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 (http://en.wikipedia.org/wiki/Order_of_operations) 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 (https://www.facebook.com/photo.php?fbid=506224432721082&set=a.192261550784040.53172.192241490786046&type=1), 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 (http://www.google.com.au/webhp?q=6-1*0%2B2%2F2%3D#hl=en&output=search&sclient=psy-ab&q=6-1*0%2B2%2F2%3D&oq=6-1*0%2B2%2F2%3D) will give you the same answer, as will Wolfram Alpha (http://www.wolframalpha.com/input/?i=6-1*0%2B2%2F2%3D&dataset=).

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.
(http://www.khanacademy.org/math/arithmetic/order-of-operations/e/order_of_operations)

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 (http://articles.cnn.com/1999-09-30/tech/9909_30_mars.metric_1_mars-orbiter-climate-orbiter-spacecraft-team?_s=PM:TECH) 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 (http://articles.cnn.com/1999-09-30/tech/9909_30_mars.metric_1_mars-orbiter-climate-orbiter-spacecraft-team?_s=PM:TECH) 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.

[DeusMortuusEst]

I haven't looked at the original facebook post, since I'm at work, but if it is true that 3/10 can't get this right then I'm shocked and a little bit upset.

This is not hard math. It is not badly written (apart from the x. I though that it was a variable, but from the rest of the thread I guess it is meant as a multiplication sign.) and failing to interpret this correctly does not depend on the equation itself.

In fact it is very straight forward. It is something that you learn how to do at a very young age and then practice for several years.

That so many are ignorant of how to do basic calculations is frightening.

[Coidzor]

One part calculator dependence and increased ubiquity of calculators at younger and younger ages...

Another part that math we need to actually do in our daily lives for most people has been greatly simplified, much like how most reading materials are at or below a 5tth grade reading level or so. So precedence rules generally don't come up and so if you've not used order of operations in over a decade or several decades in the case of older internet users, it makes sense that there would be some difficulty in recalling them.

I concur that it is rather frightening though.

[Brother Oni]

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.


I'd personally go for 95,000 (5% failure rate), since this is likely to follow a Gaussian distribution. I'm fairly sure that more than 1 in 100 people are going to get the answer wrong, either wilfully, lack of education, mis-comprehending or misreading of the question, or simple brain fart.

[Coidzor]

Plus there's being in a rush because it's facebook.

[nedz]

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.)
Calculators tend to evaluate X<op>Y left to right.


One part calculator dependence and increased ubiquity of calculators at younger and younger ages...

Its possible they (mis-) used a calculator, well see above (several posts)


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?
I don't use Facebook, is it possible to get an extract ?


Reverse Polish Notation clears everything up, but it's mostly for computer science, IIRC.
Or old HP style calculators.

The standard implementation (since the days of Fortran) is to convert infix to postfix, and then process the resultant stack.

Infix: 6,-,1,x,0,+,2,/,2,=
Postfix: 6,1,0,x,-,2,2,/,+,=

[The Extinguisher]

I honestly don't see what the harm of putting brackets in the equation here is? Really, other than making people look dumb what does it accomplish?

Also, I would like to say as someone who does a lot of maths, order of operations doesn't really come up as often as people here seem to think. It's good to know, I guess, but no mathematician is ever going to write 6-1*0+2/2, they're just going to write 7.

[AsteriskAmp]

I honestly don't see what the harm of putting brackets in the equation here is? Really, other than making people look dumb what does it accomplish?It shouldn't be needed, even if to some it would be more comfortable to work with.

Also, I would like to say as someone who does a lot of maths, order of operations doesn't really come up as often as people here seem to think. It's good to know, I guess, but no mathematician is ever going to write 6-1*0+2/2, they're just going to write 7.No, but you DO have things similar in form like
f(t)=6-v0*t+v1/v3
Evaluate at t=0 if
and x=0 for t=0
v0=1-x
v1=2x+2
v3=2

Not actual equation for anything; t isn't time, v isn't velocity, x is arbitrary variable.

[Neftren]

It shouldn't be needed, even if to some it would be more comfortable to work with.
No, but you DO have things similar in form like
f(t)=6-v0*t+v1/v3
Evaluate at t=0 if
and x=0 for t=0
v0=1-x
v1=2x+2
v3=2

Not actual equation for anything; t isn't time, v isn't velocity, x is arbitrary variable.

With the advent of computer software packages such as Mathematica and MATLAB, I don't think I've actually bothered to write out the work anymore. Pop formula into Mathematica, setup inputs, Solve[f(x) == y, 0] and so on. As The Extinguisher pointed out, people who frequently use mathematics would be more likely to either write "7" in the above example, or express things in terms of other things (e.g. 3∏/2), in which you avoid all the sticky bits of excusing our dear aunt Sally.

[The Extinguisher]

It shouldn't be needed, even if to some it would be more comfortable to work with.
No, but you DO have things similar in form like
f(t)=6-v0*t+v1/v3
Evaluate at t=0 if
and x=0 for t=0
v0=1-x
v1=2x+2
v3=2

Not actual equation for anything; t isn't time, v isn't velocity, x is arbitrary variable.

Why not though? There hasn't been a real reason why we shouldn't use brackets other than "people shouldn't be so dumb they don't know basic math." Not using brackets is great for drilling students on order of operations rules, but it's not useful for anything else. This does create confusion, don't kid yourself on that. Anything that creates confusion shouldn't be tolerated in math. It needs to strive to be as concise as possible.

[Murska]

Well, if you assume that the equation is trivial in nature, then writing additional brackets is not only more work and a waste of time and energy but can also make it more confusing, as people have to do a double-take and go "Why are there brackets here? Am I missing something?"

[AsteriskAmp]

With the advent of computer software packages such as Mathematica and MATLAB, I don't think I've actually bothered to write out the work anymore. Pop formula into Mathematica, setup inputs, Solve[f(x) == y, 0] and so on. As The Extinguisher pointed out, people who frequently use mathematics would be more likely to either write "7" in the above example, or express things in terms of other things (e.g. 3∏/2), in which you avoid all the sticky bits of excusing our dear aunt Sally.
That you can use software for making calculations faster isn't excuse to not know something as simple as Order of Operations. Why learn to read if there are text to sound programs? Or learn to read music if there are midi reading software which produce sound? Or learn another language since you have google translate?

Each time you do more than two operations you have Order of Operations, even if not so glaringly patent. As for representation, yes, someone would write seven in that case, but in the case of operations with name-worthy constants and variables you would need to use Order of Operations if you wanted to solve in relation to one or another, or just change the form to something you actually want, of which Mathematica and Matlab may not always do the last one.

Why not though? There hasn't been a real reason why we shouldn't use brackets other than "people shouldn't be so dumb they don't know basic math." Not using brackets is great for drilling students on order of operations rules, but it's not useful for anything else. This does create confusion, don't kid yourself on that. Anything that creates confusion shouldn't be tolerated in math. It needs to strive to be as concise as possible.There is no real reason to use it either other than people could get confused about something rather trivial.

It may create confusion to some, but it IS concise, there is no multiple value answer or misinterpretation on representation, it's human error which leads to a wrong result. I acknowledge it is an order of operations designed problem and normally anyone would express it differently (starting with the fact that you almost never see / or ÷ but instead it would be seen as a fraction, and ending with the fact that it indeed is just 7) but that doesn't mean it is wrong the way it is, it's another perfectly valid representation, same thing saying three is 3, 4-1, d(x^2+x+1)/dx|x=1, 3sin(Pi/2), 2.999999999..., etc.

[Tengu_temp]

Why not though? There hasn't been a real reason why we shouldn't use brackets other than "people shouldn't be so dumb they don't know basic math."

And that's a bad reason? People shouldn't be so dumb that they don't know basic math.

[nedz]

It does come up sometimes, but normally not with something so trivial.

If you just blithely type this into a calculator you might not realise that it's order of evaluation is
6,-,1,=,x,0,=,+,2,=,/,2,=
which is probably what caught most people out.

[pendell]

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.


Not so. I made my best guess based on memory, then asked questions, did the research , found my mistake, reworked to get the correct answer, solved it while noting the previous error. This is what research is all about.

I'm also, so far as I know, the only person in this thread to admit being wrong.

The equation that I pointed out was deliberately written to make people look stupid by relying on obscure rules of precedence which most people do not recall. Yes, such things are taught in grade school alongside things such as split infinitives, the use of 'who' vs. 'whom', what a gerund is, and how to diagram a sentence. Most people forget most of that but retain enough skill to function in their chosen professions.

If I were the only person who had made that mistake, then I would shut up and take it in good grace. Since I am by no means the only person who made that mistake, then I state that the equation was deliberately written to trip people up as the intended effect and did so.

I'll also thank you not to criticize me, and especially not publicly. Critique my silly ideas and statements, yes, by all means. But when you cross the line from attacking ideas to personal attack, I do not take it at all well. Most especially since :checks the birthdates on the profiles: -- I do not take being patronized at all well from someone fourteen years younger than I am.



This does create confusion, don't kid yourself on that. Anything that creates confusion shouldn't be tolerated in math. It needs to strive to be as concise as possible.


Agreed.

Respectfully,

Brian P.

[Iruka]

The equation that I pointed out was deliberately written to make people look stupid by relying on obscure rules of precedence which most people do not recall.

Wait, multiplication/division before addition/substraction is obscure now? :smallconfused:

[AsteriskAmp]

I'm also, so far as I know, the only person in this thread to admit being wrong.To a mathematical certainty. While in all other cases the discussion has been about whether it is "necessary" to include non-essential parenthesis and

The equation that I pointed out was deliberately written to make people look stupid by relying on obscure rules of precedence which most people do not recall. Yes, such things are taught in grade school alongside things such as split infinitives, the use of 'who' vs. 'whom', what a gerund is, and how to diagram a sentence. Most people forget most of that but retain enough skill to function in their chosen professions. They are not obscure, they are fairly common and required for every single mathematical operation ever (just to a different degree). What it relies on are the wrongly taught mnemonics for them which are something which should at the very least be corrected, what's wrong isn't the notation but the way the concepts are being taught.

If I were the only person who had made that mistake, then I would shut up and take it in good grace. Since I am by no means the only person who made that mistake, then I state that the equation was deliberately written to trip people up as the intended effect and did so.Or you happen to have the same reasons for making the same mistake. Mathematics is not based on population consensus. 1+1 won't become 3 because everyone makes the same mistake, otherwise we'd be voting to see which values certain series take.

[Castaras]

Wait, multiplication/division before addition/substraction is obscure now? :smallconfused:

Pretty much. There's been some examples on facebook which were deliberately misleading, but the example you posted isn't. It's basic knowledge that for those who're even slightly good at maths should be fine with doing automatically.

[TheFallenOne]

If I were the only person who had made that mistake, then I would shut up and take it in good grace. Since I am by no means the only person who made that mistake, then I state that the equation was deliberately written to trip people up as the intended effect and did so.

I'll also thank you not to criticize me, and especially not publicly. Critique my silly ideas and statements, yes, by all means. But when you cross the line from attacking ideas to personal attack, I do not take it at all well. Most especially since :checks the birthdates on the profiles: -- I do not take being patronized at all well from someone fourteen years younger than I am.

I went out of my way to point out you're NOT stupid as I think I remember you writing more agreeable things before. I'm willing to bet some people came to different conclusions in light of this thread, but don't say so for obvious reasons. I was trying to help you in dealing with this, and there is no way around it, embarrassing failure.

And despite your 'Not so', you are doing exactly what I said you do: trying to downplay the whole thing. This isn't an obscure piece of knowledge by any reasonable standard.

[pendell]

I don't use Facebook, is it possible to get an extract ?



I just checked. There are 129,953 comments. I pulled the last 156. Of which, 138 had answers

The raw data is in the spoilers.

7
1
1
1
7
5
0
7
5
0
7
7
5
5
7
7
7
3.5
1
0
5
7
1
1
1
7
2
1
1
1
7
1
1
7
7
5
5
1
1
7
1
1
0
7
1
1
1
7
7
5
1
1
5
1
1
7
4
7
7
7
7
7
0
7
7
1
7
7
1
1
5
7
1
5
0
1
7
5
5
1
5
7
1
2
7
1
5
6
5
4
7
7
7
4
7
1
1
7
1
5
7
1
1
5
4
1
7
5
5
1
1
6
6
14
7
1
1
1
7
7
1
1
1
1
1
1
1
1
7
1
1
1
7
7
7
1
1
1


I summarize:

138 answers
6 answered 0
56 answered 1
2 answered 2
1 answered 3.5
4 answered 4
20 answered 5
3 answered 6
45 answered 7
1 answered 14

so ... 45/138 = 0.32 * 100 = 32% got the correct answer. If some kind person were to take a histogram , we should see a cluster around 5 and 7, another big spike around 1, and outliers everywhere else.

Does this imply that 68% of the population is so deficient in math skills that they do not understand primary school math? Well, the mean SAT score (http://nces.ed.gov/fastfacts/display.asp?id=171) for mathematics is 501 out of a possible 800. This would imply (to me) that the average person is conversant with primary school math but at sea when it comes to secondary school mathematics.

So I conclude that the fault is with the writer of the equation. Yes, it may be technically clear and unambiguous. And if you're writing for an audience of mathematicians , that is probably good enough. But when only 1/3rd of your target audience is able to come to the correct answer, the logical conclusion is that either the person has failed to communicate clearly or that this is a deliberate trick question. Given the identity of the author, it is almost certainly the second.



1+1 won't become 3 because everyone makes the same mistake, otherwise we'd be voting to see which values certain series take.


I quite agree. Nonetheless, an equation can deliberately be written to be more obscure than strictly necessary.

1+1 = 2
is the same as
(3^0) + (natural log of e) = ((10/10) * (1000) ) / 500

but the first is much more clear. As I stated, there is an art to writing an equation such that it is as clear as possible to as many readers as possible. The equation in the OP could be immeasurably improved for general consumption by an audience of high school graduates by the addition of brackets.


Respectfully,

Brian P.

[noparlpf]

Recently I saw the following on facebook:



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.

Well, in the US it's always, always PEMDAS. BUT people forget that M and D are considered equivalent, and A and S are considered equivalent; when it comes down to them, just go left to right in order. More properly, it's PE(M/D)(A/S). It's just retained as PEMDAS because that's easy to say.
Seems like BODMAS (could equally well be written BOMDAS) is just the European way of saying it?
Anyway, I kind of hate these things floating around facebook that are about basic order of operations--one, they cause huge arguments, and people are asinine enough without provocation, and two, they're designed to make people feel stupid.
For your example, I get 6 - 1 x 0 + 2 / 2 = 6 - 0 + 1 (and here we just go left to right, so it's 6-0 first) = 6 + 1 = 7.
My TI-84 says it's 7, and WolframAlpha says it's 7.

[pendell]

I went out of my way to point out you're NOT stupid as I think I remember you writing more agreeable things before. I'm willing to bet some people came to different conclusions in light of this thread, but don't say so for obvious reasons. I was trying to help you in dealing with this, and there is no way around it, embarrassing failure.


Then I will accept your feedback in the spirit it was given and leave it at that. Thank you.

At any rate, now I know the answer is 7 and why and what I did wrong the first time. For my purposes, that is sufficient. As far as I'm concerned, if a little embarrassment leads to a more correct answer it is a price well worth paying.

Respectfully,

Brian P.

[TheFallenOne]

Does this imply that 68% of the population is so deficient in math skills that they do not understand primary school math?

If the Facebook posters there were a representative group of the general population, unable to see previous comments or use the internet before answering, and we knew they weren't drunk, high, in a hurry or trolling when answering, then yes, it would.


1+1 = 2
is the same as
(3^0) + (natural log of e) = ((10/10) * (1000) ) / 500

but the first is much more clear. As I stated, there is an art to writing an equation such that it is as clear as possible to as many readers as possible.

You do realize that reducing a complicated formula to a way simpler one is a big part of school mathematics? Of course 1+1=2 is easier to grasp at a glance. That doesn't make a more complicated equation illegitimate.

[AsteriskAmp]

I just checked. There are 129,953 comments. I pulled the last 156. Of which, 138 had answers

The raw data is in the spoilers.

7
1
1
1
7
5
0
7
5
0
7
7
5
5
7
7
7
3.5
1
0
5
7
1
1
1
7
2
1
1
1
7
1
1
7
7
5
5
1
1
7
1
1
0
7
1
1
1
7
7
5
1
1
5
1
1
7
4
7
7
7
7
7
0
7
7
1
7
7
1
1
5
7
1
5
0
1
7
5
5
1
5
7
1
2
7
1
5
6
5
4
7
7
7
4
7
1
1
7
1
5
7
1
1
5
4
1
7
5
5
1
1
6
6
14
7
1
1
1
7
7
1
1
1
1
1
1
1
1
7
1
1
1
7
7
7
1
1
1


I summarize:

138 answers
6 answered 0
56 answered 1
2 answered 2
1 answered 3.5
4 answered 4
20 answered 5
3 answered 6
45 answered 7
1 answered 14

so ... 45/138 = 0.32 * 100 = 32% got the correct answer. If some kind person were to take a histogram , we should see a cluster around 5 and 7, another big spike around 1, and outliers everywhere else.

Does this imply that 68% of the population is so deficient in math skills that they do not understand primary school math? Well, the mean SAT score (http://nces.ed.gov/fastfacts/display.asp?id=171) for mathematics is 501 out of a possible 800. This would imply (to me) that the average person is conversant with primary school math but at sea when it comes to secondary school mathematics.

So I conclude that the fault is with the writer of the equation. Yes, it may be technically clear and unambiguous. And if you're writing for an audience of mathematicians , that is probably good enough. But when only 1/3rd of your target audience is able to come to the correct answer, the logical conclusion is that either the person has failed to communicate clearly or that this is a deliberate trick question. Given the identity of the author, it is almost certainly the second. It means exactly that though. A large share of the population lacks a correct understanding of primary school math.

The SAT results (which are also not accessible to me for some reason) only prove of how well a population of United States High Schoolers can perform a standardized test most schools prepare them for, instead of actually teaching them the skills which would aid them in the test but also serve for different situations.

Or, a third interpretation, said people have learnt the concepts wrong and hence all are making mistakes. Don't blame the writer, blame the way it is almost universally taught through (incorrect) mnemonics.

I quite agree. Nonetheless, an equation can deliberately be written to be more obscure than strictly necessary.

1+1 = 2
is the same as
(3^0) + (natural log of e) = ((10/10) * (1000) ) / 500

but the first is much more clear. As I stated, there is an art to writing an equation such that it is as clear as possible to as many readers as possible. The equation in the OP could be immeasurably improved for general consumption by an audience of high school graduates by the addition of brackets. Alternatively 3^0+ln(e) = 10÷10*1000÷500

The clearest way of writing that out would have been 7=?. The purpose of those expressions are to show how a large share of the population lacks basic mathematical knowledge. Even then. odds are you'd see a part of the population get it wrong anyway.

[pendell]

That doesn't make a more complicated equation illegitimate.


Agree. But "illegitimate" and "clear/easy to follow" are not the same thing.

An equation may be legitimate and yet still be hard to follow and badly written. Likewise , an equation may be simple, clear , easy to follow, and wrong.

I think that is the heart of our disagreement. I do not view mathematics simply as whether an equation is accurate or not. I believe mathematics is in part language -- a way for people to express ideas to each other. Consequently I look for ways not just to express accurate ideas, but to express them as clearly as possible with a minimum of interpretation errors. In the above case, I believe the equation was written to maximize reader error. It is apparently simple, yes, but when 68% of the readers get it wrong, that tells me it is deceptively simple. It is a trick question, which is possible both in English and in mathematics.

Respectfully,

Brian P.

[nedz]

138 answers
6 answered 0
56 answered 1
2 answered 2
1 answered 3.5
4 answered 4
20 answered 5
3 answered 6
45 answered 7
1 answered 14

so ... 45/138 = 0.32 * 100 = 32% got the correct answer. If some kind person were to take a histogram , we should see a cluster around 5 and 7, another big spike around 1, and outliers everywhere else.

Does this imply that 68% of the population is so deficient in math skills that they do not understand primary school math? Well, the mean SAT score (http://nces.ed.gov/fastfacts/display.asp?id=171) for mathematics is 501 out of a possible 800. This would imply (to me) that the average person is conversant with primary school math but at sea when it comes to secondary school mathematics.

So I conclude that the fault is with the writer of the equation. Yes, it may be technically clear and unambiguous. And if you're writing for an audience of mathematicians , that is probably good enough. But when only 1/3rd of your target audience is able to come to the correct answer, the logical conclusion is that either the person has failed to communicate clearly or that this is a deliberate trick question. Given the identity of the author, it is almost certainly the second.

Looking at this from an error analysis point of view:

56 (40%) probably used a calculator, badly
20 (14.5%) made the sign error (sign errors are very common)
45 (23.5%) got the correct answer
11 (8%) some random error, either hopeless or just messing around.
6 (4%) entered 0 (I'm not sure why here)

I think its quite an interesting maths question in that it exposes several common errors. Is it a trick question ? Or is it trying to teach something ?

[noparlpf]

I think all the analysis of the facebook answers is flawed--you're forgetting the huge population of people who don't care and just put something random down to cause trouble, and the smaller population of people who do know but are putting down incorrect answers to cause trouble.

So we have a few groups.
1. People who can't do grade school-level math.
2. A few subgroups that boil down to people who are just being asses. (I don't like the word "trolling". Call it what it is--being a ****.)
3. People who misread the problem, or used a calculator and mistyped it.
4. People who can do grade school-level math.

[pendell]

I think its quite an interesting maths question in that it exposes several common errors. Is it a trick question ? Or is it trying to teach something ?


Most likely the first. Still, you've a good point about error analysis. If it is trying to teach something, it's probably "Don't just copy/paste an equation into a calculator and expect the right answer. THINK about it. "

ETA: I'm not familiar with error analysis. May I ask how you came to those conclusions?

Respectfully,

Brian P.

[nedz]

I'm not familiar with error analysis. May I ask how you came to those conclusions?


I'm just guessing, but it is sort of what I do for a living :smallsmile:

[Knaight]

Not so. I made my best guess based on memory, then asked questions, did the research , found my mistake, reworked to get the correct answer, solved it while noting the previous error. This is what research is all about.

I'm also, so far as I know, the only person in this thread to admit being wrong.
You also appear to have immediately jumped to the conclusion that admitting being wrong somehow grants you a moral authority over those who have not admitted being wrong on account of not being wrong in the first place. It does no such thing, and certainly doesn't substitute for an actual argument.


I'll also thank you not to criticize me, and especially not publicly. Critique my silly ideas and statements, yes, by all means. But when you cross the line from attacking ideas to personal attack, I do not take it at all well. Most especially since :checks the birthdates on the profiles: -- I do not take being patronized at all well from someone fourteen years younger than I am.
Similarly, "I'm older than you" is not a preposition from which "I'm better than you" can be directly derived from reasoning that isn't ridden with flaws. Your demand for a raised discourse rings hollow when you implicitly insult everyone younger than you.


The SAT results (which are also not accessible to me for some reason) only prove of how well a population of United States High Schoolers can perform a standardized test most schools prepare them for, instead of actually teaching them the skills which would aid them in the test but also serve for different situations.

More to the point, a 514 out of 800 on the SAT math section is an indication of mathematical deficiency given the current difficulty of the SAT math section*. It's a mixture of fairly basic arithmetic, fairly basic geometry, and extremely trivial algebra. It also briefly touches upon trigonometric functions and matrices, though there were few enough of them as recently as 2010 to reduce a score more than 40 or 50 points were every single one of them failed. That doesn't explain the loss of 286 points on average, let alone the loss of 386-401 points for approximately 15.9 percent of the test takers, assuming a roughly normal distribution. This is consistent both with a grasp of elementary math and utter inability in higher math, or with a pattern of knowledge gaps throughout mathematical levels, the latter of which would also explain the results gathered from Facebook**, particularly once one takes into account how neither the SAT nor Facebook are measures that accurately evaluate the population, due to heavy selection biases in both.

*I found data that indicate a 514 mean in 2011, with a standard deviation of approximately 100 to 115, with different sources varying. I will thus be using these numbers.

**I have not personally confirmed these, so this is a very provisional statement.

[Felyndiira]

So I conclude that the fault is with the writer of the equation. Yes, it may be technically clear and unambiguous. And if you're writing for an audience of mathematicians , that is probably good enough. But when only 1/3rd of your target audience is able to come to the correct answer, the logical conclusion is that either the person has failed to communicate clearly or that this is a deliberate trick question. Given the identity of the author, it is almost certainly the second.
If you posted an equation with an integral on facebook and forced 100,000 people to respond to it with what they think the answer is, I would bet that a similar number of people would not be able to. This does not prove that integrals are misleading or that all calculus equations should be represented with a shaded graph - it just means that those people do not understand calculus.

If you were to ask 100,000 people what the capital of Zaire is, most of them would likely not know the answer. That does not mean that the capital of Zaire is somehow ambiguous - merely, that most people do not know the capital of Zaire for whatever reason. It does not mean that whenever you write, say, "The President went to Kinshasa" you have to write "The President went to Kinshasa, Capital of Zaire, a nation in Africa located at the middle of the continent bordering, intersecting the equator, etc. etc." even if a number of people on facebook cannot pinpoint the capital or the nation on a map.

The equation is not at all ambiguous simply because there is a very clearly defined, internationally-agreed upon procedure for the order of operations; one that is taught within elementary school. The fact that 30,000 out of 100,000 people on Facebook does not know PEMDAS does not negate the fact that it exists and is one of the basics of mathematics in the same way that x number of people not knowing the capital of Zaire doesn't make Kinshasa "obscure and unclear in communication."


Does this imply that 68% of the population is so deficient in math skills that they do not understand primary school math? Well, the mean SAT score (http://nces.ed.gov/fastfacts/display.asp?id=171) for mathematics is 501 out of a possible 800. This would imply (to me) that the average person is conversant with primary school math but at sea when it comes to secondary school mathematics.
EDIT: Corrected a few things I did not remember correctly about the SATs. You can achieve a score of around 240 by leaving the entire test blank. This means that 501 as the average score means that on average, test takers answered around 50% of the math questions correctly. That does not indicate a national competency in Math by any means.


On a completely random note, I actually think that mathematicians are more likely to disregard trivial stuff and write in absolutely ambiguous manners. I can't count how many times I wrote

log x+2

to mean log (x+2) and

log x + 2

to mean (log x) + 2 on my exams for courses like probability theory.

[noparlpf]

If you posted an equation with an integral on facebook and forced 100,000 people to respond to it with what they think the answer is, I would bet that a similar number of people would not be able to. This does not prove that integrals are misleading or that all calculus equations should be represented with a shaded graph - it just means that those people do not understand calculus.

If you were to ask 100,000 people what the capital of Zaire is, most of them would likely not know the answer. That does not mean that the capital of Zaire is somehow ambiguous - merely, that most people do not know the capital of Zaire for whatever reason. It does not mean that whenever you write, say, "The President went to Kinshasa" you have to write "The President went to Kinshasa, Capital of Zaire, a nation in Africa located at the middle of the continent bordering, intersecting the equator, etc. etc." even if a number of people on facebook cannot pinpoint the capital or the nation on a map.

...Is Zaire a place?
See, I can do math, but nobody ever taught me geography.


On a completely random note, I actually think that mathematicians are more likely to disregard trivial stuff and write in absolutely ambiguous manners. I can't count how many times I wrote

log x+2

to mean log (x+2) and

log x + 2

to mean (log x) + 2 on my exams for courses like probability theory.

I make sure to be super careful about things like that, mostly so I don't confuse myself.

[Coidzor]

...Is Zaire a place?
See, I can do math, but nobody ever taught me geography.

From about the 70s to the late 90s it was what the Democratic Republic of the Congo was called, near as Google and Wikipedia can figure.

[Heliomance]

I quite agree. Nonetheless, an equation can deliberately be written to be more obscure than strictly necessary.

1+1 = 2
is the same as
(3^0) + (natural log of e) = ((10/10) * (1000) ) / 500

but the first is much more clear. As I stated, there is an art to writing an equation such that it is as clear as possible to as many readers as possible. The equation in the OP could be immeasurably improved for general consumption by an audience of high school graduates by the addition of brackets.


Respectfully,

Brian P.

http://i13.photobucket.com/albums/a265/Heliomance/Equation.jpg

[pendell]

Heliomance: You win.

Respectfully,

Brian P.

[Rawhide]

http://i13.photobucket.com/albums/a265/Heliomance/Equation.jpg

http://mrbadak.com/img/fani/exampapers.jpg

Calculators tend to evaluate X<op>Y left to right.

I know, see my earlier post - you should always use Scientific mode, which does it correctly. Windows Calculator in Standard mode will do it incorrectly regardless of brackets, Windows Calculator in Scientific mode will do it correctly regardless of brackets.


If you posted an equation with an integral on facebook and forced 100,000 people to respond to it with what they think the answer is, I would bet that a similar number of people would not be able to. This does not prove that integrals are misleading or that all calculus equations should be represented with a shaded graph - it just means that those people do not understand calculus.

If you were to ask 100,000 people what the capital of Zaire is, most of them would likely not know the answer. That does not mean that the capital of Zaire is somehow ambiguous - merely, that most people do not know the capital of Zaire for whatever reason. It does not mean that whenever you write, say, "The President went to Kinshasa" you have to write "The President went to Kinshasa, Capital of Zaire, a nation in Africa located at the middle of the continent bordering, intersecting the equator, etc. etc." even if a number of people on facebook cannot pinpoint the capital or the nation on a map.

The equation is not at all ambiguous simply because there is a very clearly defined, internationally-agreed upon procedure for the order of operations; one that is taught within elementary school. The fact that 30,000 out of 100,000 people on Facebook does not know PEMDAS does not negate the fact that it exists and is one of the basics of mathematics in the same way that x number of people not knowing the capital of Zaire doesn't make Kinshasa "obscure and unclear in communication."

This. Exactly this.

The equation is not in any ambiguous, and no amount of claiming it is so will change that. If you know and follow the very clearly defined, internationally-agreed upon procedure for the order of operations, you will always get the same answer. There is absolutely no point where something could be one thing or the other, by following the rules correctly, you will always know what to do and always get the same answer.

The original poster on Facebook observed that some people were not following the rules taught in basic maths to complete this equation and released it into the wild to trap them, but that does not make it in any way ambiguous!

It's not like the sentence "Bob met Fred at the train station, he had recently had a haircut." where either Bob or Fred could have had the haircut. Correct rules of maths dictates that it must be one particular way.

[AsteriskAmp]

http://i13.photobucket.com/albums/a265/Heliomance/Equation.jpg
Doesn't the right side end up as 2 while the left side ends up as 1?
I vaguely suspect, if my previous statement isn't wrong, that it should be a z and not a 2 in the exponent inside the first limit so that it ends up as e, whose Neperian Logarithm is 1, making the left side 1+1.

[pendell]

I know, see my earlier post - you should always use Scientific mode, which does it correctly. Windows Calculator in Standard mode will do it incorrectly regardless of brackets, Windows Calculator in Scientific mode will do it correctly regardless of brackets.



Now I'm puzzled -- why does it behave differently depending on standard mode and scientific mode? I'd have thought that a copy-pasted formula would always be evaluated the same way, and the difference between standard and scientific would simply make more functions available to the user. But that's obviously not the case.



If you know and follow the very clearly defined, internationally-agreed upon procedure for the order of operations, you will always get the same answer.

Emphasis mine. I agree with your statement. I will, however, add that it seems obvious to me that most respondants to that page did not know and follow the procedure. If my interest is for ordinary people to correctly solve the problem, it is in my interest to add brackets to make the order more obvious. Because even if it is clear and unambiguous , it's still not going to be correctly interpreted and applied by its audience.

Respectfully,

Brian P.

[Heliomance]

Doesn't the right side end up as 2 while the left side ends up as 1?
I vaguely suspect, if my previous statement isn't wrong, that it should be a z and not a 2 in the exponent inside the first limit so that it ends up as e, whose Neperian Logarithm is 1, making the left side 1+1.

On rechecking, you are in fact correct, due to ln(1) = 0. I don't think changing the 2 to a z would do anything useful, the easiest way to fix it is to get rid of the ln.

The other problem is that it's mixing matrices and real numbers - it really needs to take the determinant of (X^-1)^T-(X^T)^-1.

[AsteriskAmp]

On rechecking, you are in fact correct, due to ln(1) = 0. I don't think changing the 2 to a z would do anything useful, the easiest way to fix it is to get rid of the ln.

The other problem is that it's mixing matrices and real numbers - it really needs to take the determinant of (X^-1)^T-(X^T)^-1.Changing the 2 to a z would produce the limit's definition of e [limit of ((1+1/z)^z) as z tends towards infinity], which, inside ln, would produce 1. The only issue with the matrices is if they can in fact be inverted, otherwise it does work since the inverse of the transposed is the transposed of the inverse if the original was invertible.

So if you take that X is an invertible matrix as a hypothesis and replace the 2 in "limit of ((1+1/z)^2) as z tends towards infinity" with a z it does add up.

I'll also note Wolfram Alpha can't solve this, due to being unable to reason out through properties that the most complex calculations cancel out without needing to be performed.

[snoopy13a]

More to the point, a 514 out of 800 on the SAT math section is an indication of mathematical deficiency given the current difficulty of the SAT math section*. It's a mixture of fairly basic arithmetic, fairly basic geometry, and extremely trivial algebra. It also briefly touches upon trigonometric functions and matrices, though there were few enough of them as recently as 2010 to reduce a score more than 40 or 50 points were every single one of them failed. That doesn't explain the loss of 286 points on average, let alone the loss of 386-401 points for approximately 15.9 percent of the test takers, assuming a roughly normal distribution. This is consistent both with a grasp of elementary math and utter inability in higher math, or with a pattern of knowledge gaps throughout mathematical levels, the latter of which would also explain the results gathered from Facebook**, particularly once one takes into account how neither the SAT nor Facebook are measures that accurately evaluate the population, due to heavy selection biases in both.

*I found data that indicate a 514 mean in 2011, with a standard deviation of approximately 100 to 115, with different sources varying. I will thus be using these numbers.

**I have not personally confirmed these, so this is a very provisional statement.

The average math SAT score is not the average score of the population. It is the average score of those who take the SAT--high school students who plan on attending college. A substantial minority of people do not take the SAT, such as high school dropouts and high school students with absolutely no interest in college. Therefore, the actual average is likely lower.

Second, most people get SAT math questions wrong because of careless mistakes in a time sensitive test. The actual concepts are designed to be accessible because it is an aptitude test, not a test of one's math education.
The math section tests for speed--especially with problems that aren't hard conceptionally but require time to solve--and attention to detail more than actual math concepts.

[Rawhide]

Emphasis mine. I agree with your statement. I will, however, add that it seems obvious to me that most respondants to that page did not know and follow the procedure. If my interest is for ordinary people to correctly solve the problem, it is in my interest to add brackets to make the order more obvious. Because even if it is clear and unambiguous , it's still not going to be correctly interpreted and applied by its audience.

If you wrote that equation for people who know basic maths, then there is absolutely no reason to change it. If you wrote that equation as a test to see if people know basic maths, then there is absolutely no reason to change it. If you wrote that equation for people you know don't know basic maths, then you should not have written that equation at all, not without at least a paragraph explaining where Zaire is, what its capital is, and why it's important.

[Heliomance]

Changing the 2 to a z would produce the limit's definition of e [limit of ((1+1/z)^z) as z tends towards infinity], which, inside ln, would produce 1. The only issue with the matrices is if they can in fact be inverted, otherwise it does work since the inverse of the transposed is the transposed of the inverse if the original was invertible.

Yes, absolutely. But ((X^T)^-1)-((X^-1)^T) is not zero, it's the zero matrix, and taking the factorial of the zero matrix is nonsensical.

[AsteriskAmp]

Yes, absolutely. But ((X^T)^-1)-((X^-1)^T) is not zero, it's the zero matrix, and taking the factorial of the zero matrix is nonsensical.Oh, forgot...

Just stick a "det" to the right of the parenthesis, add another layer of parenthesis to indicate factorial of determinant and it does work out. Which is what you originally suggested.

And this is why parenthesis are really needed, in order of operations they are just trivial (and well, for playing around with a negative sign in front of parenthesis), for slightly more complex things it's to make things make sense (and exponentiation of grouped terms)

[nedz]

I know, see my earlier post - you should always use Scientific mode, which does it correctly. Windows Calculator in Standard mode will do it incorrectly regardless of brackets, Windows Calculator in Scientific mode will do it correctly regardless of brackets.


I have seen Windows Calculator produce errors
In fact pasting "6-1x0+2/2=" produces the answer 6. Hitting '=' does give 7, then 8, ...

Try 48/2(9+3) :smallbiggrin:

Oh and nice example (well almost) Heliomance :smallcool:

[Kd7sov]

Just posting here to say how much I love you guys. It's been enough years since I last took a math class that I don't understand all of Heliomance's equation, but watching you all discuss it is such a joy.


Try 48/2(9+3) :smallbiggrin:

Error. Is that 48/(2*(9+3)) or (48/2)(9+3)?

[AsteriskAmp]

Error. Is that 48/(2*(9+3)) or (48/2)(9+3)?
Implicit multiplication is prioritized.

[noparlpf]

Just posting here to say how much I love you guys. It's been enough years since I last took a math class that I don't understand all of Heliomance's equation, but watching you all discuss it is such a joy.

I only don't get the bit with X and T or whatever. The rest I could work out if I cared, or just recognise.


Error. Is that 48/(2*(9+3)) or (48/2)(9+3)?

Neither, it's 48/2(9+3).
=48/2(12) and now just left to right
=24(12)
=288 (all in my head, this is basic stuff)
WolframAlpha agrees with me, and so does my TI-84.

[AsteriskAmp]

I only don't get the bit with X and T or whatever. The rest I could work out if I cared, or just recognise.
X is a matrix, transposing and inverting it and doing that on the reverse order are the same and give a null matrix.

Neither, it's 48/2(9+3).
=48/2(12) and now just left to right
=24(12)
=288 (all in my head, this is basic stuff)
WolframAlpha agrees with me, and so does my TI-84.
The way WolframAlpha calculates implied multiplication is odd, as well as the TI series.

Implied multiplication in general is odd in terms of order of operations because it is considered above division and explicit multiplication.

It should be: 48/(2*(9+3))
Which gives off 2.

[nedz]

2, yes, but not on Windows 7 :smallbiggrin:

Ed: it ignores the 2 unless you add the extra parenthesis.

[noparlpf]

X is a matrix, transposing and inverting it and doing that on the reverse order are the same and give a null matrix.

The way WolframAlpha calculates implied multiplication is odd, as well as the TI series.

Implied multiplication in general is odd in terms of order of operations because it is considered above division and explicit multiplication.

It should be: 48/(2*(9+3))
Which gives off 2.

Ah, haven't managed to learn matrices yet.

The way I was taught it, implied multiplication is just multiplication, and in PEMDAS, M and D are equivalent, it's the order from left to right that takes precedence.

[nedz]

The way I was taught it, implied multiplication is just multiplication, and in PEMDAS, M and D are equivalent, it's the order from left to right that takes precedence.

The 2 should bind to the (), but windows just throws the 2 away.

[Rawhide]

I have seen Windows Calculator produce errors
In fact pasting "6-1x0+2/2=" produces the answer 6. Hitting '=' does give 7, then 8, ...

Try 48/2(9+3) :smallbiggrin:

Oh and nice example (well almost) Heliomance :smallcool:

That's because you're entering the wrong formula. The correct one is 6-1*0+2/2=, not 6-1x0+2/2=.

Windows Calculator does not handle implied multiplication at all. So that other example is pointless. As above, you must enter the equation into the calculator in an accepted format, or it will not produce expected results.

[ForzaFiori]

Emphasis mine. I agree with your statement. I will, however, add that it seems obvious to me that most respondants to that page did not know and follow the procedure. If my interest is for ordinary people to correctly solve the problem, it is in my interest to add brackets to make the order more obvious. Because even if it is clear and unambiguous , it's still not going to be correctly interpreted and applied by its audience.

Respectfully,

Brian P.

When something is taught in elementary school, and then used in EVERY subsequent class in that field, it should be considered common knowledge. Just like you can assume that if you ask someone where you use a period or a question mark, they should know, because they've been doing it their entire life probably, and certainly their entire school career. People should know basic grammar and punctuation, and they should know the basic rules of math.

The main problem is that sadly, in the US (and probably other nations, but not having gone through their school systems, I can't make statements about them) math is taught in an absolutely horrid manner, and being proficient in math (or any subject, really) is seen as uncool by the majority of the youth. The over dependence on calculators (which are often wrong if you don't know how to use them, and most people don't) just complicates the problem, until you have the vast majority of the population essentially illiterate in math. (and nearly illiterate in English, but that's a different problem)

basically, it's not the equation's fault. it's either the students, for not caring enough to learn the main rule in all of math, or the teachers for not stressing it's importance or teaching it well enough.

[AsteriskAmp]

Ah, haven't managed to learn matrices yet.

The way I was taught it, implied multiplication is just multiplication, and in PEMDAS, M and D are equivalent, it's the order from left to right that takes precedence.They are rather simple, much of them is intuitive except the products which are annoying to both operate and conceptualize.

Implied multiplication is above multiplication and division. The main argument is because its main use is alongside variables and constants.

So the idea is that 1/2x is not x/2 but 1/(2x). The T series and Wolfram Alpha consider it of higher precedence when no parenthesis is present between the implied multiplication terms, but disregard the priority when a parenthesis is involved in the conjoining.

Implied multiplication is an odd component of Order of Operations because it isn't used until much later, when in algebra, by that time the division sign is no longer in use due to the fact that writing fractions is infinitely more comfortable for working around with. The majority of authors give it higher precedence by citing the way it works with variables and constants.

In this case:
48/2q
Where q=9+3
So after replacement

48/(2*(9+3))
48/24
2

[Felyndiira]

Oh, forgot...

Just stick a "det" to the right of the parenthesis, add another layer of parenthesis to indicate factorial of determinant and it does work out. Which is what you originally suggested.

And this is why parenthesis are really needed, in order of operations they are just trivial (and well, for playing around with a negative sign in front of parenthesis), for slightly more complex things it's to make things make sense (and exponentiation of grouped terms)

It would be much easier to just define X as a scalar. Since a scalar is a 1x1 matrix, the transpose of a scalar is itself, and the invert is just 1/itself. This gives us (0)! without needing a scalar transform on a matrix.

Coincidentally, I am curious what the X_bar is supposed to represent. We usually denote matrices with an uppercase letter or a bold letter, and I do not recall any operations that is denoted by a bar (other than mean, although that's a different field of math).

[noparlpf]

They are rather simple, much of them is intuitive except the products which are annoying to both operate and conceptualize.

Implied multiplication is above multiplication and division. The main argument is because its main use is alongside variables and constants.

So the idea is that 1/2x is not x/2 but 1/(2x). The T series and Wolfram Alpha consider it of higher precedence when no parenthesis is present between the implied multiplication terms, but disregard the priority when a parenthesis is involved in the conjoining.

Implied multiplication is an odd component of Order of Operations because it isn't used until much later, when in algebra, by that time the division sign is no longer in use due to the fact that writing fractions is infinitely more comfortable for working around with. The majority of authors give it higher precedence by citing the way it works with variables and constants.

In this case:
48/2q
Where q=9+3
So after replacement

48/(2*(9+3))
48/24
2

Yeah, I looked it up later and realised I was remembering it wrong. So yeah, should be 2.

[Asta Kask]

When something is taught in elementary school, and then used in EVERY subsequent class in that field, it should be considered common knowledge. Just like you can assume that if you ask someone where you use a period or a question mark, they should know, because they've been doing it their entire life probably, and certainly their entire school career. People should know basic grammar and punctuation, and they should know the basic rules of math.

People know and use very advanced grammar every day. People do not use what you call the basic rules of math every day. People forget what they rarely use.

[nedz]

Coincidentally, I am curious what the X_bar is supposed to represent. We usually denote matrices with an uppercase letter or a bold letter, and I do not recall any operations that is denoted by a bar (other than mean, although that's a different field of math).

Conjugate perhaps ?

[DeusMortuusEst]

People know and use very advanced grammar every day. People do not use what you call the basic rules of math every day. People forget what they rarely use.

One would think that something that you're supposed to use frequently for several years (from about age 12 to 17, at least) is something that would stay with you...

[noparlpf]

People know and use very advanced grammar every day. People do not use what you call the basic rules of math every day. People forget what they rarely use.

XDDD
Heh, haha, wow. No, they don't. Maybe where you live people can manage grammar, but hardly anybody around here knows even some of the pretty basic stuff. The subjunctive, who vs whom, &c.


One would think that something that you're supposed to use frequently for several years (from about age 12 to 17, at least) is something that would stay with you...

Well, that's the idea of school, but...no.

[Kd7sov]

People know and use very advanced grammar every day. People do not use what you call the basic rules of math every day. People forget what they rarely use.

Don't we? How else are we to keep up our tails perpendicular, or spread out our whiskers, or cherish our pride? How else should we keep our minds nimble and active?

Now, if we were talking about full Linnaean classification of species, I would have no problem agreeing that it's not an everyday thing. It even has a broadly similar mnemonic. But the very basics of math, on a level with how to divide?

[ForzaFiori]

People know and use very advanced grammar every day. People do not use what you call the basic rules of math every day. People forget what they rarely use.

I don't think about how Germany is in Europe everyday, or that it's capital is Berlin, but I still know it, because it's BASIC geography. The basics of any field of study should be something you learn once and KNOW. I know DC is America's capital, I know where Japan and China are, where Australia is, what their capitals are, the 7 continents, etc. I can tell you that America was an English colony, I can tell you roughly when the dark ages are, who the US's first president was, the name of the person that discovered america is, what a mammal and a reptile are, I can explain how punctuation is used, and I can do basic math. I don't use ANY of those (save for punctuation) every day, or even every week. but they're BASIC knowledge that I just KNOW.

[pendell]

Now, if we were talking about full Linnaean classification of species, I would have no problem agreeing that it's not an everyday thing. It even has a broadly similar mnemonic. But the very basics of math, on a level with how to divide?


I reposted the query on my facebook page and received responses from people I know are not trolls. Only one person got it right. Most of my respondants cited PEMDAS or DMAS, but they still came up with answers of 0,1, and 5 .

They remembered their precedence mnemonic, but they'd forgotten how it worked.

I think these folks would have got it right if there had been only one precedence clash (say, 1 * 5 + 2) or two precedence clashes (1 * 5 + 2 / 3) . But when you use all four operations in the same line, only a minority of people get it right.

So I don't think it's fair to say most people can't do basic math. Most people can do the four elementary operations, and when put on the spot most of them can remember the mneumonic they were taught in school. But when push comes to shove, when the operation is at all complex there's a good chance it will be goofed.

Respectfully,

Brian P.

[Heliomance]

It's an error on similar level to mixing up it's and its, or there, their and they're. No-one should make that mistake, but plenty do.

[Asta Kask]

I don't think about how Germany is in Europe everyday, or that it's capital is Berlin, but I still know it, because it's BASIC geography. The basics of any field of study should be something you learn once and KNOW. I know DC is America's capital, I know where Japan and China are, where Australia is, what their capitals are, the 7 continents, etc. I can tell you that America was an English colony, I can tell you roughly when the dark ages are, who the US's first president was, the name of the person that discovered america is, what a mammal and a reptile are, I can explain how punctuation is used, and I can do basic math. I don't use ANY of those (save for punctuation) every day, or even every week. but they're BASIC knowledge that I just KNOW.

But that's not skills, that's facts. That's an entirely different thing. If you don't practise skills they atrophy (that may, indeed, be the important thing that separates them from facts). I studied French for three years, but that was twenty years ago. I could not hold even a simple conversation in French today without taking considerable time putting sentences together.

Saying everyone SHOULD know something doesn't make it so. Would the payoff to learn and maintain the priority rules of arithmetic be worth the time and effort? For most people? Almost certainly not, because poorly written examples are not that common. It doesn't matter for most people.

And that's another thing - this example is poorly written. It is entirely possible for something to obey all the formal rules and still be unclear. Take sentences like "The horse raced past the barn fell", or "The fat people eat accumulates". Both are perfectly well-formed English sentences, according to the rules of grammar. They are nevertheless difficult to understand.

[Elder Tsofu]

But that's not skills, that's facts. That's an entirely different thing.

Aren't rules facts, at least as much as the name of Berlin is one?

[THAC0]

As a teacher, one important skill is to know whether your students are failing an exam because you wrote the exam (incorrectly, to be too difficult, etc) or because they are (not paying attention/studying, etc).

This equation is clearly a function of the latter rather than the former.

Guess what? My thirteen year old kids, who have known that they need to capitalize the first letter in a sentence for about 7 years, sill don't always do so.

In fact, if I were (as a non language arts teacher) to grade based on that SINGLE fact, 95% of my kids would FAIL.

Guess what again. Not my fault, or their other teacher's fault, for not teaching them to capitalize the first letter in a sentence.

[THAC0]

And that's another thing - this example is poorly written. It is entirely possible for something to obey all the formal rules and still be unclear.

How can one obey the formal rules and arrive at an incorrect answer?

[Heliomance]

"The horse raced past the barn fell"

I honestly cannot see a way to parse that that works under standard grammar. The closest two interpretations I can think of are either that it's two entirely separate clauses and should be written "the horse raced past; the barn fell", or it's a poetic reversal of adjective and noun, and really ought to be "the horse raced past the fell barn".

...actually, no, I can see a third that is grammatically correct, but still wouldn't likely come up. "The horse [that was] raced past the barn fell" works. Is that the correct interpretation? Because using a verb like that (sorry, I don't know the names for the verious tenses and such involved) without the implied "that was" being explicit isn't exactly common practise.

[noparlpf]

As a teacher, one important skill is to know whether your students are failing an exam because you wrote the exam (incorrectly, to be too difficult, etc) or because they are (not paying attention/studying, etc).

This equation is clearly a function of the latter rather than the former.

Guess what? My thirteen year old kids, who have known that they need to capitalize the first letter in a sentence for about 7 years, sill don't always do so.

In fact, if I were (as a non language arts teacher) to grade based on that SINGLE fact, 95% of my kids would FAIL.

Guess what again. Not my fault, or their other teacher's fault, for not teaching them to capitalize the first letter in a sentence.

My mum is an English teacher. My little brother adamantly refuses to capitalise words when he types, and my little sister keeps making the same mistakes that my mum corrects every other day. It's frustrating.


I honestly cannot see a way to parse that that works under standard grammar. The closest two interpretations I can think of are either that it's two entirely separate clauses and should be written "the horse raced past; the barn fell", or it's a poetic reversal of adjective and noun, and really ought to be "the horse raced past the fell barn".

...actually, no, I can see a third that is grammatically correct, but still wouldn't likely come up. "The horse [that was] raced past the barn fell" works. Is that the correct interpretation? Because using a verb like that (sorry, I don't know the names for the verious tenses and such involved) without the implied "that was" being explicit isn't exactly common practise.

Well, there are a few ways to interpret it, but just because one can decipher it doesn't mean it isn't incorrect. Yuo can raed tihs, rhigt?

[THAC0]

My mum is an English teacher. My little brother adamantly refuses to capitalise words when he types, and my little sister keeps making the same mistakes that my mum corrects every other day. It's frustrating.


Indeed. And that does not mean that they have not been taught proper (whatever), simply that they have chosen to forget/not implement it. Which, to bring it back to the OP, is not the fault of the equation.

[noparlpf]

Indeed. And that does not mean that they have not been taught proper (whatever), simply that they have chosen to forget/not implement it. Which, to bring it back to the OP, is not the fault of the equation.

Well, some people are simply incapable of learning certain things (or else, just had bad teachers/teachers incompatible with their learning style). However, I put it down to laziness, in most cases.

[Pie Guy]

Bah, precedence. I just make all a-b's into a + (-b). Same with a/b into a * (1/b) There, that's like half the work done if you can remember multiplication before addition.

Yes, negatives and fractions are easier for me than precedence. I said it.

[the_druid_droid]

Speaking as someone in a fairly math-heavy field, who does quite a bit of numerical programming, I can't really see the purpose in the equation presented in the OP beyond being a test of PEMDAS memorization.

Certainly its meaning is well-defined, and there is a single correct answer, but on a practical level I think it's rather silly. If I were actually making use of such an equation, either in my notes or in a program (or any other situation where I would likely need to go back and reference earlier work) I would always use brackets to divide it up. More generally, if you have a choice between a potentially confusing but formally correct expression and a much clearer and still correct, but slightly bulkier one, I think the choice should be obvious.

After all, mathematical symbols are a form of language and the purpose of language is communication. Formally correct but poorly-communicated expressions are really only fulfilling half their purpose.

[THAC0]

Well, some people are simply incapable of learning certain things (or else, just had bad teachers/teachers incompatible with their learning style). However, I put it down to laziness, in most cases.

And even if they can't learn, that doesn't mean the equation is bad.

[noparlpf]

Speaking as someone in a fairly math-heavy field, who does quite a bit of numerical programming, I can't really see the purpose in the equation presented in the OP beyond being a test of PEMDAS memorization.

Certainly its meaning is well-defined, and there is a single correct answer, but on a practical level I think it's rather silly. If I were actually making use of such an equation, either in my notes or in a program (or any other situation where I would likely need to go back and reference earlier work) I would always use brackets to divide it up. More generally, if you have a choice between a potentially confusing but formally correct expression and a much clearer and still correct, but slightly bulkier one, I think the choice should be obvious.

After all, mathematical symbols are a form of language and the purpose of language is communication. Formally correct but poorly-communicated expressions are really only fulfilling half their purpose.

So obviously the point of that equation is to test the general population (of facebook users) on their ability to use PEMDAS. In that case, it fills it purpose perfectly.


And even if they can't learn, that doesn't mean the equation is bad.

Yeah, in that case the problem is on the user's side. On the other hand, one can probably survive without PEMDAS in "real life". My cat doesn't seem to care.

[pendell]

Speaking as someone in a fairly math-heavy field, who does quite a bit of numerical programming, I can't really see the purpose in the equation presented in the OP beyond being a test of PEMDAS memorization.

Certainly its meaning is well-defined, and there is a single correct answer, but on a practical level I think it's rather silly. If I were actually making use of such an equation, either in my notes or in a program (or any other situation where I would likely need to go back and reference earlier work) I would always use brackets to divide it up. More generally, if you have a choice between a potentially confusing but formally correct expression and a much clearer and still correct, but slightly bulkier one, I think the choice should be obvious.

After all, mathematical symbols are a form of language and the purpose of language is communication. Formally correct but poorly-communicated expressions are really only fulfilling half their purpose.

Thank you, druid_droid. You have said in three paragraphs what I have been fumbling to say for three pages.

My work with mathematics is in building software algorithms, most recently to process sales taxes (and there are diferent kind of taxes for different kinds of products -- alcohol isn't taxed the way candy is. In different states and countries). I try to make the algorithms as clear as possible, which means that any possible confusion in the human reader must be minimized. The business logic and equations must be run by accountants, then coded, then run against sample data in a variety of situations and checked by hand. I don't need to explain the consequences of error.

Before that, it was riemann sums for visual/IR frequency applications. Same thing.

Because the equations must be checked and double-checked by multiple human users, clarity and clear understanding is of primary importance. And so I prefer a bulkier equation that is less likely to be misinterpreted over one that is formally correct but confusing.

Respectfully,

Brian P.

[Asta Kask]

I honestly cannot see a way to parse that that works under standard grammar. The closest two interpretations I can think of are either that it's two entirely separate clauses and should be written "the horse raced past; the barn fell", or it's a poetic reversal of adjective and noun, and really ought to be "the horse raced past the fell barn".

...actually, no, I can see a third that is grammatically correct, but still wouldn't likely come up. "The horse [that was] raced past the barn fell" works. Is that the correct interpretation? Because using a verb like that (sorry, I don't know the names for the verious tenses and such involved) without the implied "that was" being explicit isn't exactly common practise.

That's the intended reading. It's not common practise but it is according to the rules. More examples can be found here. (http://en.wikipedia.org/wiki/Garden_path_sentence)

[Knaight]

The average math SAT score is not the average score of the population. It is the average score of those who take the SAT--high school students who plan on attending college. A substantial minority of people do not take the SAT, such as high school dropouts and high school students with absolutely no interest in college. Therefore, the actual average is likely lower.
That would be one of the reasons why I stated that the SAT is a bad measure for the population as a whole. Another is that it only tests people of a particular age, yet another would be the extent to which people study specifically for the SAT.

[Ashery]

ETA: I'm not familiar with error analysis. May I ask how you came to those conclusions?


While nedz already (sort of) responded to this, I'll fill in the blanks.

Nedz is simply answering the question that's been neglected throughout most of this discussion: How are people arriving at the wrong answer?

For the 56 that gave the incorrect answer of 1: "They" did the operations from left to right without regard to, well, anything. This mistake was blamed on the calculator because it's the exact answer some (many?) calculators will spit out if you enter the equation exactly as given and because most students, even if they don't remember everything correctly, won't simply go from left to right.

6-1*0+2/2 = 5*0+2/2 = 0+2/2 = 2/2 = 1

For the 20 that gave the incorrect answer of 5: They used the order of operations correctly but made, as nedz pointed out, what was essentially a sign error.

6-1*0+2/2 = 6-0+1 = 6-1 = 5

The primary cause for this error, I'd argue, ultimately lies with the placement of the subtraction operation. If the original question was 6+2/2-1*0, I doubt many people would've answered 5. This mistake can be mitigated, at least in this situation, by using a technique someone pointed out earlier: Replace all a - b with a + (-b).

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

It should be observed that the replacement only makes it easier because we're multiplying by 0 and therefore removing the negative/subtraction operation entirely.

As for how I'd make the argument that the subtraction operation is the primary cause: It screws with the early intuitive understanding of associativity. That is:

a+(b+c) = (a+b)+c

The problem is that the subtraction operation throws people off.

(6-0)+1 ≠ 6-(0+1)

What the right hand side correctly translates to:

6-(0+1) = 6-0-1 = (6-0)-1

Or, if we had used the a - b = a + (-b) technique:

6+((-0)+1) = (6+(-0))+1

If this question had been written as 6+2/2-0*1, there wouldn't have been an issue as you're not subtracting a set of parenthesis.

(6+1)-0 = 6+(1-0)

For the 6 that answered 0: I'm guessing they saw the *0 part of the equation and automatically assumed that the answer was 0. Maybe because they genuinely screwed up or maybe because they saw a "trick" question earlier that broke down into something like (a+(b*c/(d+e))/(f*g/h))*0 and just jumped the gun (If you feel like being a pedant: d≠-e, f,g,h≠0).

For the 11 that made other errors: Nedz has no idea why and neither do I :smalltongue:

Also, this is why you should always show your work. It makes the "How" part of the question so much easier to answer. It also makes partial credit possible :smallwink:

[Jay R]

The given expression is a problem in a math text. Its purpose is to test who does, and who does not, know how to apply the rules.

If people who don't know how to apply the precedence rule can get it right, it is a bad textbook question.

[Saposhiente]

You'd be surprised by how many people who don't work with math regularly don't know that zero is even.

[Coidzor]

You'd be surprised by how many people who don't work with math regularly don't know that zero is even.

Most of the textbooks I've run into demur on the subject, certainly.

[noparlpf]

You'd be surprised by how many people who don't work with math regularly don't know that zero is even.

I've actually never heard that, but I guess it makes sense because 0/2 isn't a fraction. (I don't know about "regularly" but I do calculus about once a week when I realise I have four hours until my P-Chem homework is due. That's a sort of math.)

[pendell]

You'd be surprised by how many people who don't work with math regularly don't know that zero is even.

*Ponders asking the question and looking stupid. Again.*

*Decides that looking stupid is as NOTHING compared to learning something *

So ... why is zero even? You can't divide by it. It seems an arbitrary classification.

Respectfully,

Brian P.

[Socratov]

[QUOTE=pendell;14033995]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?

snipQUOTE]

that depends on you expecting the caculation in integers.

For instance in integers it's the much awaited 4. However, to be overly correct one can assume that the '2's in the equation can be rounded numbers of anywhere between (for significance kept on 2 decimals) 1,50 and 2,49

this then results in a range of minimum 1,50+1,50=3,00 (or: 2+2=3 for minimal values of 2)and a maximum of 2,49+2,49=4,98 (or: 2+2=5 for maximal values of 2), resulting 2+2=y in a range of 3=<y<5 for any value of 2

When we apply rounding off again we shall see that 2+2 can be solved as 3, 4 or indeed 5.

Now if the above explanation is the correct one, the 2+2=4 is not correct (in only providing 1 solution while others exist) even though 99.000 out of 100.000 people will probably say 2+2=4.

note: this phenomenon becomes funnier if applied to slightly more complex problems. Imagine a calculation where the function is the between-2,50-and-not-quite-4-th root of (whatever). At some point you will encounter the euler root and the Pi root. have fun calculating :smallbiggrin:

[noparlpf]

*Ponders asking the question and looking stupid. Again.*

*Decides that looking stupid is as NOTHING compared to learning something *

So ... why is zero even? You can't divide by it. It seems an arbitrary classification.

Respectfully,

Brian P.

"Even" means it can be divided by two and not yield a fraction. So technically zero is even, because 0/2=0 and zero isn't a fraction.

[Castaras]

And also 2 being even fits the pattern of odd-even-odd-even, which is very much liked.

[Socratov]

"Even" means it can be divided by two and not yield a fraction. So technically zero is even, because 0/2=0 and zero isn't a fraction.

well, 0 is the black sheep of the mathematical family anyway, so...

[noparlpf]

well, 0 is the black sheep of the mathematical family anyway, so...

I usually like zero. It means I can eliminate an entire piece of an equation, or reduce it to a one, or something. Except sometimes I hate zero. Like when the equation says ln(φ)=(1/RT)(integral from 0 to p (dp/p)). Because without that actually being explained in class, when you get to it on the homework it looks like you're supposed to have ln(0).

[Socratov]

I usually like zero. It means I can eliminate an entire piece of an equation, or reduce it to a one, or something. Except sometimes I hate zero. Like when the equation says ln(φ)=(1/RT)(integral from 0 to p (dp/p)). Because without that actually being explained in class, when you get to it on the homework it looks like you're supposed to have ln(0).

it's also the very reason why differentials are way more awesome then integreals (or worse: double, triple, circle, surface and volume integrals)

[The Extinguisher]

that depends on you expecting the caculation in integers.

For instance in integers it's the much awaited 4. However, to be overly correct one can assume that the '2's in the equation can be rounded numbers of anywhere between (for significance kept on 2 decimals) 1,50 and 2,49

this then results in a range of minimum 1,50+1,50=3,00 (or: 2+2=3 for minimal values of 2)and a maximum of 2,49+2,49=4,98 (or: 2+2=5 for maximal values of 2), resulting 2+2=y in a range of 3=<y<5 for any value of 2

When we apply rounding off again we shall see that 2+2 can be solved as 3, 4 or indeed 5.

Now if the above explanation is the correct one, the 2+2=4 is not correct (in only providing 1 solution while others exist) even though 99.000 out of 100.000 people will probably say 2+2=4.

note: this phenomenon becomes funnier if applied to slightly more complex problems. Imagine a calculation where the function is the between-2,50-and-not-quite-4-th root of (whatever). At some point you will encounter the euler root and the Pi root. have fun calculating :smallbiggrin:

I'm pretty sure claiming that's not how rounding works. If you round 1.86 to 2, you're committing that value to be two later. If I have 1.86 and 2.43 and I add them, I get 4.29, but if I round them both to two and add them I get 4.

[Saposhiente]

"Even" means it can be divided by two and not yield a fraction. So technically zero is even, because 0/2=0 and zero isn't a fraction.

(Terminology nitpick: Zero is a fraction. 0/1. 0/42. So is every other number, 3/1, 6/2, etc. What it is is an integer multiple of 2, 2*0=0.) Pendell: Dividing by zero has nothing to do with it.

[noparlpf]

(Terminology nitpick: Zero is a fraction. 0/1. 0/42. So is every other number, 3/1, 6/2, etc. What it is is an integer multiple of 2, 2*0=0.) Pendell: Dividing by zero has nothing to do with it.

Eh, sort of. I don't consider n/1 a fraction. Not that it makes a difference, really.

[Heliomance]

Eh, sort of. I don't consider n/1 a fraction. Not that it makes a difference, really.

Your explanation would make pi even, because you don't get a fraction when you divide it by 2.

[noparlpf]

Your explanation would make pi even, because you don't get a fraction when you divide it by 2.

If 3/2 is a fraction then why isn't pi/2? Three is a constant, pi is a constant. Same thing, really. And anyway, if we figured out all the decimals of pi, we could very well turn it into proper fraction form, just with arbitrarily long numbers in the numerator and denominator.

[Castaras]

You can't figure out all the numbers of pi. It's irrational.

So the only way you can have pi as a fraction is pi/1.

[Urpriest]

If 3/2 is a fraction then why isn't pi/2? Three is a constant, pi is a constant. Same thing, really. And anyway, if we figured out all the decimals of pi, we could very well turn it into proper fraction form, just with arbitrarily long numbers in the numerator and denominator.

Nope! 3/2 is a fraction (the actual term is a rational number) because 3 and 2 are integers. The fact that they're constants doesn't factor into it, p/q is also rational provided p and q are integers, even if they aren't constant. Integers are not arbitrarily long, either, since infinity is a limit, not a number.

[warty goblin]

I can tell you roughly when the dark ages are, who the US's first president was, the name of the person that discovered america is

Probably not, actually. Columbus was about four hundred years after Leif Ericson, who was the first European to make landfall in the Americas. But according to one of the sagas, Leif already knew about Vinland from a guy with the impossibly marvelous name of Bjarni Herjólfsson, who, along with his crew, had been blown off course there several years previously.

Of course other sources have it being Leif who was blown off course and wound up in Vinland, so who knows...

[noparlpf]

Nope! 3/2 is a fraction (the actual term is a rational number) because 3 and 2 are integers. The fact that they're constants doesn't factor into it, p/q is also rational provided p and q are integers, even if they aren't constant. Integers are not arbitrarily long, either, since infinity is a limit, not a number.

Okay, that's fair.

[Heliomance]

You can't figure out all the digits of pi because it never ends - it has infinitely many digits, and never starts repeating. There is no way to write pi as a fraction.

[nedz]

Your explanation would make pi even, because you don't get a fraction when you divide it by 2.If 3/2 is a fraction then why isn't pi/2? Three is a constant, pi is a constant. Same thing, really.
You have answered a different question.


And anyway, if we figured out all the decimals of pi, we could very well turn it into proper fraction form, just with arbitrarily long numbers in the numerator and denominator.

This is never going to happen, π is irrational. The term irrational means that you can never write it down as a rational number. A rational number is one which can be written down as some P/Q where P and Q are integers. This has been proven, though IIRC the proof is a little subtle.

[Socratov]

I'm pretty sure claiming that's not how rounding works. If you round 1.86 to 2, you're committing that value to be two later. If I have 1.86 and 2.43 and I add them, I get 4.29, but if I round them both to two and add them I get 4.

well, rounding is only done in favor of less writing down (are you going to write every decimal? have fun with Pi) but that doesn't mean you can round off and calculate on with the rounded off numbers. You always continue with the non rounded numbers. I have been punished for not doing this in school (our teacher sometimes was slightly sadistic in making math tests like these). In maths acutally 2 things you need to know: 1) what is a function? (and how does it work and hwat can you do with it) 2)every numeral representation of a number is a range unless otherwise defined or reasoned.

[Ashery]

Nope! 3/2 is a fraction (the actual term is a rational number) because 3 and 2 are integers. The fact that they're constants doesn't factor into it, p/q is also rational provided p and q are integers, even if they aren't constant. Integers are not arbitrarily long, either, since infinity is a limit, not a number.

Hurray, one explanation I don't have to do. Although the additional requirement that q≠0 needs to be added for the definition to be complete. Ie a rational number is defined as "p/q where p, q are integers and q≠0."

Now, for even/odd:

An even number is any integer that can be expressed as 2m, where m is any integer.

An odd number is any integer that can be expressed as 2m + 1, where m is any integer.

So, for the case of proving that odd + odd = even (To bastardize the language a bit), we have:

(2m + 1) + (2n + 1) = 2m + 2n + 2 = 2(m + n + 1) = 2k

The last "step" is unnecessary, but I figured it was better to play it safe and include it. It should be clear that k = m + n + 1 and we know k to be an integer because of the fact that integers are closed under addition.

Bit of a weak explanation, but meh.

[TheFallenOne]

Probably not, actually. Columbus was about four hundred years after Leif Ericson, who was the first European to make landfall in the Americas. But according to one of the sagas, Leif already knew about Vinland from a guy with the impossibly marvelous name of Bjarni Herjólfsson, who, along with his crew, had been blown off course there several years previously.

Of course other sources have it being Leif who was blown off course and wound up in Vinland, so who knows...

Yup. And anyone using the term 'dark ages' is instantly suspected to have little clue of the Middle Ages.

Lesson being: if you give some examples of basic facts you know, be sure everything you list is in fact accurate and not just a common misconception you picked up somewhere.

[Mx.Silver]

Yup. And anyone using the term 'dark ages' is instantly suspected to have little clue of the Middle Ages.

Lesson being: if you give some examples of basic facts you know, be sure everything you list is in fact accurate and not just a common misconception you picked up somewhere.

Just to further beat this point into the ground: technically speaking, the line "who the US's first president was", is, on account of the wording, more accurately answered by 'John Hancock' than 'George Washington'*. The reason for this being that while Washington was the first person to hold the office of 'President of the United States of America', the US's first president (note lack of capital) would have been Hancock due to him being the president of the Continental Congress when independence was declared.

Pedantic? Yes, but it seemed fitting given the main topic of the thread :smalltongue:


*There are other potential answers that can be argued for as well, but Hancock is probably the strongest option.

[Knaight]

Probably not, actually. Columbus was about four hundred years after Leif Ericson, who was the first European to make landfall in the Americas. But according to one of the sagas, Leif already knew about Vinland from a guy with the impossibly marvelous name of Bjarni Herjólfsson, who, along with his crew, had been blown off course there several years previously.

Of course other sources have it being Leif who was blown off course and wound up in Vinland, so who knows...

This, of course, doesn't actually matter: When they found America, there were already people living there. It had been discovered by them first, and while either Leif, Bjarni, or some other viking did make a legitimate discovery, they were by no means the first people to do so (had they been, they wouldn't have found people there, though there is technically a possibility for that to happen that simply didn't).

[razark]

The reason for this being that while Washington was the first person to hold the office of 'President of the United States of America', the US's first president (note lack of capital) would have been Hancock due to him being the president of the Continental Congress when independence was declared.
That would depend on when the United States actually started to exist as an entity. Is the current United States under the Constitution the same as the country that existed under the Articles of Confederation or before? The Declaration of Independence mentions "the united States of America" (note lack of capital).


Pedantic? Yes, but it seemed fitting given the main topic of the thread
:smallbiggrin:

[Mx.Silver]

That would depend on when the United States actually started to exist as an entity.
Quite so. 4th of July 1776 is traditionally considered the date when the USA started officially existing as an entity, although you could also argue for 1783 (when Britain formally recognised its independence) - which would make the first president Thomas Mifflin. 1781 is a possibility, which would make the first president John Hanson.


Is the current United States under the Constitution the same as the country that existed under the Articles of Confederation or before? The Declaration of Independence mentions "the united States of America" (note lack of capital).

The capital 'u' was in use by pretty much everyone at the time though, and was set when the Articles of Confederation were ratified in 1781. Following that line would therefore be at least as likely to give you John Hanson as it would Washington.
Basing a nation's age on a constitution however can be a bit awkward in any event, e.g. one could use it to make the argument that France has only existed since 1958. Things get very wonky in the cases of nations that don't have a written constitution, such as the UK*.

Regardless of what the most accurate answer actually, the existence of this discussion just goes to show that "who the US's first president was" cannot really be considered as basic a fact as ForzaFiori assumed it could :smalltongue:


*it also leaves the proverbial door open to the 'type identity vs. token identity' discussion and how/whether that particular philosophical distinction does/should apply to countries.

[TheFallenOne]

Regardless of what the most accurate answer actually, the existence of this discussion just goes to show that "who the US's first president was" cannot really be considered as basic a fact as ForzaFiori assumed it could :smalltongue:

Well, I'd say that ambiguity is still better than just being plain wrong like about who discovered America or the Middle Ages being dark :smalltongue:
Now I do wonder about the 'capital of Australia' bit; many, or even most, mistakenly believe it to be Sydney. Including me to be honest until I was corrected in, dunno, 8th grade maybe?

That said, the little excursion on who was the first president of the USA under what criteria was quite interesting. But given the presidents are numbered, shouldn't there be an 'official' answer to it or at least one endorsed by the government?

[warty goblin]

You can't figure out all the digits of pi because it never ends - it has infinitely many digits, and never starts repeating. There is no way to write pi as a fraction.
Only when defined for Euclidean geometry in a perfect plane. Since the universe is not perfectly Euclidean* due to pesky things like gravity, the actual value of pi - the ratio of circumference to diameter fluxuates depending on where you are. Somewhere, it's rational.

*Although apparently it seems to be basically Euclidean, which is too bad. I'd much rather live in a hyperbolic universe. I want my pentagon with five interior right angles.


This, of course, doesn't actually matter: When they found America, there were already people living there. It had been discovered by them first, and while either Leif, Bjarni, or some other viking did make a legitimate discovery, they were by no means the first people to do so (had they been, they wouldn't have found people there, though there is technically a possibility for that to happen that simply didn't).

A valid point, which is why I stipulated 'European.'

(Strangely, the Norse may well not have been the first people to discover Iceland. There's some evidence that Irish monks used it as a prayer retreat before the Norse colonization. However they were certainly the first to *settle* it, as one can hardly count a seasonal population of celibate men as any sort of colonization.)

[Saposhiente]

the actual value of pi - the ratio of circumference to diameter fluxuates depending on where you are. Somewhere, it's rational.

There are infinitely more irrational numbers than rational numbers (greater cardinality), therefore the probability of pi being rational at any given location is zero. (Though it's technically possible)

[nedz]

Only when defined for Euclidean geometry in a perfect plane. Since the universe is not perfectly Euclidean* due to pesky things like gravity, the actual value of pi - the ratio of circumference to diameter fluxuates depending on where you are. Somewhere, it's rational.

*Although apparently it seems to be basically Euclidean, which is too bad. I'd much rather live in a hyperbolic universe. I want my pentagon with five interior right angles.


Assuming Einstein was right: then the universe has a parabolic geometry.

Euclidean geometry is hyperbolic. (See Euclid:postulate 5)

[Heliomance]

There are infinitely more irrational numbers than rational numbers (greater cardinality), therefore the probability of pi being rational at any given location is zero. (Though it's technically possible)

PROBABILITY DOES NOT WORK THAT WAY!

An event with P(x) = 0 cannot happen. Ever. No, not even then. If it's technically possible, the probability is not zero. It can be infinitesimally small, but not zero.

[GnomeFighter]

Yup. And anyone using the term 'dark ages' is instantly suspected to have little clue of the Middle Ages.



Well, I'd say that ambiguity is still better than just being plain wrong like about who discovered America or the Middle Ages being dark :smalltongue:


Um... The Dark Ages are not The Middle Ages... Dark Ages is still a perfectly valid name for the Early Middle ages. Normally from the Fall of Rome to around 1050. It's really a much more defined period than the Middle ages.



Basing a nation's age on a constitution however can be a bit awkward in any event, e.g. one could use it to make the argument that France has only existed since 1958. Things get very wonky in the cases of nations that don't have a written constitution, such as the UK*.

Poor example. The date the UK came in to existence is very easy. 1 January 1801, the date the twin 1800 Acts of Union came in to force (and in 1922 in its current state).

[nedz]

Um... The Dark Ages are not The Middle Ages... Dark Ages is still a perfectly valid name for the Early Middle ages. Normally from the Fall of Rome to around 1050. It's really a much more defined period than the Middle ages.

The Dark ages was so called because:
For Historians: there are very few written documents.
For the Church: Christianity retrenched as 'pagan' hordes swept through Europe.

Archaeologists don't like the term and it is considered outdated.

The term Migration period is generally preferred, which overlaps with the, so called, Age of Invasions in Britain (Romans to 1066).

The Dark ages were typically regarded as being from the end of the western Roman empire to the rise of the Carolingians. We now call that period late antiquity / early middle ages.

[Mx.Silver]

That said, the little excursion on who was the first president of the USA under what criteria was quite interesting. But given the presidents are numbered, shouldn't there be an 'official' answer to it or at least one endorsed by the government?
The 'official' numbering scheme in the USA counts holders of the office 'President of the United States of America' (which are also easier to keep track of than the earlier 'presidents' because they have fixed terms). Unfortunately it doesn't actually provide the accurate figure for how many different people have held this position either, because it counts Grover Cleveland's two terms separately.
So yeah, just because it's 'official' doesn't mean it's technically correct :smalltongue:

Funnily enough there is a similar 'official' numbering discrepancy in the British monarchy. Because monarchs only started being numbered after the Norman conquest, King Edward I was actually the second monarch to be called Edward (Edward the Confessor having been the first) and so on with all subsequent Edwards.



Poor example. The date the UK came in to existence is very easy. 1 January 1801, the date the twin 1800 Acts of Union came in to force (and in 1922 in its current state).
Yes, in much the same way that the USA is considered to have come into existence on the 4th of July 1776 with the Declaration of Independence.

[Jay R]

Only when defined for Euclidean geometry in a perfect plane. Since the universe is not perfectly Euclidean* due to pesky things like gravity, the actual value of pi - the ratio of circumference to diameter fluxuates depending on where you are. Somewhere, it's rational.

Pi is a constant, originally defined as the ratio of the circumference and the diameter of a circle on a Euclidean plane.

Its value doesn't change, even though there are non-Euclidean "circles" that are not on a Euclidean plane.

[nedz]

Funnily enough there is a similar 'official' numbering discrepancy in the British monarchy. Because monarchs only started being numbered after the Norman conquest, King Edward I was actually the second monarch to be called Edward (Edward the Confessor having been the first) and so on with all subsequent Edwards.

Its slightly worse than that since after the act of union we have two numbering systems: one English, one Scottish. Hence James I/VI etc.


Pi is a constant, originally defined as the ratio of the circumference and the diameter of a circle on a Euclidean plane.

Its value doesn't change, even though there are non-Euclidean "circles" that are not on a Euclidean plane.
See below :smallbiggrin:

[Heliomance]

Ah, but consider a 10' blast. It consists of 12 squares which have an area of 300 sq ft. Thus Pi must clearly be 3.

[warty goblin]

Assuming Einstein was right: then the universe has a parabolic geometry.

Euclidean geometry is hyperbolic. (See Euclid:postulate 5)

Rather by construction, hyperbolic geometry is non-Euclidean. Euclidean geometry extends neutral geometry with the parallel postulate: that for every line and every point not on that line, there exists a single unique line through the point parallel to the given line.

Hyperbolic geometry assumes the logical negation of this postulate given the axioms of neutral geometry: that there exists at least one line and at least one point not on that line such that there exists more than one line through the point parallel* to the given line. It's a fairly simple proof to show that in a hyperbolic geometry, for any line and any point not on the line, there exist infinite lines through the point parallel to the original line.

You can of course define pi as the limit of a sequence (and there are a lot of sequences that converge to pi) but I rather prefer the geometric definition. It's rather more aesthetically pleasing.

*In the sense 'does not intersect.' Maintaining the same distance across space, having common perpendiculars, etc turn out to be logically equivalent to the parallel postulate. As does the sum of interior angles of a triangle being 180 degrees, and the existence of rectangles.

(elliptic geometry assumes that parallel lines do not exist. Since the existence of parallel lines can be proven in neutral geometry, this makes it rather a separate beast than Euclidean/hyperbolic geometries, which simply extend the neutral axioms.)

[Saposhiente]

PROBABILITY DOES NOT WORK THAT WAY!

An event with P(x) = 0 cannot happen. Ever. No, not even then. If it's technically possible, the probability is not zero. It can be infinitesimally small, but not zero.

http://en.wikipedia.org/wiki/Almost_surely , pi will be irrational in a given location.

[nedz]

Ah, but consider a 10' blast. It consists of 12 squares which have an area of 300 sq ft. Thus Pi must clearly be 3.
Pi is the ratio between the diameter and the circumference, now look at the template and measure the circumference, Pi=4 :smallbiggrin:


Rather by construction, hyperbolic geometry is non-Euclidean. Euclidean geometry extends neutral geometry with the parallel postulate: that for every line and every point not on that line, there exists a single unique line through the point parallel to the given line.

Hyperbolic geometry assumes the logical negation of this postulate given the axioms of neutral geometry: that there exists at least one line and at least one point not on that line such that there exists more than one line through the point parallel* to the given line. It's a fairly simple proof to show that in a hyperbolic geometry, for any line and any point not on the line, there exist infinite lines through the point parallel to the original line.

You can of course define pi as the limit of a sequence (and there are a lot of sequences that converge to pi) but I rather prefer the geometric definition. It's rather more aesthetically pleasing.

*In the sense 'does not intersect.' Maintaining the same distance across space, having common perpendiculars, etc turn out to be logically equivalent to the parallel postulate. As does the sum of interior angles of a triangle being 180 degrees, and the existence of rectangles.

(elliptic geometry assumes that parallel lines do not exist. Since the existence of parallel lines can be proven in neutral geometry, this makes it rather a separate beast than Euclidean/hyperbolic geometries, which simply extend the neutral axioms.)

The analysis is generally done using an ideal line at infinity.

In a parabolic geometry parallel lines meet at infinity. This is the geometry used in computer graphics.

In Euclidean geometry they don't ever meet. Euclidean geometry is the degenerate case of a hyperbolic geometry where the number of parallel lines, through the given point, in the set = 1.

In Elliptical geometry the set of parallel lines meet before infinity.

[warty goblin]

http://en.wikipedia.org/wiki/Almost_surely , pi will be irrational in a given location.

Kolmogorov's axioms of probability don't work for all subsets of R, only Borel sets, which are generated by countably infinite unions/intersections of lengths. The Borel sets - even the extended Borel sets - are smaller in carnality than the power set of R, which means you can't make probability statements about a set without first ensuring it's in B. Strange things start to happen when you play with unmeasurable sets, like being able to dissemble solid sphere of radius one into disjoint pieces, move them through space without intersecting any of them, and assemble them into two or more disjoint solid spheres of radius one.

In this case, you can't easily. Writing I for the set of irrationals, it's not hard to see that I isn't a Borel set, since it contains an uncountable number of points - it follows that Q, the rationals, isn't either since the Borels are closed under complements and Q = I'.

With some limiting arguments that I honestly cannot remember, you can show that the Lebesque integral with respect to the Lebesque measure of f(x) = 0 (x rational), f(x) = 1 (x irrational) over a finite interval is zero. Because of the way the Lebesque integral is calculated, I'm fairly sure this can't be extended to (0, infinity) though.


Note however that showing an event has probability zero is somewhat different from saying it can't happen. Applied strictly, this would mean that any observation of a continuous variable has probability zero, despite having just been done. Usually people get around this by arguing that continuous random variables are really just approximations to a fabulously complicated discrete reality, but I don't think that really works here. If nothing else there's somewhere where gravitational curvature is continuous over an interval, which I think implies that circumference/diameter is also a continuous function. If it varies at all, it attains a rational value. You have probability zero of choosing a particular gravity such that this happens at random from all possible gravities , but that does not mean it doesn't happen.

For an easier example, consider the distribution f(x) = x over [0,1]. Y attains values 0, 1/2, and 1, but you have zero probability of choosing them at random from all values y attains.

(Unless one decides the entire universe is actually discrete, and continuity is always a vastly more tractable approximation, in which case all bets are off.)

[warty goblin]

Pi is the ratio between the diameter and the circumference, now look at the template and measure the circumference, Pi=4 :smallbiggrin:



The analysis is generally done using an ideal line at infinity.

In a parabolic geometry parallel lines meet at infinity. This is the geometry used in computer graphics.

In Euclidean geometry they don't ever meet. Euclidean geometry is the degenerate case of a hyperbolic geometry where the number of parallel lines, through the given point, in the set = 1.

In Elliptical geometry the set of parallel lines meet before infinity.
Ah, I see. We're using different, contradictory terminologies for essentially the same thing. I learned this stuff in terms of progressions of axioms, so that's how I classify it, but that way works too.

Although I'm a bit baffled by defining lines that intersect before infinity as parallel.

[The Extinguisher]

(Unless one decides the entire universe is actually discrete, and continuity is always a vastly more tractable approximation, in which case all bets are off.)

The entire universe is pretty much actually discrete. The jury is still out on how exactly it's discrete at the fundamental level, but everything else in the universe is discrete.

Just wanted to throw that out there.

[pendell]

Since we have so many good mathematicians here, can someone please explain the Gambler's fallacy (http://en.wikipedia.org/wiki/Gambler's_fallacy) to me?

Let's imagine that I flip a fair coin. I throw 48 tails in a row.

It's still a 0.5 probability that I'll get tails again on the next throw.

On the other hand, the probability that I will throw 49 tails in a row is 0.5^49 = something e-15.

So ... the coin has no memory of my previous flips. Therefore I can't say that a heads flip is "due" or bound to happen. But at the same time, it would be very unwise of me to continue betting that I will continue to throw tails successively. I've already done terrible things to the laws of probability already to get this current result, and I can't assume I will continue to beat those odds forever.

So perhaps the gambler's fallacy, although logically inaccurate , nonetheless grasps an intuitive point: A gambler can instinctively tell when his streak, winning or losing, is exceeding reasonable probability and therefore it may be time to change his bet.

After all, in my example above, there are two ways of looking at the bet: One is a bet that I will flip the coin tails once out of 1 flip. The other is to view it as a bet that I will flip 49 tails in a row. The fact that 48 of those tails have already happened doesn't change the essential fact that I'm gambling that I can flip 49 tails in a row. And that is a bad bet.

So .. anyone want to unravel this web of assumptions? Preferably somebody who is good at both math AND gambling? :)

Respectfully,

Brian P.

[factotum]

You're making the same mistake as the fallacy is talking about--you're thinking, deep in your heart, that probability must have some sort of memory and thus there is such a thing as having too much good luck. This is not the case. If you get heads 100 times while flipping a coin, the probability of the next coin being heads is still 50/50.

Casinos and other gambling establishments play on this--they set up the odds in such a way that you have a reasonable chance of winning, because not even the most foolish gambler will keep playing a game they never win at; however, the overall result will always be that the house takes more money from the gamblers than the gamblers win back from the house. In the simple heads/tails example, they might charge 50p for a coin flip and allow you to win 90p if you get heads but nothing for tails; on average, this still nets them £1 in income for every 90p they pay out, so they're happy, and the gamblers still win often enough to keep them playing.

[pendell]

I'm not sure I follow. You're right in that this is what gambling houses do. I'm told that casinos calculate their expected earnings for a given table thus:

Revenue = House Edge * Bets.

Where "bets" is the total money put down and "house edge" is the probability that the house will win any given bet.

So from a Casino's point of view, the gambler's fallacy would encourage me. If I were to walk the floor and see someone winning twenty hands of blackjack in a row, I would first send Vinnie and Louis the Louse to ensure he wasn't cheating. Once I was sure this really was the result of blind luck, I'd let him continue playing. After all, the laws of probability can't be mocked forever. If he keeps playing at that probability, not only will I eventually win it all back but I'll get his starting stake too. It may not happen on the next hand or the next twenty hands. It may take several years, but it will happen. Meanwhile the winnings will encourage OTHER people to put down money at bad bets, resulting in even more cash for me.

How am I , as the Casino owner, not succumbing to the gambler's fallacy?

This leads up to my sure-fire foolproof system to winning at gambling: Buy a casino and make the laws of probability work for, rather than against, you :).

Respectfully,

Brian P.

[razark]

It's still a 0.5 probability that I'll get tails again on the next throw.
Assuming a fair coin and a fair thrower, yes, the odds will always be 50% of heads or tails, no matter what has happened before. (Ignoring the possibility that the coin could land on it's edge, or for some reason, not land at all.)


On the other hand, the probability that I will throw 49 tails in a row is 0.5^49 = something e-15.
The odds of throwing any specific combination of 49 results would be the same. The odds of getting a random-seeming result were exactly the same, you happen to exist in the universe where that one particular sequence came up.

[nedz]

Kolmogorov's axioms of probability don't work for all subsets of R, only Borel sets, which are generated by countably infinite unions/intersections of lengths. The Borel sets - even the extended Borel sets - are smaller in carnality than the power set of R, which means you can't make probability statements about a set without first ensuring it's in B. Strange things start to happen when you play with unmeasurable sets, like being able to dissemble solid sphere of radius one into disjoint pieces, move them through space without intersecting any of them, and assemble them into two or more disjoint solid spheres of radius one.

In this case, you can't easily. Writing I for the set of irrationals, it's not hard to see that I isn't a Borel set, since it contains an uncountable number of points - it follows that Q, the rationals, isn't either since the Borels are closed under complements and Q = I'.

With some limiting arguments that I honestly cannot remember, you can show that the Lebesque integral with respect to the Lebesque measure of f(x) = 0 (x rational), f(x) = 1 (x irrational) over a finite interval is zero. Because of the way the Lebesque integral is calculated, I'm fairly sure this can't be extended to (0, infinity) though.


Note however that showing an event has probability zero is somewhat different from saying it can't happen. Applied strictly, this would mean that any observation of a continuous variable has probability zero, despite having just been done. Usually people get around this by arguing that continuous random variables are really just approximations to a fabulously complicated discrete reality, but I don't think that really works here. If nothing else there's somewhere where gravitational curvature is continuous over an interval, which I think implies that circumference/diameter is also a continuous function. If it varies at all, it attains a rational value. You have probability zero of choosing a particular gravity such that this happens at random from all possible gravities , but that does not mean it doesn't happen.

For an easier example, consider the distribution f(x) = x over [0,1]. Y attains values 0, 1/2, and 1, but you have zero probability of choosing them at random from all values y attains.

(Unless one decides the entire universe is actually discrete, and continuity is always a vastly more tractable approximation, in which case all bets are off.)
Surely its easier to just use elementary analysis ?

If f(x) is continuous and i1 < p/q < i2; where i1, i2 ϵ I and p/q ϵ R then f(p/q) must exist.


Although I'm a bit baffled by defining lines that intersect before infinity as parallel.
Welcome to Elliptical geometry.:smallcool:

[warty goblin]

I'm not sure I follow. You're right in that this is what gambling houses do. I'm told that casinos calculate their expected earnings for a given table thus:

Revenue = House Edge * Bets.

Where "bets" is the total money put down and "house edge" is the probability that the house will win any given bet.

So from a Casino's point of view, the gambler's fallacy would encourage me. If I were to walk the floor and see someone winning twenty hands of blackjack in a row, I would first send Vinnie and Louis the Louse to ensure he wasn't cheating. Once I was sure this really was the result of blind luck, I'd let him continue playing. After all, the laws of probability can't be mocked forever. If he keeps playing at that probability, not only will I eventually win it all back but I'll get his starting stake too. It may not happen on the next hand or the next twenty hands. It may take several years, but it will happen. Meanwhile the winnings will encourage OTHER people to put down money at bad bets, resulting in even more cash for me.

How am I , as the Casino owner, not succumbing to the gambler's fallacy?

This leads up to my sure-fire foolproof system to winning at gambling: Buy a casino and make the laws of probability work for, rather than against, you :).

Respectfully,

Brian P.
Ah, probability confusion! Something I'm actually somewhat qualified to talk about!

So your confusion from your first post stems from not taking conditional probabilities into account.

In your first example, you are correct that the odds of your next flip being heads is 1/2. You are also right that your probability of getting 49 heads in a row is very, very bad. But your chances of getting 49 heads in a row given you have gotten 48 heads in a row is the same as the probability of getting heads on any flip of the coin: 1/2

You can think about this intuitively. At this point in time, you have flipped 48 heads. After the forty-ninth flip, you can have either of two outcomes: 49 heads in a row, or 48 heads followed by one tail. Since heads or tails are equally likely, both have probability 1/2 of happening, given you already have the 48 heads. Ergo, given 48 heads, the probability of getting a 49th is 1/2.

In more theoretical terms, conditional probability for events is defined as P(A | B) = P(A and B)/P(B), where A is any event (such as getting 49 heads), B is any event with non-zero probability. The | is read 'given' here.

The intuition for this weird definition is that since B has happened, you are only interested in the parts of A that can happen concurrently with B. You divide by B because you have to adjust for the change in P(A and B) due to B having happened - intuitively you are counting the number of outcomes in A and B, then dividing by the number in B. (This last sentence is technically incorrect in many cases, but don't worry about that here)

So for this case, let A be 49 heads, B be 48 heads. Then P(A|B) = P(49 heads and 48 heads)/P(48 heads) = P(49 heads)/P(48 heads) = (.5)^49/(.5)^48 = 1/2.

Take-away: You aren't betting on getting 49 heads, you're betting on getting one more head.

OK, for your second confusion, casinos care mostly about expected values, which are probability weighted averages. Simply put, you take each outcome of a game, multiply that by the probability, and add 'em all up. If all outcomes are equally likely, this is just your bog standard arithmetic mean. It's easiest to think about it as the value the average winnings approach as you play lots and lots of games.

So here's an example. Suppose the buy-in for a game is $5. Suppose furthermore than the gambler will win $10 with probability 1/4, and therefore lose with probability 3/4. The casino's expected income is
5*P(gambler loses) - 10*P(gambler wins)
= 5(3/4) - 10(1/4) = 5/4, or $1.25

Assuming the casino has a reasonable amount of money in the vault, it won't care about whether or not somebody wins or loses an individual game. It won't even matter if somebody always wins (as long as they aren't cheating). They are, on average, making money.

The difference between this and the first example is that the casino is basically playing the long game. The gambler only cares about the next flip. You shouldn't gamble not because you will lose money, but because you will probably eventually lose money, and the more you gamble, the more likely it is that you come out behind.

Hope that helps.

[Heliomance]

http://en.wikipedia.org/wiki/Almost_surely , pi will be irrational in a given location.

...nothing to see here. I didn't just make a fool of myself, no sir.
*flees*


Pi is the ratio between the diameter and the circumference, now look at the template and measure the circumference, Pi=4 :smallbiggrin:

Alternatively, it's the ratio between the square of the radius and the area. In Euclidean space, that definition works just as well.

[noparlpf]

Since we have so many good mathematicians here, can someone please explain the Gambler's fallacy (http://en.wikipedia.org/wiki/Gambler's_fallacy) to me?

Let's imagine that I flip a fair coin. I throw 48 tails in a row.

It's still a 0.5 probability that I'll get tails again on the next throw.

On the other hand, the probability that I will throw 49 tails in a row is 0.5^49 = something e-15.

So ... the coin has no memory of my previous flips. Therefore I can't say that a heads flip is "due" or bound to happen. But at the same time, it would be very unwise of me to continue betting that I will continue to throw tails successively. I've already done terrible things to the laws of probability already to get this current result, and I can't assume I will continue to beat those odds forever.

So perhaps the gambler's fallacy, although logically inaccurate , nonetheless grasps an intuitive point: A gambler can instinctively tell when his streak, winning or losing, is exceeding reasonable probability and therefore it may be time to change his bet.

After all, in my example above, there are two ways of looking at the bet: One is a bet that I will flip the coin tails once out of 1 flip. The other is to view it as a bet that I will flip 49 tails in a row. The fact that 48 of those tails have already happened doesn't change the essential fact that I'm gambling that I can flip 49 tails in a row. And that is a bad bet.

So .. anyone want to unravel this web of assumptions? Preferably somebody who is good at both math AND gambling? :)

Respectfully,

Brian P.

Well...I dropped stats, I'm just going by common sense, but if you had bet at the beginning that you would get 49 tails in a row, that would have been a long shot. But if you've already gotten 48 tails in a row, and you're betting on the outcome of the next flip, odds are even either way. The issue is that if you're betting on the single flip, people tend to factor in the previous 48 when that actually has nothing to do with the next flip. There's no cosmic balancing book that says the number of heads and tails absolutely has to balance; it's just highly likely that it will after a while. Even so, I once sat and flipped a coin a thousand times and got something like 560-something heads, if I remember that rightly.

[nedz]

Alternatively, it's the ratio between the square of the radius and the area. In Euclidean space, that definition works just as well.

But for a 5' blast, by the area method, Pi = 4 :smallbiggrin:

More seriously: I find it interesting that in the square board geometry Pi is always an integer ?

[Heliomance]

I don't think it is, not if you calculate it by the area. On a square board, the ration of diameter to circumference is always exactly 4. I think that the ration of the radius squared to area approaches pi as the "circle" increases in size, though, if memory serves.

[nedz]

OK - that's what I get for just using 2 samples.

two more examples

20' radius = 44 squares = 1100 square feet so Pi = 2.75

40' radius = 172 squares = 4300 square feet so Pi = 2.6875

I think that this may require more thought ?

[Saposhiente]

Note however that showing an event has probability zero is somewhat different from saying it can't happen. Applied strictly, [...]
Umm yeah, that's kind of exactly what I said.

Gambler's fallacy stuff

Here is the difference:
The probability of a fair coin coming up heads 50 times in a row is 1/250 (Very small). The probability of a fair coin coming up heads 50 times in a row given that the first 49 flips are guaranteed to come up heads (In this case, because you have already observed that they have. But you can also pretend that you're a time traveler and looked into the future before flipping. Etc...) is 50/50.

One thing to note though, is that while the number of times heads has come up in a row doesn't change the probability of the next flip being heads on a fair coin, it does increase the probability that the coin is not actually fair. So if given even odds on a real-life coin, after 49 headflips, you should bet on heads. Thus, the gambler's fallacy suggests the opposite of the correct move*

*Unless you're using a guaranteed-fair pseudorandom computer RNG. RNGs use an algorithm to effectively just cycle through a long list of random numbers where each number comes up the same number of times. Thus, if a bunch of the numbers in the list that cause one thing to happen come up at the same time, those instances of those numbers aren't going to come up again until the list is run through (which is almost never; instead it's generally when you restart your computer/program). Of course the list is so large that the effect is negligible, and other more complicated things can negate the effect; thus unless you are absolutely certain that the coin is indeed fair you should still take the advice opposite of the Gambler's Fallacy.
(If none of that made any sense, try replacing the coin with a deck of cards with face cards removed, with heads being even numbers and tails being odd numbers. The Gambler's fallacy will hold true with this for similar reasons.)

Casino stuff: House edge isn't the probability that the house wins; it's how much more likely it is for the house to win (Adjusted as necessary if the house gets less money when it whens than it has to give out if the player wins). Eg. even the best blackjack player can still only win ~47% of the time without cheating. The 47% where the player wins cancels out with 47% that the house wins, leaving 53%-47%=6% where the house gains the bet in money on average.

nedz: It should approach pi, however rounding methods can change this.

[nedz]

nedz: It should approach pi, however rounding methods can change this.

Yes I know. What I'm interested in is why it doesn't.
I suspect that the issue is with the WotC circle templates.

[Heliomance]

Can we establish what it does approach, I wonder. How big do the templates go, and how easy is it to extrapolate larger?

[nedz]

The only other one is the 80 foot one, which is: 684 squares = 17100 feet2
So Pi = 2.671875 here

Time to get the compasses out I think :smallsmile:

[nedz]

OK - they are rounding very badly.
The 20' radius should look like

OOOOOOOO
OOOOOOOO
OOOOOOOO
OOOOOOOO
OOOOOOOO
OOOOOOOO
OOOOOOOO
OOOOOOOO

Rather than

OOOOOOOO
OOOOOOOO
OOOOOOOO
OOOOOOOO
OOOOOOOO
OOOOOOOO
OOOOOOOO
OOOOOOOO

i.e.
52 squares instead of 44 squares
Yielding a Pi of 3.25.
Which is a lot better than the 2.75 we got previously.

[noparlpf]

OK - they are rounding very badly.
The 20' radius should look like

OOOOOOOO
OOOOOOOO
OOOOOOOO
OOOOOOOO
OOOOOOOO
OOOOOOOO
OOOOOOOO
OOOOOOOO

Rather than
OOOOOOOO
OOOOOOOO
OOOOOOOO
OOOOOOOO
OOOOOOOO
OOOOOOOO
OOOOOOOO
OOOOOOOO

i.e.
52 squares instead of 44 squares
Yielding a Pi of 3.25.
Which is a lot better than the 2.75 we got previously.

Wait since when did we think WotC knew how to do things right?

[warty goblin]

Wait since when did we think WotC knew how to do things right?

Given that they're modeling things that often should be following the inverse square law, rounding area down seems pretty reasonable. Let's face it, most people don't play RPGs for their accurate approximations of mathematical constants.

If they did, pretty much everybody would just play as 'e' anyway, and have irrational fears about forests. It's well known that pieces of wood can reduce an exponentially good time to a linear railroaded crawl. Fortunately since default mathematical adventures are set in the real plane, trees are only supported in 'applied' computer science sorts of modules scorned by most 'pure' math players. There would of course be the occasional annoying dude who insists on playing as 'i' and complains when magnitude monsters conjugate all his imaginary abilities away, leaving him perfectly flat.


...dear god, that may be the nerdiest thing I've ever written.

[nedz]

Given that they're modeling things that often should be following the inverse square law, rounding area down seems pretty reasonable. Let's face it, most people don't play RPGs for their accurate approximations of mathematical constants.
Its not the constant that's the problem, its the badly drawn circle.

If they did, pretty much everybody would just play as 'e' anyway, and have irrational fears about forests. It's well known that pieces of wood can reduce an exponentially good time to a linear railroaded crawl. Fortunately since default mathematical adventures are set in the real plane, trees are only supported in 'applied' computer science sorts of modules scorned by most 'pure' math players. There would of course be the occasional annoying dude who insists on playing as 'i' and complains when magnitude monsters conjugate all his imaginary abilities away, leaving him perfectly flat.
I'm sure that the default is Complex, definitely something imaginary involved.

[warty goblin]

Its not the constant that's the problem, its the badly drawn circle.

Well as I said, rounding downwards makes sense for blast templates.


I'm sure that the default is Complex, definitely something imaginary involved.
Imagination in numbers is highly overrated*. Except for eleventeen and thirtytwelve, but you have to be a tiger to use them.

*Complex Analysis: where joy goes to die.

[nedz]

Well as I said, rounding downwards makes sense for blast templates.
I haven't done the integration, though its not hard so I might, but by eye they have rounded 0.85 down !


Imagination in numbers is highly overrated*. Except for eleventeen and thirtytwelve, but you have to be a tiger to use them.

*Complex Analysis: where joy goes to die.

Euler's Identity says "Hi"

[pendell]

Ah, probability confusion! Something I'm actually somewhat qualified to talk about!


So your confusion from your first post stems from not taking conditional probabilities into account.

In your first example, you are correct that the odds of your next flip being heads is 1/2. You are also right that your probability of getting 49 heads in a row is very, very bad. But your chances of getting 49 heads in a row given you have gotten 48 heads in a row is the same as the probability of getting heads on any flip of the coin: 1/2

You can think about this intuitively. At this point in time, you have flipped 48 heads. After the forty-ninth flip, you can have either of two outcomes: 49 heads in a row, or 48 heads followed by one tail. Since heads or tails are equally likely, both have probability 1/2 of happening, given you already have the 48 heads. Ergo, given 48 heads, the probability of getting a 49th is 1/2.

In more theoretical terms, conditional probability for events is defined as P(A | B) = P(A and B)/P(B), where A is any event (such as getting 49 heads), B is any event with non-zero probability. The | is read 'given' here.

The intuition for this weird definition is that since B has happened, you are only interested in the parts of A that can happen concurrently with B. You divide by B because you have to adjust for the change in P(A and B) due to B having happened - intuitively you are counting the number of outcomes in A and B, then dividing by the number in B. (This last sentence is technically incorrect in many cases, but don't worry about that here)

So for this case, let A be 49 heads, B be 48 heads. Then P(A|B) = P(49 heads and 48 heads)/P(48 heads) = P(49 heads)/P(48 heads) = (.5)^49/(.5)^48 = 1/2.

Take-away: You aren't betting on getting 49 heads, you're betting on getting one more head.

OK, for your second confusion, casinos care mostly about expected values, which are probability weighted averages. Simply put, you take each outcome of a game, multiply that by the probability, and add 'em all up. If all outcomes are equally likely, this is just your bog standard arithmetic mean. It's easiest to think about it as the value the average winnings approach as you play lots and lots of games.

So here's an example. Suppose the buy-in for a game is $5. Suppose furthermore than the gambler will win $10 with probability 1/4, and therefore lose with probability 3/4. The casino's expected income is
5*P(gambler loses) - 10*P(gambler wins)
= 5(3/4) - 10(1/4) = 5/4, or $1.25

Assuming the casino has a reasonable amount of money in the vault, it won't care about whether or not somebody wins or loses an individual game. It won't even matter if somebody always wins (as long as they aren't cheating). They are, on average, making money.

The difference between this and the first example is that the casino is basically playing the long game. The gambler only cares about the next flip. You shouldn't gamble not because you will lose money, but because you will probably eventually lose money, and the more you gamble, the more likely it is that you come out behind.

Hope that helps.



It does, thank you.

The "gambler's fallacy" is the assumption that the next event is dependent when, in fact, it is independent. So a bet on a coin flip should be made without regard to previous history, assuming a fair coin.

By contrast, the casino isn't using the gambler's fallacy to predict a single event . It is, instead, playing the averages. It can reasonably expect that, if the probability of a casino win is 0.75 and a player win is 0.25, then the casino will make money *in the long run*, because the averages will smooth out deviations caused by someone being 'lucky'.

The casino is correctly using probability to predict an expected outcome over a statistically significant sample. The gambler is not, because he is attempting to use probability that works over long averages to predict a single event.

Incidentally, I once did some research to find out how professional gamblers make their living. Short answer: By having a great tolerance for debt and by being subsidized by casinos, as a professional gambler can be a sort of marketing stunt, since he advertises by example that gambling can be a 'success', even though it isn't.

Respectfully,

Brian P.

[Jay R]

The quick answer is this. The probability (now) that the coin already landed the way it did is 100%. It isn't a random event after it has happened.

So the probability that I will flip 49 heads in a row is 0.5^49 (roughly 0.0000000000000018).

But if I already flipped the first 48 heads, the probability that the 49th will also be heads is 0.5.

So Prob(49 heads in a row, given that 48 have already happened) is not the same thing as Prob(49 heads in a row, starting now).

If you want to understand it well, pick up a book on Bayesian probability (http://en.wikipedia.org/wiki/Bayesian_probability). Budget a fair amount of time.

[Douglas]

Incidentally, I once did some research to find out how professional gamblers make their living. Short answer: By having a great tolerance for debt and by being subsidized by casinos, as a professional gambler can be a sort of marketing stunt, since he advertises by example that gambling can be a 'success', even though it isn't.
It is possible to make a living as a professional gambler without being subsidized in that fashion, but it places heavy constraints on how you gamble. You do need a high tolerance for temporary debt regardless, but the main key is that you have to play games where skill can tip the odds and you have to have enough skill to tip the odds past the 50% point.

Perhaps the most famous example of this is black jack. It is possible, with absolutely statistically perfect play, to get the odds in black jack very close to 50-50 in general. It is further possible, assuming a standard 52-card deck, for variations in which cards have not yet been dealt to tip the odds past that point, and careful calculations can detect when this has happened. Most importantly, this happens frequently enough and by large enough margins that increasing the size of your bets when it happens can shift the overall odds in your favor. Do it well enough, and you can reliably beat the house.

A group of students from MIT famously took advantage of this in the 1980s to make quite a bit of money, but it suffers from the drawbacks of requiring extremely rigorous adherence to calculated strategy and provoking a hostile reaction from the casinos. Poker and its many variations offer the same possibility of skill tipping the odds, while being more interesting (in my opinion) and not offending the casinos because the losers are the other players. On the other hand, the standard of skill required is relative to your competition rather than an absolute, which puts you in an eternal arms race with all the other people with the same ambition.

[pendell]

Poker and its many variations offer the same possibility of skill tipping the odds, while being more interesting (in my opinion) and not offending the casinos because the losers are the other players.


I forget the book I read it in, but the comment in it was that , while poker is a game of skill which the house has no interest in because it gets a cut regardless, the other players at the table are typically professionals out to make easy money from the tourists. If you ARE a professional, it might be worth a try. But an amateur sitting down at the table should probably be resigned to losing most of what they put down.

Respectfully,

Brian P.

[Douglas]

If you ARE a professional, it might be worth a try. But an amateur sitting down at the table should probably be resigned to losing most of what they put down.
That goes for pretty much every serious gambling game, though. If a casino is running it, then it's almost certainly A) purely luck based with the odds against you (e.g. slot machines); B) affectable by skill, but with even the best possible odds still against you to the best of the casino's knowledge (e.g. black jack); or C) pitting your skill against that of other gamblers (e.g. poker).

In case A, you're just screwed. In case B, you have to find a strategy the casino hasn't countered yet (most casinos have long since taken measures to prevent black jack from tipping over the break even point), and it's by no means certain that any such strategy even exists. In case C, either you're good enough to do it professionally and probably do, or you're not and will lose to the people who are. You can try to join the professionals in case C, but there's only so much money from amateurs to split among them and they will try - hard - to prevent you from joining their ranks and reducing their share by outplaying you.