Math question help

[Duke: A Dynasty]

On another thread there is a huge argument going on over the answer to this. Everyone is saying its 2 or 288.



48/2(9+3)

[dukebluelemur]

On another thread there is a huge argument going on over the answer to this. Everyone is saying its 2 or 288.



48/2(9+3)

Parenthesis always come first. Then left to right. Would people be confused if it was written like this? 48 ÷ 2 x (9+3)

So the answer is 288.

The problem is that the slash is somewhat ambiguous. As well as being a division symbol, it is a fraction symbol, so people are seeing it as:

48
2x(9+3)

But as I see it, that's an incorrect understanding.

http://en.wikipedia.org/wiki/Order_of_operations

[Duke: A Dynasty]

I dont have a division sign to use so had to put the /.

But I thought it went like this:

48 / 2(9+3)
Parenthesis comes first so 9 + 3=12
48 / 2(12)
Your parenthesis did not magically disappear so now you got 2x12=24
48 / (24)
then 48/24
2

[InSpades]

I dont have a division sign to use so had to put the /.

But I thought it went like this:

48 / 2(9+3)
Parenthesis comes first so 9 + 3=12
48 / 2(12)
Your parenthesis did not magically disappear so now you got 2x12=24
48 / (24)
then 48/24
2

Only what is *inside* the parenthesis is evaluated 1st. So once you have 12 inside the parenthesis you proceed to the next step in order of operations. Multiplication and division occur at the same time and are performed left to right. So 288 is correct.

[callmecrazy]

PEMDAS
I teach 6th grade math and oder of operations is one of our topics.
parenthesis exponent multiplication division addition subtraction
The multiplication and division are whichever comes first working left to right. Same with addition and subtraction
288

[dukestheheat]

48 divided by 2 times 9 plus 3.

Moving left to right through PEMDAS, we take the parenthesis first.

48 divided by 2 times 12.

Next up would be multiplication, so 2 times 12 is 24.

48/24, then, is 2, I guess, based on the acronym: parenthesis, no exponents, multiply, then divide, no addition or subtraction.

I ain't no math wizard and clearly my strength is English, which you can see. ;)

dth, on a limb but feeling brave in front of the brethren.

[Duke: A Dynasty]

http://cdn2.knowyourmeme.com/i/000/112/837/original/16h6ja8.jpg?1302454815

[Duke: A Dynasty]

PEMDAS
I teach 6th grade math and oder of operations is one of our topics.
parenthesis exponent multiplication division addition subtraction
The multiplication and division are whichever comes first working left to right. Same with addition and subtraction
288

Its not that I dont believe you I just find it kind of confusing.

What about distribution?

48/ 2 (9+3)

48/ 18 + 6

48/ 24

2

[darthur]

You could ask this to the 2 greatest mathematicians in the world and get 2 different answers. Which means it isn't a math question, it's a language question. And the correct answer is everybody loses, and you should just never use this notation. :D

PS: I would read this as 2.

[BCGroup]

TrY wolfram alpha website. Put in question, it gives you nswe. Kids are loving it for homework these days.

[DevilBen02]

Discussion of this problem:
http://knowyourmeme.com/memes/48293

According to this analysis, the big issue here is whether division or multiplication is performed first. It is explained that even though PEMDAS has multiplication listed first, multiplication and division hold equal precedence. As a result, moving from left to right, the answer is 288.

[snowdenscold]

I always thought M and D has same precedence, just like A and S. Otherwise things get realllly strange.

If you some write 11 - 4 + 2, you wouldn't say the answer was 5, you would say it was 9.

The fact that some versions above have whitespace and others don't also adds confusion.

This is probably only an issue in typed posts (which I suppose will be more much common going forward) rather than hand-written, since you would put a ------ for division and it would be clear where factors go.

[94duke]

PEMDAS
I teach 6th grade math and oder of operations is one of our topics.
parenthesis exponent multiplication division addition subtraction
The multiplication and division are whichever comes first working left to right. Same with addition and subtraction
288

callmecrazy is correct. * and / are treated equally, moving left to right


Its not that I dont believe you I just find it kind of confusing.

What about distribution?

48/ 2 (9+3)

48/ 18 + 6

48/ 24

2

Distribution is really just multiplication. You still have to follow order of operations, moving left to right. It would go like this:

48 / 2 (9 + 3)
24 (9 +3)
(216 + 72)
(288)

To get an answer of 2, you would need another set of parentheses. Without them, the rules say you perform 48/2, first. The only way to multiply 2(12) first is to have parentheses force that operation first.
48 / (2(9+3))
48 / (2(12)) or 48 / (18 + 6)
48 / (24)
2

I liked what darthur said bout this being a language question.
It is a language question. You have to understand the language to interpret it correctly and get the correct answer. (which is 288) <wink> ;)

[anon]

Your parenthesis did not magically disappear

This post needs to be brought to a teacher's convention. I guess it's not being emphasized that the reason parentheses are there is to prioritize the evaluation of what is *inside* them. Multiplication outside them is treated no differently however it is expressed, be it with *, ×, ·, or even juxtaposition (the "parentheses" case everyone is referring to).



I liked what darthur said bout this being a language question.
It is a language question. You have to understand the language to interpret it correctly and get the correct answer.

Math *is* language. It's the study of a special kind of language with strict rules. And unlike the rules of the English language, which invariably change as people break them (see the "mute vs. moot" post for a particularly painful example), these rules can't change just because of an Internet argument.

[hurleyfor3]

Real geeks use HP calaulators, not Casio or TI crap.

48 (enter) 2 (divide) 9 (enter) 3 (plus) (times). Answer, 288.

Anyway, does no one remember the invert and multiply rule? 48 times 1/2 times 12. Two-eighty-eight.

[OldPhiKap]

Math *is* language. It's the study of a special kind of language with strict rules. And unlike the rules of the English language, which invariably change as people break them (see the "mute vs. moot" post for a particularly painful example), these rules can't change just because of an Internet argument.

Unless something gets lost in the translation from binary, of course.

[BD80]

Real geeks use HP calaulators, not Casio or TI crap.

48 (enter) 2 (divide) 9 (enter) 3 (plus) (times). Answer, 288.

Anyway, does no one remember the invert and multiply rule? 48 times 1/2 times 12. Two-eighty-eight.

Ah, RPN, a convention only a Duke coach could love.

[snowdenscold]

I was about to bring up reverse polish notation as a 'solution' to OO ambiguity, but it looks like someone beat me to it.

48 2 % 9 3 + * FTW

[Deslok]

Just to toss some fuel on the fire and be amused. What is:

48/2x

when x is equal to 12?

[Jarhead]

You could ask this to the 2 greatest mathematicians in the world and get 2 different answers. Which means it isn't a math question, it's a language question. And the correct answer is everybody loses, and you should just never use this notation. :D

PS: I would read this as 2.

No matter how hard I try, I cannot see an approach that properly results in 288. Do this calculation on a chalk board or a sheet of paper, without a keyboard, and if you follow the correct procedure or notation, the answer will always be 2. An answer of 288 can only result from bad structure, the kind offered by a keyboard and a line of characters.

Another approach is to start from the slash symbol. (That is the divide symbol on my keyboard.) Everything to the right of the slash is the divisor which must be evaluated separately from the multiplier. Then the correct result can be evaluated. That's the way I've always done it since the fourth grade, and if I ever made an error, it was because of forgetting that rule.

...48
------= 2
2(9+3)

[snowdenscold]

Just to toss some fuel on the fire and be amused. What is:

48/2x

when x is equal to 12?

I think this come back down to a typed-language question again.

IMO, there's a difference between 48/2x and 48 / 2x.

[snowdenscold]

Another approach is to start from the slash symbol. (That is the divide symbol on my keyboard.) Everything to the right of the slash is the divisor which must be evaluated separately from the multiplier. Then the correct result can be evaluated. That's the way I've always done it since the fourth grade, and if I ever made an error, it was because of forgetting that rule.

...48
------= 2
2(9+3)

The '/' is what begins the problem is the first place. You assume everything to the right of a '/' is the denominator, but it doesn't have to be read that way. Others see it as

48
---- (9+3)
2

Using a '/' makes the problem read 1-dimensional instead of 2, hence the ambiguity.

[OldPhiKap]

Using a '/' makes the problem read 1-dimensional instead of 2, hence the ambiguity.

So the problem isn't language. It's Euclid.

[hurleyfor3]

I think this come back down to a typed-language question again.

IMO, there's a difference between 48/2x and 48 / 2x.

There is a slightly different convention when variables are involved, especially on the Internet. If the x was meant to multiplied it would normally be written as 48x/2. Personally I see "48/2x" and, without additional information, presume the writer intends x to be in the denominator. A similar case is something like "sqrt 2x". No one would interperet that to first take (sqrt 2), then multiply by x. That would be "x sqrt 2".

When written on paper, or in a format such as TeX that supports standard math notation, the meaning is unambiguous.

[snowdenscold]

There is a slightly different convention when variables are involved, especially on the Internet. If the x was meant to multiplied it would normally be written as 48x/2. Personally I see "48/2x" and, without additional information, presume the writer intends x to be in the denominator. A similar case is something like "sqrt 2x". No one would interperet that to first take (sqrt 2), then multiply by x. That would be "x sqrt 2".

When written on paper, or in a format such as TeX that supports standard math notation, the meaning is unambiguous.

You're right, if I saw 48/2x written on the internet, I would assume they mean the equivalent of 48/(2x).
My poorly-made point was that on the internet, [enough] white space becomes a new order of operation that functions like ()'s.

[mkirsh]

http://cdn2.knowyourmeme.com/i/000/112/837/original/16h6ja8.jpg?1302454815


On my old school TI-81, I get the following:

48/2*(9+3) = 288
48/2(9+3) = 2

So TI thinks that putting in the * breaks the (9+3) out of the denominator but removing it leaves (9+3) in the denominator. Funny

I'm in the camp of 288 being correct however.

[hudlow]

Just another reason for hating math.

[BD80]

Just another reason for hating math.

Wonder how many different answer would be posted if the question were posed on IC?

I'd guess closer to 288 than 2!

[Jim3k]

I've been away from algebra way too long; mid Fifties. Even so, if 288 is the correct answer, I think I must have been taught incorrectly. The phrase PEDMAS has absolutely no meaning to me. It is an entirely foreign mnemonic device. I'm like Jar, I get 2 every time. I was never taught about the order of operation--and I went to two high schools and two junior high schools. None of the teachers taught the order of operation as described here. So far as I am aware, and have always thought, you were supposed to reduce to whole integers before operating them. Thus, anything inside a paren (never called bracket, to my knowledge) was to be processed first, to the extent it could be. If there was an unknown inside the parens, it couldn't be, but you went as far as you could.

I am about 6 years older than my wife, who went to three different junior high schools, and one high school, all in entirely different parts of the country from me. Me: Colorado and Connecticut; her: Washington state, Florida, Maine and Minnesota.

Like me, her answer was 2. Plus she is defensive about it; in her mind the PEDMAS rule is totally foreign.

So not only is this a language problem, it is a teaching problem. Did some sort of change in pertinent math conventions occur in the relatively recent past? Is that an issue? Or were older people simply taught incorrectly? It's not like problems like this are frequently confronted, so most people never use orders of operation.

How can so many people be wrong?

(And, I confess, my math skills are terrible.)

[Jarhead]

The problem is simple. I am right. Prove me wrong. The answer is 2. 2 what? I don't know is on third. http://crazietalk.net/ourhouse/images/smilies/21.gif

[hurleyfor3]

I first learned formally learned order of operations around the age of 10, in an introductory computer class. Maybe my dad taught it to me even earlier. It was a standard part of the curriculum in junior high, which would have been the mid-1980s.

I think the increasing use of computers caused order of operations to become more important. With the invention of spreadsheets (I used to "play" on the original VisiCalc!) and programming languages, it became common to specify mathematical expressions in lines of code. Before this, expressions were usually written on paper or typeset in books, where it was much easier to specify big fraction bars, exponents, and so on, and where the O of O was unambiguous.

[OldPhiKap]

[lawyer opk] Can't we just split the difference at 145 and move on? [/lawyer opk]

[hurleyfor3]

[lawyer opk] Can't we just split the difference at 145 and move on? [/lawyer opk]

That's only like 90 after your fee though, right?

[OldPhiKap]

That's only like 90 after your fee though, right?

Give or take.

Of course, if you're in the "2" camp, you've still come out very well.

[BD80]

[lawyer opk] Can't we just split the difference at 145 and move on? [/lawyer opk]

Well, if you want to go lawyerly, the real questions are: what does the client want the answer to be, and how much is he willing to spend?

My first inclination would be to investigate a change of venue, perhaps to a base nine system that uses a right-to-left convention of reading. I think a judge with a liberal arts background would be more willing to allow this to go to a jury, where I could dazzle.

Don't even get me started about the ambiguity of the "/" : I foresee weeks of depositions on the intent of the drafter and common standard of usage in the industry as well as between the drafter and recipient. There may also be questions regarding the history of dyslexia in the drafter's family, and lingering effects of drug usage in college.

Experts. Oh the experts. As useless and mercenary as lawyers may be, is there a lower form of life on the planet than the hired gun? (yes, I have been an expert on occasion) We could have experts in mathematics, computers, the history of numerology, psychologists, "educators" ... what was our budget again?

I could see this going to the high court where a flood of attorneys could earn serious $ filing amicus briefs preserving/challenging a convention to benefit clients with $ to defend or make.

[OldPhiKap]

Well, if you want to go lawyerly, the real questions are: what does the client want the answer to be, and how much is he willing to spend?

My first inclination would be to investigate a change of venue, perhaps to a base nine system that uses a right-to-left convention of reading. I think a judge with a liberal arts background would be more willing to allow this to go to a jury, where I could dazzle.

Don't even get me started about the ambiguity of the "/" : I foresee weeks of depositions on the intent of the drafter and common standard of usage in the industry as well as between the drafter and recipient. There may also be questions regarding the history of dyslexia in the drafter's family, and lingering effects of drug usage in college.

Experts. Oh the experts. As useless and mercenary as lawyers may be, is there a lower form of life on the planet than the hired gun? (yes, I have been an expert on occasion) We could have experts in mathematics, computers, the history of numerology, psychologists, "educators" ... what was our budget again?

I could see this going to the high court where a flood of attorneys could earn serious $ filing amicus briefs preserving/challenging a convention to benefit clients with $ to defend or make.

BD clearly doesn't bring a knife to a gun fight.

The big problem is: aren't damages in this case, be definition . . . .


{um}





{wait for it}



. . . nominal?

[BD80]

BD clearly doesn't bring a knife to a gun fight.

The big problem is: aren't damages in this case, by definition . . . .

{um}

{wait for it}

. . . nominal?

Scorched earth is not sufficient. Plow, salt, crossplow, salt again. Then submit bill.

Haven't heard much from Carthage recently have you?

As for damages, we'll certainly be going for punitives ... think of the mental anguish caused by mathematic equations over the years! A few discrete inquiries in the jury questionnaire, and I bet we could get a jury willing to go for the death penalty!

[OldPhiKap]

Scorched earth is not sufficient. Plow, salt, crossplow, salt again. Then submit bill.

Haven't heard much from Carthage recently have you?

As for damages, we'll certainly be going for punitives ... think of the mental anguish caused by mathematic equations over the years! A few discrete inquiries in the jury questionnaire, and I bet we could get a jury willing to go for the death penalty!

I haven't heard much from Rome recently, etiher.

Better get a big retainer either way.

[BD80]

I haven't heard much from Rome recently, either. ...

Didn't pay their legal bills, the Visigoths were among the waves of collectors. There was trouble pursuing collection in Constantinople.

[Jarhead]

This thread is getting close to public policy. http://crazietalk.net/ourhouse/images/smilies/icon_tease.gif

[OldPhiKap]

This thread is getting close to public policy. http://crazietalk.net/ourhouse/images/smilies/icon_tease.gif

If having a policy against raiding mongrel hordes is wrong, I don't want to be right.

The Vandals really wrecked the place, after all.

[rthomas]

The problem is simple. I am right. Prove me wrong. The answer is 2. 2 what? I don't know is on third. http://crazietalk.net/ourhouse/images/smilies/21.gif

OK, so Donald Trump comes to you and says that he is gonna give you a job and he will pay you a thousand times 48/2(9+3) for your first year. You say, hell yea, $288K.

[OldPhiKap]

OK, so Donald Trump comes to you and says that he is gonna give you a job and he will pay you a thousand times 48/2(9+3) for your first year. You say, hell yea, $288K.

But only $2k if you don't have your long form. Or if you're Gary Bussey.

[sagegrouse]

I have three answers:


The answer is 288 based on the first computer programming course ever offered by the Duke math department (spring 1963). We were taught FORTRAN, where the divisions are done first. And also, IIRC, the same is true in COBOL and BASIC.

In Microsoft Excel the answer is also 288.

The real answer is -- USE ENOUGH PARENTHESES THAT PEOPLE KNOW WHAT YOU ARE TALKING ABOUT!


Over and out --

sagegrouse

[Jarhead]

OK, so Donald Trump comes to you and says that he is gonna give you a job and he will pay you a thousand times 48/2(9+3) for your first year. You say, hell yea, $288K.

Nah! I say he's a fool. See, I told you we are getting into politics. Oh, sorry. He's not a politco. Just a fool. Okay, let's say he offers me a room for 48/2(9+3) in one of his hotels. Hell, yeah, for $2.00 plus tax.http://crazietalk.net/ourhouse/images/smilies/wizard.gif

[ArnieMc]

The answer is 288 based on the first computer programming course ever offered by the Duke math department (spring 1963). We were taught FORTRAN, where the divisions are done first.
I was in that course. We were taught ALGOL and ALGOL60 because FORTRAN was obsolete. They are still teaching that FORTRAN is obsolete, and it is still the language of choice for number crunching.

[sagegrouse]

I was in that course. We were taught ALGOL and ALGOL60 because FORTRAN was obsolete. They are still teaching that FORTRAN is obsolete, and it is still the language of choice for number crunching.

In the spring of 1963 the computer languages taught in Math 221, or whatever, were machine language (ten digits and a +/- sign), Autocoder (one-to-one with machine language), and then FORTRAN. In 1964 ALGOL was taught, which was probably the class you took.

What I remember best about the course is the prof's lecture before launching us on FORTRAN. "You are about to learn a very ineffieicnt computer programming language. You should not use it, but use Autocoder instead. You see, people are very cheap, but computers are very expensive."

And, of course, in 1963 he was right.

I still used FORTRAN for a couple of decades off and on (mostly off) for multivariate statistical techniques of my own device. When I went to the Congressional Budget Office, it acquired a Wang computer preloaded with BASIC. I had an occasion to build a statistical or mathematical model. So, I wrote it in FORTRAN and then learned BASIC by looking up the error messages ("FOR loops" instead of "DO loops").

And then I spent 30 years in the computer services industry, where I discovered how dangerous my code-slinging techniques were.

sagegrouse

[Newton_14]

Discussed this with a few folks at work today and all agreed the answer is 288. One guy found the following on Wiki. Interesting.


MnemonicsMnemonics are often used to help students remember the rules, but the rules taught by the use of acronyms can be misleading. In the United States the acronym PEMDAS is common. It stands for Parentheses, Exponents, Multiplication, Division, Addition, Subtraction. If PEMDAS is followed without remembering that multiplication and division have the same weight, and addition and subtraction have the same weight. Doing multiplication before division, can give the wrong answer. So can doing addition before subtraction. Some grade school books teach this incorrectly. For example: 6÷2×3 = 9, not 1. 6-2+3 = 7, not 1.

[ArnieMc]

In the spring of 1963 the computer languages taught in Math 221, or whatever, were machine language (ten digits and a +/- sign), Autocoder (one-to-one with machine language), and then FORTRAN. In 1964 ALGOL was taught, which was probably the class you took.

What I remember best about the course is the prof's lecture before launching us on FORTRAN. "You are about to learn a very ineffieicnt computer programming language. You should not use it, but use Autocoder instead. You see, people are very cheap, but computers are very expensive."

And, of course, in 1963 he was right.

I still used FORTRAN for a couple of decades off and on (mostly off) for multivariate statistical techniques of my own device. When I went to the Congressional Budget Office, it acquired a Wang computer preloaded with BASIC. I had an occasion to build a statistical or mathematical model. So, I wrote it in FORTRAN and then learned BASIC by looking up the error messages ("FOR loops" instead of "DO loops").

And then I spent 30 years in the computer services industry, where I discovered how dangerous my code-slinging techniques were.

sagegrouse

I think you're right. I must have taken programming my sophomore year instead of my freshman year.

I always enjoy talking to new aerospace engineers about codes. The first thing I ask them is "Do you know Fortran?" The answer is always "No, Fortran's obsolete. They don't teach it anymore. We learned _______ ." Unfortunately for them, the "legacy" codes that do most of the engineering computations are written in Fortran. There will be some problems when the last of our generation retires (or keels over at our computers).

[darthur]

Discussed this with a few folks at work today and all agreed the answer is 288. One guy found the following on Wiki. Interesting.

The quote you mentioned isn't really the point. Nobody would argue about 6÷2×3 = 9. The ambiguity comes from whether you should read a(b) as implicitly being a single term. For example, someone mentioned 48/2x with x=12 earlier. I think the vast majority of people would agree that should be read as 48 / (2x), rather than as 48÷2×x. It has nothing to do with PEDMAS - it has to do with how you interpret what is written.

Anyway, sagegrouse expressed the answer better than I did: "The real answer is -- USE ENOUGH PARENTHESES THAT PEOPLE KNOW WHAT YOU ARE TALKING ABOUT!". I did a fair amount of research mathematics before going into compsci in the end, and no mathematician worth his salt would ever, ever use an expression like this in a paper: it is ambiguous and the only person who is wrong is the person who wrote it.

[hughgs]

The quote you mentioned isn't really the point. Nobody would argue about 6÷2×3 = 9. The ambiguity comes from whether you should read a(b) as implicitly being a single term. For example, someone mentioned 48/2x with x=12 earlier. I think the vast majority of people would agree that should be read as 48 / (2x), rather than as 48÷2×x. It has nothing to do with PEDMAS - it has to do with how you interpret what is written.

The expression 48/2x is only ambiguous if you don't follow the rules. The lack of an operator between the "2" and the "x" has always implied multiplication, a fact that you clearly agree with. If people add parentheses when none are indicated then that's not an issue with the ambiguity of the statement. I agree that the issue is independent of the PEMDAS (the way I was told it) mnemonic but it is also independent of how the majority interpret the expression.

Rather say that the expression is ambiguous I would argue that the expression is simply poorly written in that it isn't intuitively clear what one is calculating. But, that is very different than saying that the expression is ambiguous. I can't think of any example where mathematics is ambiguous.

[77devil]

Angels on a pin head-we are doomed.

[BD80]

Angels on a pin head-we are doomed.

Flat or round pin head?

[Deslok]

The expression 48/2x is only ambiguous if you don't follow the rules. The lack of an operator between the "2" and the "x" has always implied multiplication, a fact that you clearly agree with. If people add parentheses when none are indicated then that's not an issue with the ambiguity of the statement. I agree that the issue is independent of the PEMDAS (the way I was told it) mnemonic but it is also independent of how the majority interpret the expression.

Rather say that the expression is ambiguous I would argue that the expression is simply poorly written in that it isn't intuitively clear what one is calculating. But, that is very different than saying that the expression is ambiguous. I can't think of any example where mathematics is ambiguous.

Well, since you asked ;)
What is 0^0 (or perhaps more accurately, is it defined, since it is generally agreed upon what the numerical value of it would be, were it defined).
And of course, the obvious ambiguous case. In triangle ABC, AB = 10, BC = 6, and angle A = 30 degrees. What is the length of side AC?

[snowdenscold]

And of course, the obvious ambiguous case. In triangle ABC, AB = 10, BC = 6, and angle A = 30 degrees. What is the length of side AC?

It's been awhile since geometry and trig, but is that ambiguous in the technical sense?

I thought any time you knew two lengths and an angle (which is NOT the angle between the sides you know) you have two answers. So not ambiguous but expected?

[Deslok]

Basically, so long as BC is smaller than AB, but bigger than AB sin A, you have what is, by definition, the ambiguous case, in that either triangle is correct according to the parameters given. So, unless more information is given, the result is expectedly ambiguous.

You can also get into some ambiguities in Godelian logic, but, its been too long for me to remember details and present/explain them(essentially the incompleteness theorems).

[snowdenscold]

Basically, so long as BC is smaller than AB, but bigger than AB sin A, you have what is, by definition, the ambiguous case, in that either triangle is correct according to the parameters given. So, unless more information is given, the result is expectedly ambiguous.


I guess I just was thrown by the use of ambiguous in that case. I was thinking it was analogous to saying x^2 - 1 = 0, solve for x. Is x = {-1,1} ambiguous?

[Dukiedevil]

288 is the correct answer. I think most people who have written code (or worked extensively with spreadsheets) know how should be interpreted. I was always taught to be very careful with the denominators of fractions when coding for this very reason. If you mean for this answer to be 2, then you need more parentheses (as someone mentioned above).

[OldPhiKap]

Angels on a pin head-we are doomed.

California, or Victoria's Secret?

[hughgs]

Well, since you asked ;)
What is 0^0 (or perhaps more accurately, is it defined, since it is generally agreed upon what the numerical value of it would be, were it defined).
And of course, the obvious ambiguous case. In triangle ABC, AB = 10, BC = 6, and angle A = 30 degrees. What is the length of side AC?

0^0 is undefined, the same as 0/0. To solve those cases you have to use limits.

Your triangle example may be ambiguous but the math behind it is not. You're comparing apples to oranges. Show me how the math behind your triangle case is ambiguous. Just because there is more than one answer to a problem doesn't make the math ambiguous. I can come up with millions of examples of problems that have more than one answer.

There's a big difference between ambiguous math and ambiguous answers. Here's one example. At t=0, a ball is 100 m above the ground with velocity of 10 m/s upward. When is the ball on the ground?

If you go through the math you should get a positive and negative time. Most people would say then say that the positive time is correct. But, if you read the problem carefully, you'll notice that I didn't say that the ball started at 100 m, I simply said that at t=0 the ball was at 100 m. So, both times are correct.

The same thing holds for your triangle. The length of AC can have an infinite number of lengths. Draw any triangle that meets your criteria and the answers provided by the math will contain your triangle. Multiple answers is not due to ambiguous math but due to the problem statement.

[darthur]

The expression 48/2x is only ambiguous if you don't follow the rules. The lack of an operator between the "2" and the "x" has always implied multiplication, a fact that you clearly agree with. If people add parentheses when none are indicated then that's not an issue with the ambiguity of the statement. I agree that the issue is independent of the PEMDAS (the way I was told it) mnemonic but it is also independent of how the majority interpret the expression.

Rather say that the expression is ambiguous I would argue that the expression is simply poorly written in that it isn't intuitively clear what one is calculating. But, that is very different than saying that the expression is ambiguous. I can't think of any example where mathematics is ambiguous.

It's not the mathematics that's ambiguous, it's the notation that's ambiguous. This is why I said earlier that it's a language problem. Your entire argument is built on the fact that 2x implies multiplication but not parentheses. Says whom?

And it's not just intuition that's at stake here. Math is *not* a formal language that can be fed into a computer. Once you get far enough, proofs and papers tend to be more prose than equations anyway. There are formal rules to define exactly how you should interpret common expressions, but when you deviate from the common expressions, it is the writer's responsibility to explicitly say how it should be interpreted. Certainly the original post falls into that category (although there is no reason to be using uncommon notation for an expression this simple), but I'd argue that few mathematicians would ever read 48/2x as anything but 2. See here for a particularly common example of this usage: http://www.jimloy.com/algebra/quad.htm.

Deslok's 0^0 example is actually quite apt. Yes, it's undefined mathematically, but it is very useful to have the notation mean something sometimes. It is extremely common to write a polynomial as sum c_i * x^i, but this is not technically well defined for i = x = 0, so the author will tell us to interpret 0^0 as being equal to 1. It is just a convention, like many other things, and in other contexts, people might tell you to interpret it as being equal to 0. This is perfectly fine, and neither is wrong.


The same thing holds for your triangle. The length of AC can have an infinite number of lengths.

Actually it can only have 2 :D.

[Indoor66]

Actually it can only have 2 :D.

I know that one - longer or shorter! :cool:

[mkirsh]

I don't understand the 2 answer. Not sure if the missing function is messing things up or the division sign/order of operations, but this is how I see it:


48/2(9+3)

With no function between 2 and (9+3) the assumed function is multiplication, so we have:

48/2*(9+3)

The next problem I think people have is with the order of division vs multiplication. But if we replace /2 with *0.5 we get:

48*0.5*(9+3)

Which is 48*0.5*12 = 288

[OldPhiKap]

I don't understand the 2 answer.

You only get there if you read the "/" as creating a fraction, with 2(9+3) being the denominator.

48
2 (9+3)

=

48
24

=

2

[hughgs]

It's not the mathematics that's ambiguous, it's the notation that's ambiguous. This is why I said earlier that it's a language problem. Your entire argument is built on the fact that 2x implies multiplication but not parentheses. Says whom?

The lack of an operator is called an implied multiplication operation. I suppose that we could ask every mathematician in the world if it exists but I suspect you're real question is why can't the implied operation be parentheses. That's a good hypothesis so show me a reputable source that says discusses the implied parentheses operation.


And it's not just intuition that's at stake here. Math is *not* a formal language that can be fed into a computer. Once you get far enough, proofs and papers tend to be more prose than equations anyway. There are formal rules to define exactly how you should interpret common expressions, but when you deviate from the common expressions, it is the writer's responsibility to explicitly say how it should be interpreted. Certainly the original post falls into that category (although there is no reason to be using uncommon notation for an expression this simple), but I'd argue that few mathematicians would ever read 48/2x as anything but 2. See here for a particularly common example of this usage: http://www.jimloy.com/algebra/quad.htm.

I agree that the expression is poorly written, but not because the expression is ambiguous but because it's not intuitive. And I completely disagree with your assertion that "... few mathematicians would ever read 48/2x as anything but 2." You may find some that read it that way but I suspect that if you told them the answer was 288 they would realize their mistake. Just because an expert gets it wrong doesn't mean the notation is ambiguous (and I'm not sure I consider Jim Loy a mathematician). The notation is completely unambiguous. It may not be intuitive but that's another argument.


Deslok's 0^0 example is actually quite apt. Yes, it's undefined mathematically, but it is very useful to have the notation mean something sometimes. It is extremely common to write a polynomial as sum c_i * x^i, but this is not technically well defined for i = x = 0, so the author will tell us to interpret 0^0 as being equal to 1. It is just a convention, like many other things, and in other contexts, people might tell you to interpret it as being equal to 0. This is perfectly fine, and neither is wrong.

In the polynomial case you cite at no time does anyone actually evaluate x^0. As you are probably aware the polynomial is actually c_0 + c_1*x + .... The fact that it gets written as a summation is simply an abuse of notation that everyone (including myself) uses and understands. In this particular case the author allows the abuse of notation by defining 0^0 = 1 to make it easy for the readers and to keep a terse notation. But, there is no evaluating 0^0 and hence
there is no ambiguity.

In fact, in quite a number of math texts you'll find that authors separate out the constant from the other polynomial factors to avoid this very issue. Boyce and diPrima in the Duke MTH108 course keep the constant out of the summation during the definition of the Fourier series (which is kind of an ancillary case). The authors of the Duke MTH032L book also keep the constant out of the summation when defining the Taylor series. I personally keep the constant in the summation and have had students question the usage. I happily admit that I'm abusing the notation and give them the same explanation.

In either case just because authors abuse notation doesn't mean that the mathematics is ambiguous, they're simply trying to make things easier for the reader.


Actually it can only have 2 :D.

DOHHH! Yeah, you're right. I was trying to do the solution set in my head and rotated the wrong leg of the triangle. I needed to rotate BC which can only meet the leg AC in two possible points.

[darthur]

I agree that the expression is poorly written, but not because the expression is ambiguous but because it's not intuitive. And I completely disagree with your assertion that "... few mathematicians would ever read 48/2x as anything but 2." You may find some that read it that way but I suspect that if you told them the answer was 288 they would realize their mistake. Just because an expert gets it wrong doesn't mean the notation is ambiguous (and I'm not sure I consider Jim Loy a mathematician). The notation is completely unambiguous. It may not be intuitive but that's another argument.

Let me put it this way. I am not a mathematician right now because I decided to get a PhD in theoretical computer science instead of math. I did however come to Duke under a full scholarship for my math abilities and I did graduate as one of two faculty award nominees from the math department, so I hope you believe that I could just as easily be a mathematician. In either case, my Dad is a former president of the American Math Society, and I count many of the most talented mathematicians of my generation as very close friends.

So please believe me when I say that it is simply not true that a mathematician would be convinced it is unreasonable to interpret 48 / 2x as 48 / (2 * x) when you explained PEDMAS to them.

PS: I have no idea who Jim Loy is. I just happened to remember that the quadratic formula has "2x" in the denominator, so I used Google to prove that people often do use 2x to mean (2*x) rather 2*x. I now challenge you to find a single example of someone using X/2Y to mean (X/2) * Y :D.

[darthur]

Random followup: I did Google "implied multiplication" for fun. One of the top hits was a Dr. Math response who did prefer (48/2)*x over 48/(2*x) but specifically said both are used in text books, both interpretations are correct, and the question itself which prompted the discussion was therefore wrong.

So yes, I may have to retract my claim that ab = (a*b) is actually more common than ab = a*b, but I have now been vindicated by the internets in claiming that both are completely correct :D.

[hughgs]

So please believe me when I say that it is simply not true that a mathematician would be convinced it is unreasonable to interpret 48 / 2x as 48 / (2 * x) when you explained PEDMAS to them.

I never said that a mathematician (or anyone) wouldn't think it was unreasonable to put the "x" in the denominator. Reasonableness is a completely separate argument from ambiguity. I agree that a reasonable person would say that the expression should be 48/(2x). That is why the expression is poorly written. It's too easy to interpret the expression the wrong way. I've said that multiple times.

But, the original argument is whether the expression is ambiguous. The rules are simple. The expression is 48/2x = 48/2*x. There is no ambiguity.

[darthur]

But, the original argument is whether the expression is ambiguous. The rules are simple. The expression is 48/2x = 48/2*x. There is no ambiguity.

There is no universal authority that dictates the meaning of math notation. There is only general consensus, and in this case, there is no consensus either in books, in calculators, or in people. Hence, there is no rule.

Anyway, we're talking in circles. Believe what you will.

[Jarhead]

We are definitely talking in circles. Can anyone create a real problem that gives us the same irregularity that we are playing with here? Would that help us?

[OldPhiKap]

Anyway, we're talking in circles.

Frustra laborant quotquot se calculationibus fatigant pro inventione quadraturae circuli.

“Futile is the labor of those who fatigue themselves with calculations to square the circle."



(Still a great thread, btw, to all).

[hughgs]

There is no universal authority that dictates the meaning of math notation. There is only general consensus, and in this case, there is no consensus either in books, in calculators, or in people. Hence, there is no rule.

Anyway, we're talking in circles. Believe what you will.

I don't agree that we're talking in circles. But, I think you've hit the crux of our argument, that the idea of the order of operation is NOT a universal rule. Whether there's a consensus or not simply demonstrates whether people follow some sort of rule. However, the idea that the order of operation is not a universal rule is something that I haven't heard before.

So, can you show me a link or something that says that there is no universal rule? I'm not trying to be a smart I'm a real wanker for saying this.I'm a real wanker for saying this.I'm a real wanker for saying this. about this but everything I've seen or done has always followed the rules outlined by PEMDAS and if that idea is wrong I would like to know that it's wrong. Thanks.

[darthur]

I don't agree that we're talking in circles. But, I think you've hit the crux of our argument, that the idea of the order of operation is NOT a universal rule. Whether there's a consensus or not simply demonstrates whether people follow some sort of rule. However, the idea that the order of operation is not a universal rule is something that I haven't heard before.

So, can you show me a link or something that says that there is no universal rule? I'm not trying to be a smart I'm a real wanker for saying this.I'm a real wanker for saying this.I'm a real wanker for saying this. about this but everything I've seen or done has always followed the rules outlined by PEMDAS and if that idea is wrong I would like to know that it's wrong. Thanks.

http://mathforum.org/library/drmath/view/54341.html

"Some texts make a rule, as in your second solution, that multiplication without a symbol ("implied multiplication") should be done before any other operations in an expression, including "explicit multiplication" using a symbol."

"So to answer your question, I think both answers can be considered right - which means, of course, that the question itself is wrong."

He prefers your definition, but nonetheless concedes there is zero consensus, which means there is no universal definition and hence no rule. He does says he doesn't know mathematicians who favor the implied multiplication rule, but *I* know plenty - I'm guessing he's just not well connected with mathematicians :). Dr. Math is a site for high-school teachers and students after all.

[hughgs]

http://mathforum.org/library/drmath/view/54341.html

"Some texts make a rule, as in your second solution, that multiplication without a symbol ("implied multiplication") should be done before any other operations in an expression, including "explicit multiplication" using a symbol."

"So to answer your question, I think both answers can be considered right - which means, of course, that the question itself is wrong."

He prefers your definition, but nonetheless concedes there is zero consensus, which means there is no universal definition and hence no rule. He does says he doesn't know mathematicians who favor the implied multiplication rule, but *I* know plenty - I'm guessing he's just not well connected with mathematicians :). Dr. Math is a site for high-school teachers and students after all.

Let me first address your major point. Zero consensus is completely independent from a universal definition. There are plenty of universal rules that have zero consensus. So, you're example is not an example that rejects the idea of a universal rule. In fact he addresses that point:

"... the idea of Order of Operations (or precedence, as it is called in the computer world) is supposed to be to ensure that everyone will interpret an otherwise ambiguous expression the same way ..."

I also reject your conclusion that Dr. Math says he "doesn't know any mathematicians who who favor the implied multiplication rule". What he says is that "... general rule among mathematicians that implied multiplication should be done before explicit multiplication, ...". Again, those are two very different statements.

Finally, if you don't think Dr. Math is "not well connected with mathematicians", a point I'm more than willing to conceded, then why should I use take his word as an authority?

So, I don't think you've shown me a site where states there is no rule. In fact, I would argue that if we agree that the site is written by an expert then he argues my point, that while the expression is poorly written it is ambiguous.

[darthur]

Let me first address your major point. Zero consensus is completely independent from a universal definition. There are plenty of universal rules that have zero consensus. So, you're example is not an example that rejects the idea of a universal rule. In fact he addresses that point:

"... the idea of Order of Operations (or precedence, as it is called in the computer world) is supposed to be to ensure that everyone will interpret an otherwise ambiguous expression the same way ..."

I also reject your conclusion that Dr. Math says he "doesn't know any mathematicians who who favor the implied multiplication rule". What he says is that "... general rule among mathematicians that implied multiplication should be done before explicit multiplication, ...". Again, those are two very different statements.

Finally, if you don't think Dr. Math is "not well connected with mathematicians", a point I'm more than willing to conceded, then why should I use take his word as an authority?

So, I don't think you've shown me a site where states there is no rule. In fact, I would argue that if we agree that the site is written by an expert then he argues my point, that while the expression is poorly written it is ambiguous.

Ugh. We've been over all this before.

Pretty much only high school math teachers think this is a universal definition, even though there is not even consensus among other high school teachers and/or books.

Anyway, forget I posted - I do not really care whether you believe me. I *promise* you this is not a definition that is used by mathematicians, and there is nothing further from the spirit of mathematics than arguing about what is or isn't widely accepted notation. That Dr. Math author may not have worked many mathematicians, but I have, and it's true. Too bad that you don't believe me.

[Jarhead]

Earlier on this thread I asked that someone state an actual problem that would encompass this same dichotomy of solutions. So far, no one has offered to do so. Math is supposed to be one of the natural sciences. Please, somebody explain how this problem could actually happen.

[OldPhiKap]

There are plenty of universal rules that have zero consensus. So, you're example is not an example that rejects the idea of a universal rule. . . .

What he says is that "... general rule among mathematicians that implied multiplication should be done before explicit multiplication, ...". Again, those are two very different statements..

So there are universal rules AND general rules?

[Bostondevil]

If this many people disagree then the statement is ambiguous. Unintentionally ambiguous statements are just bad writing, whether it's an English paper or a math problem.

It has been my experience though, that ambiguously gay duos will make 14 year old boys laugh.

[Hancock 4 Duke]

We learned in pre-algebra, when a question like this is asked, always start with distribution to the parentheses. So 48/2(9+3), you would distribute the 2 to the 9 and the 3.
48/(18+6)
48/24
2

[Udaman]

The correct answer is whatever the Wall Street CEO tells the Wall Street CFO it should be.

[snowdenscold]

We learned in pre-algebra, when a question like this is asked, always start with distribution to the parentheses. So 48/2(9+3), you would distribute the 2 to the 9 and the 3.
48/(18+6)
48/24
2

That's not really the issue at all. IF it were just 2(9+3) sure that makes sense, but we have mix of mulitplication and division here.

It all comes back to the issue created by using a '/' in typed text. On a whiteboard (or using LaTeX) there's no issue.

[Jarhead]

That's not really the issue at all. IF it were just 2(9+3) sure that makes sense, but we have mix of mulitplication and division here.

It all comes back to the issue created by using a '/' in typed text. On a whiteboard (or using LaTeX) there's no issue.
48/2(9+3)
How would it appear on a black board? That's the medium I used when demonstrating my math skills to my classmates. On a black board I would show:

__48__
2(9+3)

The Nun would have rapped my knuckles with a ruler if I did otherwise.

[snowdenscold]

48/2(9+3)
How would it appear on a black board? That's the medium I used when demonstrating my math skills to my classmates. On a black board I would show:

__48__
2(9+3)

The Nun would have rapped my knuckles with a ruler if I did otherwise.

But that's going backward. You would have to ask, "What on a blackboard would show up as 48/2(9+3) when typed?"

And the problem is if you saw
48 (9+3)
2
and
__48__
2(9+3)

hopefully neither would be translated to solely 48/2(9+3) to avoid confusion. But if it was, which one was it?

[darthur]

But that's going backward. You would have to ask, "What on a blackboard would show up as 48/2(9+3) when typed?"

And the problem is if you saw
48 (9+3)
2
and
__48__
2(9+3)

hopefully neither would be translated to solely 48/2(9+3) to avoid confusion. But if it was, which one was it?

Yeah, I can't imagine transcribing the first as anything but 48(9+3) / 2. The second I really would transcribe as 48/2(9+3) if I was being lazy.