understood by the audience. If he is writing over their heads, it is his responsibility, not theirs, to lower the exchange until they understand what he's saying.

Almost. You are responsible to make what you write understandable to the target audience. You are NOT obligated to pander to the lowest common denominator so everyone who happens to read what you write can understand it.

Various class/optimization guides would be completely incomprehensible to someone who only knows D&D as 'that weird thing geeks do'. But that's alright, they are not written for them. A book dealing with university-level math, physics, chemistry etc will be too much for someone who just heard about integrals for the first time. That's OK, it's not for him.

Unless you are writing a children's book, assuming the target audience has the understanding of math expected of a 10 year old in school is perfectly acceptable.

Well, it's not that any brackets/parentheses are trivial; it's that whatever brackets/parentheses you add makes it a different equation.

Funnily enough, you neglected the word I placed in brackets. Any bracket you correctly add to an equation is trivial; if you can add it without changing the equation, you can remove it again without doing so either.

[Rawhide]

When thousands of people can copy-paste that equation into one of the most popular calculator programs and get the wrong answer , I contend it requires revision.

If you don't run Windows Calculator in Scientific mode, you will always get the wrong answer - with or without brackets.

Windows Calculator in Standard mode:
6-1*0+2/2=1
6-(1*0)+(2/2)=1

(Note, both are the wrong answer.)

[pendell]

Almost. You are responsible to make what you write understandable to the target audience. You are NOT obligated to pander to the lowest common denominator so everyone who happens to read what you write can understand it.

Agreed.

Unless you are writing a children's book, assuming the target audience has the understanding of math expected of a 10 year old in school is perfectly acceptable.

Note that in the OP I myself messed up the equation.

I contend that there are things which stick with most people and there are minutiae that doesn't. Order of precedence operation is one of the things that go by the boards because most people don't practice with them regularly. Correction. Order of precedence operations which involve multiple precedence orders in the same equation as in the OP. I think most people could correctly figure out 2+3*4 = 14. But when you have three or four precedence rules on the same line, ordinary people get confused.

When writing for a general audience of adults in the age range of 20-40, I would assume the reader can do basic arithmetic well enough to purchase products in a store. I can't even assume they know how to balance their checkbooks. At a previous church there was a guy who did that as part of his ministry -- visit people at their homes and balance their checkbook for them, because they didn't know how .

I would introduce algebra for a hard science fiction novel.

If anything, this is a social experiment that shows a person on average does not know mathematics very well.

Agreed. And I don't see this changing howsoever much we try. I suspect that's one of the reason roleplaying isn't popular outside computer games -- because average people simply don't have the time or the patience to calculate d20 probabilities.

ETA: Incidentally, the original poster of the equation on facebook is "rofl". Which suggests to me that the original author knew damn well that it would confuse most of his audience.
Respectfully,

Brian P.

[WalkingTarget]

Funnily enough, you neglected the word I placed in brackets. Any bracket you correctly add to an equation is trivial; if you can add it without changing the equation, you can remove it again without doing so either.

Actually, what I missed was the word "that" - as in this specific equation, not equations in general. This changed my understanding of your point. Mea culpa.

[Elder Tsofu]

Can just say that I didn't have any problem with the equation except reading your reasoning. ^^
It is an easy question, although the "0" seem unnecessary if it isn't an "hidden" x or something like that. Basic mathematics.

I suppose people can forget, but its such a basic thing that I can't really see how people have managed to get passing grades in the later stages of early math-education without a firm grasp of it.
Although I suppose that being on facebook is in no way a guarantee that the person in question passed any of their math tests. ^^

Regardless the most basic way of writing would nonetheless be 7=?
I wonder how many people would give the wrong answer to that one...

Edit: Good text below.

[TheFallenOne]

Let me share a little anecdote about a classmate. She always was good in school, teachers liked her, great grades(though I contest this was more a measure of the amount of work she invested than actual intelligence, but that's beside the point).

Anyway, 10th grade, we were reading and interpreting a poem called Hiroshima. Middle of the lesson she raises her hand and asked what Hiroshima is. She honestly didn't have the slightest idea. I don't think I've ever been that speechless in my life.

Point being, even supposedly intelligent and educated people can have shocking, I daresay inexcusable gaps in knowledge.

When this happens to us we should feel bad, even ashamed, and then move to better ourselves. One such failure, embarrassing as it it, doesn't make us an idiot for all time. Neglecting to learn from it, strive to avoid such things in the future does. Pendell, you did something very bad when your lack of basic math was pointed out - you got defensive about it. You tried to find fault in something else, namely the structure of the equation. You tried to downplay the importance of knowing it or that it's a matter of course in the first place. I've seen people with reasoning like that before. If you don't know something that's common knowledge, you're an idiot. I'm no idiot and I didn't know it, so people can't be expected to know it.

Don't do that. Be ashamed. Yes, be very ashamed. Then grab a math book, fix the poblem and let it never happen again. It will make you a better and more educated person. We can overcome failure. But we can't overcome ignorance of our failure.

[Coidzor]

Reverse Polish Notation clears everything up, but it's mostly for computer science, IIRC.