What is the answer?

[battlewraith]

6÷2(1+2) = ?

[PeregrineFalcon]

Nein.

 

😁

 

No, seriously. The actual answer is 9.

.

Edited May 25, 2022 by PeregrineFalcon

[Six-Six]

According to Bob Dylan, it's blowing in the wind.

[ThaOGDreamWeaver]

2 hours ago, Six-Six said:

According to Bob Dylan, it's blowing in the wind.

Except in New Jersey, where what's blowing in the wind smells funny.

[battlewraith]

5 hours ago, PeregrineFalcon said:

Nein.

 

😁

 

No, seriously. The actual answer is 9.

.

Yeah apparently what is considered the correct answer is 9. 

The mathematical order of operations that I learned in school gave me an answer of 1. And I'm not the only one. The comments section of the video where I saw this issue discussed was full of people who arrived at the same conclusion. So now I'm wondering if the "old" way was some sort of fluke of the American educational system at the time. I'm also wondering if the "new" way is the result of the influence of programmers on the field of mathematics. I read comments to the effect that the order of operations that yields 9 as an answer is more reliable and less ambiguous than the order that gives 1 as a result--thus more preferable from a coding standpoint.

[Techwright]

2 hours ago, battlewraith said:

Yeah apparently what is considered the correct answer is 9. 

The mathematical order of operations that I learned in school gave me an answer of 1. And I'm not the only one. The comments section of the video where I saw this issue discussed was full of people who arrived at the same conclusion. So now I'm wondering if the "old" way was some sort of fluke of the American educational system at the time. I'm also wondering if the "new" way is the result of the influence of programmers on the field of mathematics. I read comments to the effect that the order of operations that yields 9 as an answer is more reliable and less ambiguous than the order that gives 1 as a result--thus more preferable from a coding standpoint.

You're not alone, I too, thought it was 1.

 

Just to be sure, are the parties that posted the answer reliable?  They're not some flippant video-obsessed teenagers just spouting answers, or scam callers branching out into a new line of work are they?  They're honest-to-goodness math professors, or at least stayed at a Holiday Inn Express last night?

[PeregrineFalcon]

Ok, this has been bugging me, so I went and found this:

 

- What is the Order of Operations?

Operations have a specific order, and this is what “Please Excuse My Dear Aunt Sally” helps us to understand. It’s an acronym that tells us in which order we should solve a mathematical problem.

 

“Please” stands for “Parentheses,” so we solve everything inside of the parentheses first.

Then, “Excuse,” which is for “Exponents.” We solve that after we solve everything in parentheses.

Multiplication, which is the “My,” and this happens from left to right.

And then division, which is the “Dear,” which also happens left to right.

And then we have addition and subtraction, which also happens from left to right, and this is “Aunt” and “Sally.” -

 

It's been awhile since I've been in school, but I do vaguely remember the acronym PEMDAS. So using that order of operations the formula should be solved in this order:

6÷2(1+2) = ?

6÷2*3 = ?

6÷6 = ?

? = 1

 

Which means that I was wrong and you all were correct.

[TygerDarkstorm]

7 minutes ago, PeregrineFalcon said:

Ok, this has been bugging me, so I went and found this:

 

- What is the Order of Operations?

Operations have a specific order, and this is what “Please Excuse My Dear Aunt Sally” helps us to understand. It’s an acronym that tells us in which order we should solve a mathematical problem.

 

“Please” stands for “Parentheses,” so we solve everything inside of the parentheses first.

Then, “Excuse,” which is for “Exponents.” We solve that after we solve everything in parentheses.

Multiplication, which is the “My,” and this happens from left to right.

And then division, which is the “Dear,” which also happens left to right.

And then we have addition and subtraction, which also happens from left to right, and this is “Aunt” and “Sally.” -

 

It's been awhile since I've been in school, but I do vaguely remember the acronym PEMDAS. So using that order of operations the formula should be solved in this order:

6÷2(1+2) = ?

6÷2*3 = ?

6÷6 = ?

? = 1

 

Which means that I was wrong and you all were correct.

I had to redo it this way myself as originally I got 9. Then I went to type out an explanation similar to this (but how it gets the answer 9) and as I was realizing I was wrong, I saw your post come in. 🤣 

 

Although looking at it, I was under the impression that once you hit just multiplication and division (or addition/subtraction) you just go from left to right, not necessarily in the order of PEMDAS. Edit: That is to say, that if the problem is only multiplication/division left, it doesn't matter which order you do it in; multiplication does not have to be the first one. Same as Addition does not have to be done before subtraction. e.g., The problem would look more like this:

6/2(1+2) = ?

6/2*3 = ?

3*3 = ?

? = 9

Edited May 26, 2022 by TygerDarkstorm
Clarification

[battlewraith]

So I wanted to get people's responses before I posted the video which is actually from 2016. The youtuber studied math and economics from Stanford.

 

The comment section is interesting because there are people arguing for both interpretations. There is a general consensus that we shouldn't write equations like this.

[TheOtherTed]

@PeregrineFalcon was right the first time, as was @TygerDarkstorm's intuition.  PEMDAS applies, but neither multiplication nor division have operative precedence over the other.  The same is true with addition and subtraction.  In both cases, they are calculated left to right (but with MD together having operative precedence over AS).

 

The reason AS is calculated left and right is clear - you're going to get the same number whether you prioritize addition, or calculate left to right.  The reason MD is calculated left to right is less clear, but you can see from this thread that you get different results when you prioritize multiplication, as opposed to calculating left to right.  Thus, the left-to-right rule for MD is pretty much there to avoid these ambiguities.  So...

 

6÷2(1+2) =

6÷2(3) =

3(3) =

9

 

The punchline is that If you really want to prioritize multiplication over division, PEMDAS lets you force it with parentheses, for example, 6÷(2(1+2)) .

 

...or, you know, you could watch the video, which I may or may not have done.

 

Edited May 26, 2022 by TheOtherTed
bleargh

[WanderingAries]

But, that's NOT how that works...

 

The P in PEMDAS means that you solve the equation inside of the Parentheses BEFORE doing anything else. PEMDAS is actually better written as PE[M/D][A/S] where the brackets mean that they are equal, but performed in left to right priority with the possibility of a reciprocal PE depending on the equation (IE: nested PE). So that means you end up with 6 / 2 * 3, not the nested thing I see just above. The fuzzy part is where you run into "Do I convert the division into a fraction?". Converting it to a fraction gives you the different answer only if the condition far below is true, but once you've completed the Parenthesis, then you're actually left with simple Division and Multiplication. In this case, the Division goes first.

 

Based on the provided equation, you'd get the following:

 

6 / 2(1+2)

6 / 2(3)

6 / 2 * 3

3 * 3

9

 

The key here is that anything adjacent to a Parenthesis is implicit Multiplication

 

What's throwing everyone is that at some point or another we were all taught that division can be rewritten as a fraction.

 

With that logic, you get the following:

 

6 / 2(1+2)

 

     6

--------

 2(1+2)

 

     6

--------

   2(3)

 

     6

--------

     6

 

     1

 

But in reality, that only works if it were written like this where an additional set of brackets was originally included:

 

6 / [2(1+2)]

 

I tried to find the rule for this, but IDK what to search for.

[PeregrineFalcon]

30 minutes ago, WanderingAries said:

I tried to find the rule for this, but IDK what to search for.

The problem with your method is that the M in PEMDAS says very specifically that all multiplication is done and THEN division, which is the D in PEMDAS. Your solution ignores that and treats multiplication and division as equal, which they are not.

[TygerDarkstorm]

2 hours ago, PeregrineFalcon said:

The problem with your method is that the M in PEMDAS says very specifically that all multiplication is done and THEN division, which is the D in PEMDAS. Your solution ignores that and treats multiplication and division as equal, which they are not.

Incorrect: https://www.infoplease.com/math-science/mathematics/numbers-formulas/order-of-operations#:~:text=There are no Exponents.,done in the same step.

Once you're left with just multiplication/division, you treat them as equals and solve from left to right.

[huang3721]

On 5/25/2022 at 11:53 PM, battlewraith said:

6÷2(1+2) = ?

 

According to you guys, is this:

 

Equals to A): 

 

Or B):  ?

 

Choose one

[battlewraith]

The way you've rewritten it is equal to A. The way you've rewritten it is the way I was taught to solve it in school--resolve the brackets and then solve everything on the right of the division symbol. So 6÷2(3)  would be read as "six divided by the product of 2 times 3".  If the equation was written 6÷2 x 3, that would be read as "six divided by two times 3" and the answer would be 9. Under the current understanding of PEMDAS these two versions of the equation are considered the same, both with an answer of 9. The way I was taught to do it, the 2(3) implies a grouping that needs to be resolved before the division happens. 

 

If you enter  6÷2(1+2) into Google search, it automatically changes it to (6/2)*(1+2)

[huang3721]

57 minutes ago, battlewraith said:

PEMDAS

 

That question up there was my childhood. Back then, my math teacher loved giving similar questions. Somehow, our class passed.

 

I am not following you guys about PEMDAS. I don't think we need PEMDAS for these problems since they are pure multiplication. In my opinion, ÷2(3) is a dead giveaway that part is a denominator. At least, it is the convention I learned decades ago. My teacher taught me to differentiate between 2x3 and 2(3). Maybe this convention is not observed in the western world?

 

Also, both equations are not similar:
6÷2x3 = 6÷2(3)
6 x 2^-1 x 3 = 6 x 2^-1 x 3^-1
9 = 1 (?)

[battlewraith]

1 hour ago, huang3721 said:

Maybe this convention is not observed in the western world?

 

I don't know. That's part of the reason I posted this. I was taught to answer this question the same way you were, but that was decades ago. 

I suspect that there has been a push to streamline mathematical conventions to make things simpler and more consistent for programming. But that's just a hunch. 

I'm not a mathematician or a programmer.

 

[huang3721]

That or whoever originally posted that problem years ago was a troll. They should put more parenthesis.

Edited May 27, 2022 by huang3721
Typo

[Bionic_Flea]

42.

[battlewraith]

Ok so the plot thickens!

 

Here's a video explaining this mess:

 

 

The people who arrive at the answer of 1 are following PEJMDAS (Parenthesis-exponents-juxtaposition-multiplication-division-addition-subtraction).

As she states  in the video, the majority of mathematicians, scientists, researchers, etc. follow this rule. Historically, juxtaposition was widely assumed before PEMDAS became a thing.

 

So what's the issue? Educators in North America started pushing PEMDAS and it got picked up by calculator manufacturers. So depending on which calculator you were using, the answer to the original equation would be 1 or 9. That is a legitimate problem as she points out in the video. 

 

So I guess the takeaway is this: If you, like me, learned PEJMDAS--make sure that the calculator you're using is doing so as well.

If you are strictly using PEMDAS--you have to make sure that your calculator does the same AND you have to make sure that the equation is properly constructed for PEMDAS. Which is something modern calculators are apparently doing. If you enter a PEJMDAS style of equation, the calculator will add brackets for you. But I guess you should still pay attention to make sure that is still the actual equation you want to solve.

 

The youtuber I posted above has an earlier video on PEMDAS with more examples if people are interested:

 

 

[TheOtherTed]

I'm not young, and I was never taught the acronym PEMDAS in school, but I was also never taught the "juxtaposition before multiplication/division" rule either.

 

That said, as someone whose job demands a lot of on-the-fly math, I never would have typed the original formula as presented.  Depending on the context, I'd either type (6/2)*(1+2) or 6/(2*(1+2)) [but with unit markers of various sciencey types].  That said, I usually do math on paper towels or kim-wipes or on the back of a co-worker's lab glove, so it would all look like fraction multiplication, which is far more intuitive to my old eyes.

 

So on the plus side, I learned that juxtaposition has precedence over MD, so that's cool.  On the not-necessarily-a-down-side, in turns out that all the versions of Excel to which I have access try to auto-correct any attempt at juxtaposition to make it fit a strict left-to-right format.

[WanderingAries]

On 5/26/2022 at 5:42 PM, TygerDarkstorm said:

Incorrect: https://www.infoplease.com/math-science/mathematics/numbers-formulas/order-of-operations#:~:text=There are no Exponents.,done in the same step.

Once you're left with just multiplication/division, you treat them as equals and solve from left to right.

 

Correct, every definition of PEMDAS I've ever seen states (as I tried to show) that P and E are treated separate, but M/D and A/S are equal once the P&E are resolved. Keeping in mind that you solve Left -> Right.

[PeregrineFalcon]

5 minutes ago, WanderingAries said:

Correct, every definition of PEMDAS I've ever seen states (as I tried to show) that P and E are treated separate, but M/D and A/S are equal once the P&E are resolved. Keeping in mind that you solve Left -> Right.

So what you're saying is that I was right the first time? 😉

[WanderingAries]

22 minutes ago, PeregrineFalcon said:

So what you're saying is that I was right the first time? 😉

 

At this point, IDK if this whole thing is throwing me due to remembering wrong (or maybe just forgetting a rule) or the fact that I've gone through both American and British based schools (with a hint of French). I'm constantly having trouble deciding between gray/grey for this reason alone and often interchange theater/theatre. X.X

 

What I Distinctly remember from all schooling was that I should Always be careful and show my brackets when intentionally separating things as to indicate when something like 1/2x is really the fraction of 1/(2x) so that when my answers would get checked, that the teacher would know exactly what I meant. I also grew up during the introduction of the calculators mentioned above and always found it odd that teachers of all types specifically asked for certain calculators and was often confused why. It was explained that it was because that's what They were going to use, so as not to have to figure out the keys on one vs another during their explanation. That and the TI calculators had cool mods that they used to present with.

 

So I guess that I can say I was being shown the PEJMDAS as being what to follow growing up and didn't realize it even though I distinctly recall Hearing PEMDAS as the statement? (See note on "maybe I just forgot a rule")

[battlewraith]

In the second video I posted above ("PEMDAS is wrong"), she looks at examples from different sources in different fields using PEJMDAS.

In physics, she looked at Richard Feynman's math in his famous lecture series. In the description for that video, she asks:

 

"When it comes to order of operations (PEMDAS, BODMAS, BEDMAS, BIDMAS), who are you going to believe: your primary school teacher, or Richard Feynman?"

 

That kind of settles it for me. "1" is the correct answer. The reason being that if we don't accept that answer, due to PEMDAS, then we would have to deem a vast quantity of theory by mathematicians, scientists, etc. that worked, AND was the foundation of subsequent theory, as incorrect. I understand that from a programming perspective PEMDAS is preferable, but maybe that preference shouldn't bleed into other areas.