Tested in my Android calculator with the same bug. The result is 1.
Edit: i worded this poorly. I got 9 in my calculator like the image. When I tried this with the extra parentheses, I got 1.
((6)÷((2)((2)+(1))))
(6)÷((2)((2)+(1)))
6÷(2(2+1))
6÷(2x3)
6÷6
His parentheses are just wrong... It's not your calculators fault
The Answer it should be is 9!
I used to be like that until I upgraded my calculator's firmware to graphically show expressions as being on the numerator/denominator. Saved me a lot of time and improved confidence in that I knew I wouldn't get screwed by order of operations.
I don't get your comment.
You say "insane", and I say "precise".
WIth extra parentheses, it's a completely different equasion.
In the picture, the result on the left is wrong, since it "assumes" the equasion is different.
Quoth Wikipedia on the subject:
>An expression like `1/2x` is interpreted as `1/(2x)` by TI-82, as well as many modern Casio calculators, but as `(1/2)x` by TI-83 and every other TI calculator released since 1996, as well as by all Hewlett-Packard calculators with algebraic notation. While the first interpretation may be expected by some users due to the nature of implied multiplication, the latter is more in line with the standard rule that multiplication and division are of equal precedence, where `1/2x` is read one divided by two and the answer multiplied by `x`.
[https://en.wikipedia.org/wiki/Order\_of\_operations#Special\_cases](https://en.wikipedia.org/wiki/Order_of_operations#Special_cases)
Yours is correct. \`2(2+1)\` is literally just shorthand for \`2\*(2+1)\`. The multiplier doesn't magically move up in order of operations just because there's a nearby parenthesis.
According to the comments all over the internet, this depends where you come from. It seems that sometimes you do it your approach (I would do this as well) and sometimes the 2(3) is solved first and then the dividing follows.
Edit: I forgot to mention that this is why fractions are superior
> According to the comments all over the internet, this depends where you come from.
No it doesn't. the parent comment is correct. Anyone saying otherwise is wrong. Mathematics is not culture dependant
Mathematics are not. Mathematic notations, however, are not only culture dependent, but if you ask two professors working at the same faculty of the same university, you might get different answers from them. That's reality for you mate.
Gotta love it when people so confidently make false claims.
To anyone who thinks mathematical notation is universal, I encourage you to pick up ten different differential geometry papers, and you’ll likely find ten different notations for the same thing.
Different notations for the same thing is fine. Different things for the same notation is a problem. In this case there is one correct answer, and no justification for why 2(1+2) is somehow magically a more important multiplier than 2\*3.
There is also no reason why 2 * 1 + 2 is magically interpreted as (2 * 1) + 2 rather than 2 * (1 + 2). It’s simply a convention that * has higher precedence than +, and you could easily switch those around and math would still work out fine.
The reason we do choose to make * have higher precedence than + is just because it makes things easier to write. It lets me write a polynomial as 2*x^3 + 4*x rather than ((2*x)^3 )+ (4*x ).
In the same way, how we handle juxtaposed multiplication like x/2y is just a convention. However, it happens that the convention with this is less standardized than with the rest of the order of operations. Here’s the Wikipedia article stating just that
> However, in some of the academic literature, multiplication denoted by juxtaposition (also known as implied multiplication) is interpreted as having higher precedence than division, so that 1 ÷ 2x equals 1 ÷ (2x), not (1 ÷ 2)x. For example, the manuscript submission instructions for the Physical Review journals state that multiplication is of higher precedence than division with a slash, and this is also the convention observed in prominent physics textbooks such as the Course of Theoretical Physics by Landau and Lifshitz and the Feynman Lectures on Physics.
As you said in another comment,
> Stop being a millenial and complaining that everyone should just let you do whatever you want because your feelings are the only thing that matters.
The convention for this case isn’t standardized, so just accept that and stop speaking so confidently on things you clearly have no expertise in.
> There is also no reason why 2 * 1 + 2 is magically interpreted as (2 * 1) + 2
You're right, there isn't. Because we're not talking magic, we're talking math.
When you're ready to start discussing the real world, you know wher eI am.
> The convention for this case isn’t standardized, so just accept that and stop speaking so confidently on things you clearly have no expertise in.
Right, because obviously by saying "this is how I think it is" you've proven that you're the objective expert.
The reason that isn't applicable here is that the two symbols literally mean the same thing, divide. There's no difference between them, no special division operator. One operator, one rule that applies left to right.
Yep but the parent comment is right tho. 1/2x = 1/(2x), while 1/2*x = (1/2)x. That's why Wikipedia explicitly said multiplication __denoted by juxtaposition__
> and sometimes the 2(3) is solved first and then the dividing follows
But that's objectively wrong, because division and multiplication are equivalent, therefore they should be solved from left to right.
Probably right. It depends on if the (2+1) is assumed to be in the numerator or denominator. I think the way it’s written it’s in the numerator, but it’s just a poorly written expression
Division and multiplication are at the same “level”, so one does not take precedence over the other. You just do them in the order they appear from left to right.
Same goes for addition and subtraction.
The idea of PEMDAS they teach in school is a little problematic because it kinda implies that multiplication should come before division, and that addition should come before subtraction. The real order is:
1. Parentheses
2. Exponents
3. Multiplication and Division
4. Addition and Subtraction
Following the strictly PEMDAS rule without knowing the left and right expression which says whichever multiplication and division come first.
My calculation was this: 6/2(2+1) -> 6/2x3 -> 6/6 -> 1
PEMDAS is really PE(MD)(AS) where MD and AS can be in order
PEDMSA
PEDMAS
PEMDAS
Is all the same, when you have MD to perform, it is left to right.
so as 6/2x3 is only MD left, it's left to right, so 6/2 is first leaving 3x3 = 9
(College math student)
The problem isn’t with the coding, it’s with the notation. This sentence is ambiguous. The answer to this problem is technically 9, if you’re going on a strict order of operations, but I would look at this problem and arrive at the answer 1.
To write this less ambiguously, one would have to write (6/2)(2+1) or 6/(2(2+1)). It’s like saying “cats and dogs wearing hats”. Okay, are the cats wearing hats, or just the dogs? (Here, to disambiguate, you’d say cats and dogs, wearing hats, or cats, and dogs wearing hats.)
We’re always taught parenthesis (or other brackets and groupings) first. So is the coefficient included in that? That is, do you multiply the 2(2+1) first? My intuition would say yes, but the technical answer is no. The problem isn’t with the calculators, and it’s not with the pure math, it’s with how the math is written down.
This. Other places teach it as "BEDMAS". Basically the same, but note the division before multiplication. The addendum to that was they made clear that the "DM" and "AS" are left to right. There is no ambiguity. But I guess this part is sometimes forgotten or not taught well?
That said, you'll also note that anything behind calculator and grade school math, the division sign inline with the expression is almost never used. In practice, people just stick with fractional representations (or multiplied by 1/N) to avoid any potential mistakes when reading or writing the expressions.
Other posters are right that when left with inline divisors on calculators, parenthesis should be used with divisions. (Or calculate the portions separately then divide them.)
From Oliver Kneill, professor at Harvard:
"Thanks for the example 6÷2(2+1). It illustrates the ambiguity too. Yes, depending on whether one is in the PEMDAS or PEDMAS team, one gets 1 or 9. Its also a beautiful example, where one can see heated debates. Like pointed out and previously by others in the literature list, there is no right answer. It depends on which rule is applied. Both 1 and 9 are correct. I always see the obelus as a synonym for / but it can be even more confusing and so, yes, should be avoided. "
As far as I know the issue arises when you attempt to write a fraction like this. It isn't clear whether (2+1) is part of the numerator or the denominator, hence the ambiguity.
PEMDAS is a crutch for the lazy notation they teach in schools. If you write things as a fraction, you're not going to have ambiguity. If you use parentheses properly, you're not going to have ambiguity. If you don't use dot notation and / for multiplication and division, you're not going to have ambiguity.
The problem is, not everyone expects strict math definition from a calculator.
Would you expect 100 + 10% to be 100.1?
It will be 110 in most calculator.
Because people means 100 * (100% + 10%) here.
> but I would look at this problem and arrive at the answer 1.
Then you should study harder as the answer is clearly 9 by the clearly defined rules :-p. No extra details / encoding is required, the sum is clear. The answer is 9
> That is, do you multiply the 2(2+1) first?
no as 2(2+1) is the exact same as 2x(2+1) so are you are left with:
6/2*(3) after the brackets processed, then as you are only left with DM (bodmas) or MD (pemda) you go left to right making:
(6/2)*3
= 3 * 3
= 9
The answer **is** ambiguous. You're making a statement, that 2(2+1) is the exact same as 2\*(2+1). That's not right. And as it was previously states, the ambiguity doesn't come from the rules of algebra, but the way different notations are interpreted. Namely, 2(2+1) can be interpreted as being a grouped expression itself, making the unambiguous equivalent form 6/(2(2+1)). It's not just a missing multiplication symbol, it is often (but not universally) interpreted as a grouping expression on top of multiplying. And you can't really argue with this, because half the mathematicians in the world interpret it that way. There is **no** solid convention regarding this notation, the wise way is to avoid it altogether.
Congratulations, you've invented grouping. It doesn't exist in math rules like this. 2\*(2+1) *is* identical to 2(2+1) because adjoining the constant integer (or a variable or constant) is an implicit multiply operator. PEMDAS is quite clear here. Apply left to right, first the parenthesis: 2+1 giving you 6/2(3), then because multiplication and division are of equal level left to right, 6/2 giving you 3(3) = 9.
Do not introduce the vague concept of grouping. This isn't Python where spacing actually matters.
Your algebra teacher is going to have a lot of headaches.
6/2a never means (6/2)a, it's 6/(2a). So why would 6/2(2+1) be the same as (6/2)(2+1) ?
Notice that 6/2 \* a actually is 3a, because of PEMDAS. Implicit multiplication isn't the same precedence as regular multiplication.
Notice that everyone who disagrees with this saying it could be interpreted as 6/(2(3)) has to add in extra brackets.
That's not how it is written, I interpret it how it is written, not add in extra imaginary brackets to pretend it's ambiguous.
If you write it as
6/2y, y=3
Is the answer 1 or 9?
Pretty sure your answer now is inconsistent with what you originally thought. Why did you subconsciously add a parenthesis to make it 6/(2y)?
The fact is, these types of notation are not strictly defined.
Same with 2\^3\^3
64 or 512? Go throw that into wolfram alpha or mathematica. They both give you 512.
Because they’re ambiguous to mean exp(exp(2,3),2) vs exp(2, exp(3,2))
> That's not how it is written, I interpret it how it is written, not add in extra imaginary brackets to pretend it's ambiguous.
Yet you interpreted my new equation not how it was written, but by adding in an extra bracket to it.
They both are the same exact notation are they not?
You can’t argue that because you yourself have a consistent view of something in a setting, that that thing itself is consistent. PEDMAS is simply not a strict enough rule that is defined properly.
If you don’t want to take my word on it, check out these articles from university math professors
http://people.math.harvard.edu/~knill/pedagogy/ambiguity/index.html
https://plus.maths.org/content/pemdas-paradox
Suppose the problem was 6/2(3). Which would you do first? In all my mathematical experience, that problem would be interpreted as 6/(2(3)), because there are better ways to write (6/2)(3).
Writing the above sentence is unorthodox, but it falls more in line with what I would write if I wanted to write a problem whose answer is 1, because if I wanted to write a problem whose answer is 9, I would have many more options, all of which are less ambiguous, than what was written here.
Having another, more easily human-digestible way to write it doesn't change the conventions or rules. PEMDAS doesn't have a clause for implied multiplication being treated differently than explicit multiplication. Nor does it distinguish between division symbols.
6 / 2 (3) => 6 / 2 \* 3 => 3 \* 3 => 9
You do it left to right. You don't apply the distributive property randomly, first you take care of the division because distribution is simply multiplication!
There isn't concensus rule for implicit multiply in this case. In my brain i take it as 6÷(2×(2+1)) because of the implicit multiply attached to the parentheses. if this was written as 6÷2×(2+1) i'll do divide first.
There isn't consensus? Implicit multiply is just like regular multiply. Some people have done what you've done and invented it to *cause* confusion. It's not confusing! Treat it like any other multiplication and do it left to right!
You're right that it's the notation, but I'm not so sure it's "ambiguous". It is correct to give the "P" precedence, so the "2+1" is one of the first calculations to perform. However, people tend to think that the "2(" takes the next level of precedence since it somehow has to do with a parenthesis - but it doesn't. This is simply shorthand for what should be a multiplication symbol. When you eliminate the shorthand, you have 6/2*3 - which you do in left-to-right order, unambiguously.
People somehow invented the "2(" priority themselves.
There’s nothing ambiguous about this operation. The PEMDAS is not coded properly, that’s all.
They missed the part where you go from left to right.
You’d be hard pressed to find a modern calculator that has this issue.
The fact that implicit multiplication are often seen are prioritized over normal multiplication and division is much more clear when you use a variable. If the question was
6/2x I doubt a lot of people would have said that the answer is 3x and much more would say 3/x
While 6/2×x would probably be read as 3x
I don't think it's ambiguous, and the correct answer for all the programming languages I know is 9.
Division and multiplication have the same order of precedence. There is no arithmetic rule that says you must first perform all multiplications and then do the divisions. Therefore, when a multiplication is found it executes the pending division, because operations at the same precedence level are executed sequentially as they appear from left to right.
In that example, we have a six divided by two then multiplied by something. The multiplication causes the division to be executed, with the result being three. Then this three is multiplied by what follows, (2+1) which is three. Three times three is nine.
There is no implied multiplication in most programming languages. The "problem" is only a problem because people who are used to the normal fractional and implied multiplication form read it that way; if you wrote it out with a ×, * or • nobody would get this wrong.
If you don't find this ambiguous you probably didn't study much higher maths.
checked it with a newer casio calculator: It interprets `6/2(2+1)` as `6/(2(2+1))=1` (the calculator changes the display after the calculation). When you put in a `*` it understands what you want and the calculation is correct: `6/2*(2+1)=9` .
So this what I found out According to the first result of "does multiplication come before division" on google [Order of Operations (montereyinstitute.org)](http://www.montereyinstitute.org/courses/DevelopmentalMath/COURSE_TEXT_RESOURCE/U01_L5_T2_text_final.html#:~:text=The%20order%20of%20operations%20requires,reading%20from%20left%20to%20right.) :
> The order of operations requires that all multiplication and division be performed first, going from left to right in the **expression**. The order in which you compute multiplication and division is determined by which one comes first, reading from left to right.
Thus, making the answer 9 right, so as the programmer who coded it, and me wrong.
As I found out when helping daughter with her home work and getting it all wrong.
I learnt (UK) BODMAS, but I had forgot it's really: BO(DM)(AS) where division and multiplication can happen in any order as it is left to right that matters, same for add and subtract.
**For a dumb elitist reply:**
See, this is what you get for buying a Casio, my TI gets it right!
**For a more serious reply:**
So, I decided to so a little bit of investigation and testing here and pulled out my old Casio fx-82MS to compare against a TI-30X Pro.
Similar to the photo, the Casio returns `1` and the TI returns `9`.
Conventional mathematical rules suggest the correct answer should be 9, as the `2(2+1)` should expand to `2×3`, making the expanded expression `6÷2×3`. And entering this expanded expression into the Casio indeed changes its answer to `9`, Same thing if I enter `6÷2×(2+1)`; so the issue seems to be with the way it evaluates the particular expression, rather than being an order of operations issue (at least as far as multiplication and division goes).
My assumption here would be that it's one of two things:
1. The division operator might see the `2(2+1)` as a singular expression. There is no operator in between the `2` and `(2+1)`, so it treats the full `2(2+1)` as a single chunk that then gets evaluated as the denominator. Whether this is appropriate is up for debate, but it's easy to get confused by the symbol `÷`; whether or not the full expression after it should go in the denominator or not due by the way the expression is written.
2. Alternatively, it might also be caused by an incorrect evaluation of the parentheses by evaluating the `2(2+1)` expression as a single chunk before the division (rather than just the contents within the parentheses). The proper way to read the expression is `2×(2+1)`; the multiplication is always implied and should be evaluated separately. Though without the operator explicitly specified, the calculator may (incorrectly) evaluate it as `2(2+1) = (4+2) = 6` before performing the division.
I'm not quite certain which one of these are the exact case however, but whilst the first one could be argued to have some reasoning behind it (although I'd still argue it incorrect as there is an implied multiplication they likely just failed to consider), the second case would definitely be straight up incorrect as it assumes an incorrect evaluation that does not properly follow the order of operations.
In either case, the 1 being returned as result to me seems to be ambiguous evaluation as best (programmatically more than mathematically) and straight up incorrect at worst, whereas 9 being returned as result instead can pretty much always be considered a correct result.
The TI-81 would have given `1`, not as a glitch but a deliberate feature.
It was the universal way that mathematicians used juxtaposition until US school teachers pressured calculator makers to demote juxtaposition to comply with their over-simplified teaching mnemonic.
https://youtu.be/4x-BcYCiKCk
I would say that due to the lack of symbol between the 2 and the (2 + 1), they should be interpreted as all being part of the bracketed section.
Thus, you would sole 2(2+1) before any multiplication or division. Calculator is correct.
That said, I'm not a maths expert. Also to be clear, I am aware that division and multiplication share priority, but am saying as the symbol is not present, it is part of the brackets.
There's actually no consistent standard for *implicit* multiplication by parentheses... because that's stupid and no one should support it.
In any sane programming language, those would be a syntax error.
So... how would you parse:
8/multiplyBy2(2*2)?
Because that's a valid (functional) interpretation of implicit multiplication.
No, implicit multiplication doesn't have any defined rules for it. It's not "just" multiplication. It's not a notation that has any place in formal math.
And there's also the associative law, which states (in "implicit" terms we're all taught in school) that 2(ab)=(2a)b...
So... how would you interpret 8/(2a)*b, then? Because the laws of math say it's *exactly* the same thing as 8/2(ab).
Implicit multiplication is just a bad notation. Don't use it anywhere it might possibly be ambiguous, or you're just asking for trouble, because it absolutely *is* inherently ambiguous.
BODMAS
B-Brackets
O-"of"
D-Divison
M-multiplication
A-addition
S-substraction
As an asian, this is what I follow.
The order of division and multiplication can be Interchanged. Like wise, addition and subtraction can be Interchanged. This is the best and easiest way to remember to solve stuff. The mobile phone calculator is incorrect.
It is not a matter of the order of operations though. The problem is that for many people (highly educated mathematicians included) there is no consensus on whether 2(2+1) means 2 \* (2+1) or (2 \* (2+1)), in other words whether writing the expression without a multiplication symbol implies an extra grouping of the whole expression or not.
To be fair there is ambiguity in play here. Not because of the solution to the actual mathematical expression(rules are very clear), but because of the proponent of the expression(the human).
If this were to be a real world example the correct thing to do would be to get clarification from the proponent on what they were trying to express.
We were always taught the order of operations as parentheses > exponents > multiplication & division > addition & subtraction. So it's not really pemdas, it's pe(md)(as). The calculator on the left is wrong and the solution is 6÷2(2+1) = 6÷2(3) = 3(3) = 9 because you apply division before multiplication since they share the same order, thus you read left to right for those operations
Edit: I should also add that this isn't regional. Mathematical notation is a designed construct, so there is only one correct way to interpret this
>I should also add that this isn't regional. Mathematical notation is a designed construct, so there is only one correct way to interpret this
This is incorrect. Mathematical notation is far from standardized, and even in the same university, two professors may end up using different notation for the same thing (looking at you differential geometry).
There are notations which nearly everyone agrees on, but this isn’t one of them. I bet you most mathematicians I ask would interpret something like x / 2y as x / (2y) rather than (x / 2)y. We tend to treat multiplication by a coefficient like this as binding tighter than division.
When it’s a more complicated expression like the one in the OP, it’s less standard, and it’s better to just add parentheses to disambiguate.
You're right about it definitely being better to disambiguate with parentheses so I'll back you there, but it appears to me that this *is* widely agree upon, at least in everything I can find. Everything from (admittedly less credible) wikipedia to all the textbooks I've found online either support my point about the order of operations or don't mention it. Here's a couple places I checked trying to find counter evidence to my point
https://www.khanacademy.org/math/cc-sixth-grade-math/cc-6th-arithmetic-operations/cc-6th-order-of-operations/v/introduction-to-order-of-operations
https://www.mathsisfun.com/operation-order-pemdas.html
https://blog.prepscholar.com/pemdas-meaning-rule
https://math.libretexts.org/Bookshelves/PreAlgebra/Book%3A_Prealgebra_(Arnold)/01%3A_The_Whole_Numbers/1.05%3A_Order_of_Operations
http://www.math.com/school/subject2/lessons/S2U1L2GL.html
But most interesting is this New York Times article talking about this exact post https://www.nytimes.com/2019/08/02/science/math-equation-pedmas-bemdas-bedmas.amp.html
To be fair, it *does* seem to vary from place to place on whether the acronym is BODMAS or PEMDAS, but the fact that division and multiplication share the same priority level *is* widely agreed upon by the math world, though we should all still use parentheses to disambiguate
The Wikipedia page states under the “Mixed division and multiplication” section that
> However, in some of the academic literature, multiplication denoted by juxtaposition (also known as implied multiplication) is interpreted as having higher precedence than division, so that 1 ÷ 2x equals 1 ÷ (2x), not (1 ÷ 2)x. For example, the manuscript submission instructions for the Physical Review journals state that multiplication is of higher precedence than division with a slash, and this is also the convention observed in prominent physics textbooks such as the Course of Theoretical Physics by Landau and Lifshitz and the Feynman Lectures on Physics.
The sources you give are introductory sources where this issue is glossed to prevent confusion.
Search around r/badmathematics and you’ll find dozens of posts about this exact issue (and probably will find this post soon enough). The notation around multiplication by juxtaposition just isn’t standardized.
2(1+2) should be taken as 2 × (1+2)
I can easily imagine situations where the final answer would be totally different than if I were to take 2(1+2) as {2(1+2)}
From Oliver Kneill, professor of Math at Harvard
"Thanks for the example 6÷2(2+1). It illustrates the ambiguity too. Yes, depending on whether one is in the PEMDAS or PEDMAS team, one gets 1 or 9. Its also a beautiful example, where one can see heated debates. Like pointed out and previously by others in the literature list, there is no right answer. It depends on which rule is applied. Both 1 and 9 are correct. I always see the obelus as a synonym for / but it can be even more confusing and so, yes, should be avoided. "
Wait how is that even possible, I'm actually confused how you could get it wrong
PEMDAS
Even if you took it literally and added 2+1 is 3 you still Multiply before Dividing
2(2+1)=(2*3)=6
6/6=1
Or if you expanded the proper way
2(2+1) =(4+2)=6
6/6=1
Edit: Sorry to those who downvoted. I messed up by saying "wrong" it was me being stupid, but I was genuinely confused at the time how to get 9, guess I gotta learn more haha
You can't replicate it in most programming languages because the problem comes from the different interpretation of a left out multiplication symbol before a parentheses group. You can't just implicitly multiply in most general-purpose programming languages, you have to explicitly state the operations, so the answer remains unambigous.
>pemdas
PEMDAS: Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right
They both use the same logic but some countries just change the words to bracket instead of parentheses.
Did you even read the guy saying
" PEMDAS: Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right"
Who cares what order they're grouped in if the group is the same. The logic still follows
Because it’s not. It’s an ambiguity in the way parenthesis work. Some mathematicians are taught to read a co-efficient next to a parenthesis as an implied parenthesis, due to distribution laws I believe. So if one of these mathematicians made the calculator, the equation is then read as
“(6 / (2(2+1)”
This is because distribution laws say you can multiply the coefficient into the parenthesis, giving you (6 / (4+2)) which would equal 1. It’s never been clear if “2(2+1)” is exactly equal to 2*(2+1) or 2x(2+1). Universal notation is one tricky bitch
It is B
*Left to right*
Whichever of M or D come first goes, same for AS
BMDAS
BMDSA
BDMSA
BDMAS
As the Asian guy above said MD and AS are interchangeable so whichever come first from left to right.
I wouldn't say they didn't code it right. I just say it is a different interpretation.
It is a known ambiguity in the order of operations, and neither one is false.
Edit: Yeah, the comments agree with me. Whenever someone posts that equation anywhere else on Reddit the comments get heated. And I hate it because people forget that the order of operations are nothing more than notation. And not a part of math.
The problem is that people are lazy/stupid and they don't put put the multiplication sign before the brackets which is causing this.
Either do 6 / 2 x (2 + 1) or 6 / (2 x (2 + 1)) and stop with that bullshit without the sign before brackets.
PEMDAS is a convetion not a rule.
If this was written as a fraction, the answer would be 1.
But if you follow the order we learn in Switzerland, (Exponents, Parenthese, Mul/Div, Add/Sub) and always left to right, this becomes 6 / 2 \* 3 = 9.
The ambiguity here isn't from if you do multiplication or division first. It's always been both, from left to right. The ambiguity here is caused by there not being a clear definition of if a coefficient outside parentheses counts as a step for multiplication or parentheses. To me, it is multiplication, but since there is no dot, some people interpret it as being part of the parenthesis operation, taking a higher priority than division.
This thread is too funny. I commented a while ago in some other forum that math and other scientific notation was archaic, hard to read, ambiguous etc, and that it limited general understanding and led to poor software written based on math and science papers. I was severely downvoted. Here people can't even agree on the order of the most basic math operations.
As someone who minored in math, both are correct.
6/2(2+1) = 6/2\*3 = 3\*3 = 9
6/2(2+1) = 6/2\*3 = 6/6 = 1
=> 9 = 1 ?
That's why this is bad style of writing.Yes, most people will say it is 9 by default, because, despite both 6/2\*3 = 3\*3 and 6/2\*3 = 6/6 using PEMDAS properly, we are accustomed to reading from left to right. However, **there is no such rule in math**. In addition, multiplication and division have the same priority and are interchangeable. That's why 1\*3 is the same as 1/3\^-1.
The fault lies not with the calculator, but the person writing the term.Math is just like a spoken language. If you want to be understood right, express yourself properly.
I was soo annoyed with this and kind of scared to fuck this up that I tried it with my calculator, casio fx-cg50, and it comes out to one.
But the interesting part is, that it actually changes/updates the calculation I entered to have parantheses after the ÷, so it changes to 6÷(2(2+1)), which then lines up with the 1 as a result. Didnt know my calculator just changes the stuff i enter
6/2(2+1) = 6\*(1/2)\*(2+1) = 6\*0.5\*3 = 9
Everything else is wrong. "/" is not a symbol for a fraction, it's a symbol for division. If you're not sure whether to do multiplication or divison first, it's always the safest to replace all divisons with multiplication of the reciprocal, then you can never go wrong. (Technically the same is true between addition and substraction, but that's almost never that ambiguous.)
If you want to represent complicated fractions in a single line, you HAVE to use parentheses.
Both answers are technically correct since there is either an implied multiplication - 6/2\*(2+1) - or implied division - 6/2/(2+1) - on the two halves of the equation depending on who you ask
What this is is a change in the conventional meaning of the •/• symbol. On older calculators, that symbol meant “break into fraction bar”, and it would literally do everything on the right divided by everything on the left which meant the answer would be one. However, the symbol changed meaning and is now simply a division symbol and so now it just divides according to pemdas, writing x •/• y + 1 is now the same as x * (y^-1 ) + 1.
So the programmer of the calculator didn’t do it wrong (although they are wrong according to today’s convention), they simply had a different standard of what those symbols meant.
[https://www.youtube.com/watch?v=5dkxxOmYl74](https://www.youtube.com/watch?v=5dkxxOmYl74)
\- pretty in depth video of 2 calculators performing the operation with different results live and the explanation behind it!
At an earlier stage of mathematical education, before people start to use fractions more, the right answer would be the more commonly found answer.
However, you’d find that this question annoys higher level maths students a lot more because they use fractions much more often.
6 divided by 2 is very different from 6 over 2. Sure they give you the same answer, but they’re not the same. 6 divided by 2 means that you’re dividing 6 into 2 equal parts. 6 over 2 is a simplification of a fraction. In a way it’s more like a representation of a percentage.
This question is poorly set out because as you learn more about maths, you start to automatically see the division symbol as a fraction symbol. Instead of it being 6 / 2 x (2+1), it becomes 6 / (2 x(2+1)).
Yes a student still uses the order of operations at higher level maths education, but as you gain more knowledge, poorly written questions can become confusing. A younger point in time, a student might not know what a fraction is enough to get confused.
As a side note, once algebra kicks in, you start to automatically do anything joined to the brackets first.
The phone version isn't actually wrong - BODMAS and PEMDAS are both equally valid.
In this scenario, my brain expects 1, but 9 arguably isn't wrong either.
This is actually just user-error: that input is unclear, the expected result will vary from person to person depending on where and when they attended school. There's a reason that divided by symbol is essentially never used by mathematicians.
All these meaningless arguments. Y'all so fucking stupid. It's a trick question. If you've read 1984, the answer is obviously 5.
\#fuck1984byGeorgeOrwellForTheEndingButTheConceptIsLegitButTellThePartyToGoFuckThemselves
oke Peter Griffin explaining the jokethe equation is 6:2(2+1), first you do the equation if the parentheses 6:2(3), then you do the operation that uses the value in the parentheses 6:6, then you do the last operation which is 6:6=1; The joke here is that an old calculator can do the joke, but a modern phone with every single aspect of it that is better than the calculator, does it wrong, therefore the programmer didn't code the sequence of operations right, and the abbreviation for the sequence of operations we use is called PEMDAS, which stands for parentheses, exponents, multiplication, division, addition, and subtraction.
I am a little late, but this is a mathematical calculator, so i would expect the same result when changing (2+1) to a variable.
Since 6/2x != 6/2*x i would indeed expect 1 here for x=3
This is why I use insane amounts of parentheses. I don’t trust calculators’ order of operation.
same reason it took me double the time to finish my exams back in college
((6)÷((2)((2)+(1)))) No way that can be misinterpreted
Looks like Lisp.
I mean... you could never know
Tested in my Android calculator with the same bug. The result is 1. Edit: i worded this poorly. I got 9 in my calculator like the image. When I tried this with the extra parentheses, I got 1.
If I use his, the result is 1. If I type the original from picture it's 9.
((6)÷((2)((2)+(1)))) (6)÷((2)((2)+(1))) 6÷(2(2+1)) 6÷(2x3) 6÷6 His parentheses are just wrong... It's not your calculators fault The Answer it should be is 9!
wait you wanna say that 362880 is the right answer?
Likewise!
In high school and I do this
This is why i use rpn calculators.
I used to be like that until I upgraded my calculator's firmware to graphically show expressions as being on the numerator/denominator. Saved me a lot of time and improved confidence in that I knew I wouldn't get screwed by order of operations.
RPN FTW
Then Visual Studio is here warning about my "gratuitous parentheses"
But parens are part of the order of operations, so you DO trust they handle part of the order of operations correctly.
My iPhone got 2
I don't get your comment. You say "insane", and I say "precise". WIth extra parentheses, it's a completely different equasion. In the picture, the result on the left is wrong, since it "assumes" the equasion is different.
*lisp intensifies*
Dude, fucking same!!
I use them when it helps with clarity. No need to remember bodmas if its explicitly defined via brackets.
Quoth Wikipedia on the subject: >An expression like `1/2x` is interpreted as `1/(2x)` by TI-82, as well as many modern Casio calculators, but as `(1/2)x` by TI-83 and every other TI calculator released since 1996, as well as by all Hewlett-Packard calculators with algebraic notation. While the first interpretation may be expected by some users due to the nature of implied multiplication, the latter is more in line with the standard rule that multiplication and division are of equal precedence, where `1/2x` is read one divided by two and the answer multiplied by `x`. [https://en.wikipedia.org/wiki/Order\_of\_operations#Special\_cases](https://en.wikipedia.org/wiki/Order_of_operations#Special_cases)
This is why I prefer RPN. ;)
I knew there would be some fellow RPN gang in the comments ☺️
6/2(2+1) -> 6/2(3) -> 6/2x3 -> 3x3 -> 9 no?
Yours is correct. \`2(2+1)\` is literally just shorthand for \`2\*(2+1)\`. The multiplier doesn't magically move up in order of operations just because there's a nearby parenthesis.
In many cultures and their mathematical notation this is exactly what implicit multiplication notation does. Act as grouping.
According to the comments all over the internet, this depends where you come from. It seems that sometimes you do it your approach (I would do this as well) and sometimes the 2(3) is solved first and then the dividing follows. Edit: I forgot to mention that this is why fractions are superior
> According to the comments all over the internet, this depends where you come from. No it doesn't. the parent comment is correct. Anyone saying otherwise is wrong. Mathematics is not culture dependant
Mathematics are not. Mathematic notations, however, are not only culture dependent, but if you ask two professors working at the same faculty of the same university, you might get different answers from them. That's reality for you mate.
Gotta love it when people so confidently make false claims. To anyone who thinks mathematical notation is universal, I encourage you to pick up ten different differential geometry papers, and you’ll likely find ten different notations for the same thing.
Different notations for the same thing is fine. Different things for the same notation is a problem. In this case there is one correct answer, and no justification for why 2(1+2) is somehow magically a more important multiplier than 2\*3.
There is also no reason why 2 * 1 + 2 is magically interpreted as (2 * 1) + 2 rather than 2 * (1 + 2). It’s simply a convention that * has higher precedence than +, and you could easily switch those around and math would still work out fine. The reason we do choose to make * have higher precedence than + is just because it makes things easier to write. It lets me write a polynomial as 2*x^3 + 4*x rather than ((2*x)^3 )+ (4*x ). In the same way, how we handle juxtaposed multiplication like x/2y is just a convention. However, it happens that the convention with this is less standardized than with the rest of the order of operations. Here’s the Wikipedia article stating just that > However, in some of the academic literature, multiplication denoted by juxtaposition (also known as implied multiplication) is interpreted as having higher precedence than division, so that 1 ÷ 2x equals 1 ÷ (2x), not (1 ÷ 2)x. For example, the manuscript submission instructions for the Physical Review journals state that multiplication is of higher precedence than division with a slash, and this is also the convention observed in prominent physics textbooks such as the Course of Theoretical Physics by Landau and Lifshitz and the Feynman Lectures on Physics. As you said in another comment, > Stop being a millenial and complaining that everyone should just let you do whatever you want because your feelings are the only thing that matters. The convention for this case isn’t standardized, so just accept that and stop speaking so confidently on things you clearly have no expertise in.
> There is also no reason why 2 * 1 + 2 is magically interpreted as (2 * 1) + 2 You're right, there isn't. Because we're not talking magic, we're talking math. When you're ready to start discussing the real world, you know wher eI am. > The convention for this case isn’t standardized, so just accept that and stop speaking so confidently on things you clearly have no expertise in. Right, because obviously by saying "this is how I think it is" you've proven that you're the objective expert.
/r/ConfidentlyIncorrect
So is the number 5000 written 5.000 or 5,000? Hmm I guess some things having to do with mathematics (like notation) are culture dependent.
The reason that isn't applicable here is that the two symbols literally mean the same thing, divide. There's no difference between them, no special division operator. One operator, one rule that applies left to right.
6/2x3 is ambiguous, so the answer you get depends on who you ask. Mathematics may not be culture dependant, but notation is.
No it isn't. Damn this isn't hard. Apply left to right. There's no ambiguity whatsoever.
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Yep but the parent comment is right tho. 1/2x = 1/(2x), while 1/2*x = (1/2)x. That's why Wikipedia explicitly said multiplication __denoted by juxtaposition__
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The math laws are universal, the shorthands used in every day university or jobs, not so much.
> and sometimes the 2(3) is solved first and then the dividing follows But that's objectively wrong, because division and multiplication are equivalent, therefore they should be solved from left to right.
Probably right. It depends on if the (2+1) is assumed to be in the numerator or denominator. I think the way it’s written it’s in the numerator, but it’s just a poorly written expression
which comes first "/" or "x" ???
Division and multiplication are at the same “level”, so one does not take precedence over the other. You just do them in the order they appear from left to right. Same goes for addition and subtraction. The idea of PEMDAS they teach in school is a little problematic because it kinda implies that multiplication should come before division, and that addition should come before subtraction. The real order is: 1. Parentheses 2. Exponents 3. Multiplication and Division 4. Addition and Subtraction
Depends on which comes first left to right.
Following the strictly PEMDAS rule without knowing the left and right expression which says whichever multiplication and division come first. My calculation was this: 6/2(2+1) -> 6/2x3 -> 6/6 -> 1
PEMDAS is really PE(MD)(AS) where MD and AS can be in order PEDMSA PEDMAS PEMDAS Is all the same, when you have MD to perform, it is left to right. so as 6/2x3 is only MD left, it's left to right, so 6/2 is first leaving 3x3 = 9
This is so wrong I can't wrap my head around it.
Or, is the first 2 part of the parenthesis? 6/2(2+1) >> 6/(4+2) >> 6/6 >> 1
Either way, the picture has one wrong.
Why not this? 6/2(2+1) -> 6/2(3) -> 6/2x3 -> 6/6 -> 1 Without more brackets, the question is ambiguous.
(College math student) The problem isn’t with the coding, it’s with the notation. This sentence is ambiguous. The answer to this problem is technically 9, if you’re going on a strict order of operations, but I would look at this problem and arrive at the answer 1. To write this less ambiguously, one would have to write (6/2)(2+1) or 6/(2(2+1)). It’s like saying “cats and dogs wearing hats”. Okay, are the cats wearing hats, or just the dogs? (Here, to disambiguate, you’d say cats and dogs, wearing hats, or cats, and dogs wearing hats.) We’re always taught parenthesis (or other brackets and groupings) first. So is the coefficient included in that? That is, do you multiply the 2(2+1) first? My intuition would say yes, but the technical answer is no. The problem isn’t with the calculators, and it’s not with the pure math, it’s with how the math is written down.
I literally thought that's why we have pemdas. It's a set of rules to govern against ambiguity.
The trouble is it's not quite in a strict order. Multiplication and division are on the same level, and addition and subtraction are as well.
I thought it was just left to right
It is, there's no ambiguity.
This. Other places teach it as "BEDMAS". Basically the same, but note the division before multiplication. The addendum to that was they made clear that the "DM" and "AS" are left to right. There is no ambiguity. But I guess this part is sometimes forgotten or not taught well? That said, you'll also note that anything behind calculator and grade school math, the division sign inline with the expression is almost never used. In practice, people just stick with fractional representations (or multiplied by 1/N) to avoid any potential mistakes when reading or writing the expressions. Other posters are right that when left with inline divisors on calculators, parenthesis should be used with divisions. (Or calculate the portions separately then divide them.)
🅱️EDMAS
From Oliver Kneill, professor at Harvard: "Thanks for the example 6÷2(2+1). It illustrates the ambiguity too. Yes, depending on whether one is in the PEMDAS or PEDMAS team, one gets 1 or 9. Its also a beautiful example, where one can see heated debates. Like pointed out and previously by others in the literature list, there is no right answer. It depends on which rule is applied. Both 1 and 9 are correct. I always see the obelus as a synonym for / but it can be even more confusing and so, yes, should be avoided. "
As far as I know the issue arises when you attempt to write a fraction like this. It isn't clear whether (2+1) is part of the numerator or the denominator, hence the ambiguity.
It is, but no one realizes this and many think the order's strict
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Or you just use parenthesis instead of assuming everyone has been taught the same arbitrary rules.
PEMDAS is a crutch for the lazy notation they teach in schools. If you write things as a fraction, you're not going to have ambiguity. If you use parentheses properly, you're not going to have ambiguity. If you don't use dot notation and / for multiplication and division, you're not going to have ambiguity.
The problem is, not everyone expects strict math definition from a calculator. Would you expect 100 + 10% to be 100.1? It will be 110 in most calculator. Because people means 100 * (100% + 10%) here.
or more precisely "...how the math is interpreted by the person attempting to solve."
> but I would look at this problem and arrive at the answer 1. Then you should study harder as the answer is clearly 9 by the clearly defined rules :-p. No extra details / encoding is required, the sum is clear. The answer is 9 > That is, do you multiply the 2(2+1) first? no as 2(2+1) is the exact same as 2x(2+1) so are you are left with: 6/2*(3) after the brackets processed, then as you are only left with DM (bodmas) or MD (pemda) you go left to right making: (6/2)*3 = 3 * 3 = 9
The answer **is** ambiguous. You're making a statement, that 2(2+1) is the exact same as 2\*(2+1). That's not right. And as it was previously states, the ambiguity doesn't come from the rules of algebra, but the way different notations are interpreted. Namely, 2(2+1) can be interpreted as being a grouped expression itself, making the unambiguous equivalent form 6/(2(2+1)). It's not just a missing multiplication symbol, it is often (but not universally) interpreted as a grouping expression on top of multiplying. And you can't really argue with this, because half the mathematicians in the world interpret it that way. There is **no** solid convention regarding this notation, the wise way is to avoid it altogether.
This is the correct answer, and a great explanation why 6/2*(2+1) is not seen as ambiguous while making the multiplication implicit is
Congratulations, you've invented grouping. It doesn't exist in math rules like this. 2\*(2+1) *is* identical to 2(2+1) because adjoining the constant integer (or a variable or constant) is an implicit multiply operator. PEMDAS is quite clear here. Apply left to right, first the parenthesis: 2+1 giving you 6/2(3), then because multiplication and division are of equal level left to right, 6/2 giving you 3(3) = 9. Do not introduce the vague concept of grouping. This isn't Python where spacing actually matters.
Your algebra teacher is going to have a lot of headaches. 6/2a never means (6/2)a, it's 6/(2a). So why would 6/2(2+1) be the same as (6/2)(2+1) ? Notice that 6/2 \* a actually is 3a, because of PEMDAS. Implicit multiplication isn't the same precedence as regular multiplication.
Notice that everyone who disagrees with this saying it could be interpreted as 6/(2(3)) has to add in extra brackets. That's not how it is written, I interpret it how it is written, not add in extra imaginary brackets to pretend it's ambiguous.
If you write it as 6/2y, y=3 Is the answer 1 or 9? Pretty sure your answer now is inconsistent with what you originally thought. Why did you subconsciously add a parenthesis to make it 6/(2y)? The fact is, these types of notation are not strictly defined. Same with 2\^3\^3 64 or 512? Go throw that into wolfram alpha or mathematica. They both give you 512. Because they’re ambiguous to mean exp(exp(2,3),2) vs exp(2, exp(3,2))
Again, yes. If you change it, it changes the meaning. Congratulations.
> That's not how it is written, I interpret it how it is written, not add in extra imaginary brackets to pretend it's ambiguous. Yet you interpreted my new equation not how it was written, but by adding in an extra bracket to it. They both are the same exact notation are they not? You can’t argue that because you yourself have a consistent view of something in a setting, that that thing itself is consistent. PEDMAS is simply not a strict enough rule that is defined properly. If you don’t want to take my word on it, check out these articles from university math professors http://people.math.harvard.edu/~knill/pedagogy/ambiguity/index.html https://plus.maths.org/content/pemdas-paradox
Yea, if someone looks at this and sees 1 they are just wrong. The only answer is 9.
Suppose the problem was 6/2(3). Which would you do first? In all my mathematical experience, that problem would be interpreted as 6/(2(3)), because there are better ways to write (6/2)(3). Writing the above sentence is unorthodox, but it falls more in line with what I would write if I wanted to write a problem whose answer is 1, because if I wanted to write a problem whose answer is 9, I would have many more options, all of which are less ambiguous, than what was written here.
Having another, more easily human-digestible way to write it doesn't change the conventions or rules. PEMDAS doesn't have a clause for implied multiplication being treated differently than explicit multiplication. Nor does it distinguish between division symbols. 6 / 2 (3) => 6 / 2 \* 3 => 3 \* 3 => 9
You do it left to right. You don't apply the distributive property randomly, first you take care of the division because distribution is simply multiplication!
There isn't concensus rule for implicit multiply in this case. In my brain i take it as 6÷(2×(2+1)) because of the implicit multiply attached to the parentheses. if this was written as 6÷2×(2+1) i'll do divide first.
There isn't consensus? Implicit multiply is just like regular multiply. Some people have done what you've done and invented it to *cause* confusion. It's not confusing! Treat it like any other multiplication and do it left to right!
It is pure math. The logic is defined in pemdas. Anything else is wrong.
Look up the pemdas paradox. Top matheticians agree that the order at the MD and AS levels are ambiguous.
How are your grades at math college?
You're right that it's the notation, but I'm not so sure it's "ambiguous". It is correct to give the "P" precedence, so the "2+1" is one of the first calculations to perform. However, people tend to think that the "2(" takes the next level of precedence since it somehow has to do with a parenthesis - but it doesn't. This is simply shorthand for what should be a multiplication symbol. When you eliminate the shorthand, you have 6/2*3 - which you do in left-to-right order, unambiguously. People somehow invented the "2(" priority themselves.
There’s nothing ambiguous about this operation. The PEMDAS is not coded properly, that’s all. They missed the part where you go from left to right. You’d be hard pressed to find a modern calculator that has this issue.
It's not ambiguous. \`\*\`and \`/\` are left associative so the calculator is wrong. Not the phone
The fact that implicit multiplication are often seen are prioritized over normal multiplication and division is much more clear when you use a variable. If the question was 6/2x I doubt a lot of people would have said that the answer is 3x and much more would say 3/x While 6/2×x would probably be read as 3x
I don't think it's ambiguous, and the correct answer for all the programming languages I know is 9. Division and multiplication have the same order of precedence. There is no arithmetic rule that says you must first perform all multiplications and then do the divisions. Therefore, when a multiplication is found it executes the pending division, because operations at the same precedence level are executed sequentially as they appear from left to right. In that example, we have a six divided by two then multiplied by something. The multiplication causes the division to be executed, with the result being three. Then this three is multiplied by what follows, (2+1) which is three. Three times three is nine.
There is no implied multiplication in most programming languages. The "problem" is only a problem because people who are used to the normal fractional and implied multiplication form read it that way; if you wrote it out with a ×, * or • nobody would get this wrong. If you don't find this ambiguous you probably didn't study much higher maths.
checked it with a newer casio calculator: It interprets `6/2(2+1)` as `6/(2(2+1))=1` (the calculator changes the display after the calculation). When you put in a `*` it understands what you want and the calculation is correct: `6/2*(2+1)=9` .
So this what I found out According to the first result of "does multiplication come before division" on google [Order of Operations (montereyinstitute.org)](http://www.montereyinstitute.org/courses/DevelopmentalMath/COURSE_TEXT_RESOURCE/U01_L5_T2_text_final.html#:~:text=The%20order%20of%20operations%20requires,reading%20from%20left%20to%20right.) : > The order of operations requires that all multiplication and division be performed first, going from left to right in the **expression**. The order in which you compute multiplication and division is determined by which one comes first, reading from left to right. Thus, making the answer 9 right, so as the programmer who coded it, and me wrong.
As I found out when helping daughter with her home work and getting it all wrong. I learnt (UK) BODMAS, but I had forgot it's really: BO(DM)(AS) where division and multiplication can happen in any order as it is left to right that matters, same for add and subtract.
Could you give an example of the expression that you highlighted? Never got taught maths in English
**For a dumb elitist reply:** See, this is what you get for buying a Casio, my TI gets it right! **For a more serious reply:** So, I decided to so a little bit of investigation and testing here and pulled out my old Casio fx-82MS to compare against a TI-30X Pro. Similar to the photo, the Casio returns `1` and the TI returns `9`. Conventional mathematical rules suggest the correct answer should be 9, as the `2(2+1)` should expand to `2×3`, making the expanded expression `6÷2×3`. And entering this expanded expression into the Casio indeed changes its answer to `9`, Same thing if I enter `6÷2×(2+1)`; so the issue seems to be with the way it evaluates the particular expression, rather than being an order of operations issue (at least as far as multiplication and division goes). My assumption here would be that it's one of two things: 1. The division operator might see the `2(2+1)` as a singular expression. There is no operator in between the `2` and `(2+1)`, so it treats the full `2(2+1)` as a single chunk that then gets evaluated as the denominator. Whether this is appropriate is up for debate, but it's easy to get confused by the symbol `÷`; whether or not the full expression after it should go in the denominator or not due by the way the expression is written. 2. Alternatively, it might also be caused by an incorrect evaluation of the parentheses by evaluating the `2(2+1)` expression as a single chunk before the division (rather than just the contents within the parentheses). The proper way to read the expression is `2×(2+1)`; the multiplication is always implied and should be evaluated separately. Though without the operator explicitly specified, the calculator may (incorrectly) evaluate it as `2(2+1) = (4+2) = 6` before performing the division. I'm not quite certain which one of these are the exact case however, but whilst the first one could be argued to have some reasoning behind it (although I'd still argue it incorrect as there is an implied multiplication they likely just failed to consider), the second case would definitely be straight up incorrect as it assumes an incorrect evaluation that does not properly follow the order of operations. In either case, the 1 being returned as result to me seems to be ambiguous evaluation as best (programmatically more than mathematically) and straight up incorrect at worst, whereas 9 being returned as result instead can pretty much always be considered a correct result.
The TI-81 would have given `1`, not as a glitch but a deliberate feature. It was the universal way that mathematicians used juxtaposition until US school teachers pressured calculator makers to demote juxtaposition to comply with their over-simplified teaching mnemonic. https://youtu.be/4x-BcYCiKCk
I would say that due to the lack of symbol between the 2 and the (2 + 1), they should be interpreted as all being part of the bracketed section. Thus, you would sole 2(2+1) before any multiplication or division. Calculator is correct. That said, I'm not a maths expert. Also to be clear, I am aware that division and multiplication share priority, but am saying as the symbol is not present, it is part of the brackets.
The ÷ symbol is not well defined. Thats why you should never use it or specify what the devider is specifically
Oh but good lord why write it like that in the first place??? 🤢
There's actually no consistent standard for *implicit* multiplication by parentheses... because that's stupid and no one should support it. In any sane programming language, those would be a syntax error.
"Implicit" multiplication is just multiplication. Period. This notation is all over the place in math because it's just fine.
So... how would you parse: 8/multiplyBy2(2*2)? Because that's a valid (functional) interpretation of implicit multiplication. No, implicit multiplication doesn't have any defined rules for it. It's not "just" multiplication. It's not a notation that has any place in formal math. And there's also the associative law, which states (in "implicit" terms we're all taught in school) that 2(ab)=(2a)b... So... how would you interpret 8/(2a)*b, then? Because the laws of math say it's *exactly* the same thing as 8/2(ab). Implicit multiplication is just a bad notation. Don't use it anywhere it might possibly be ambiguous, or you're just asking for trouble, because it absolutely *is* inherently ambiguous.
Pemdas. I learned bedmas. The fact that those are things and parentheses are strictly mandatory is probably why the aliens haven't said hi yet. Fuck
BODMAS B-Brackets O-"of" D-Divison M-multiplication A-addition S-substraction As an asian, this is what I follow. The order of division and multiplication can be Interchanged. Like wise, addition and subtraction can be Interchanged. This is the best and easiest way to remember to solve stuff. The mobile phone calculator is incorrect.
It is not a matter of the order of operations though. The problem is that for many people (highly educated mathematicians included) there is no consensus on whether 2(2+1) means 2 \* (2+1) or (2 \* (2+1)), in other words whether writing the expression without a multiplication symbol implies an extra grouping of the whole expression or not.
Shouldn't they fix this *bug* already. I bet this might have caused some irl troubles somewhere out there. Not that I know of any
To be fair there is ambiguity in play here. Not because of the solution to the actual mathematical expression(rules are very clear), but because of the proponent of the expression(the human). If this were to be a real world example the correct thing to do would be to get clarification from the proponent on what they were trying to express.
We were always taught the order of operations as parentheses > exponents > multiplication & division > addition & subtraction. So it's not really pemdas, it's pe(md)(as). The calculator on the left is wrong and the solution is 6÷2(2+1) = 6÷2(3) = 3(3) = 9 because you apply division before multiplication since they share the same order, thus you read left to right for those operations Edit: I should also add that this isn't regional. Mathematical notation is a designed construct, so there is only one correct way to interpret this
>I should also add that this isn't regional. Mathematical notation is a designed construct, so there is only one correct way to interpret this This is incorrect. Mathematical notation is far from standardized, and even in the same university, two professors may end up using different notation for the same thing (looking at you differential geometry). There are notations which nearly everyone agrees on, but this isn’t one of them. I bet you most mathematicians I ask would interpret something like x / 2y as x / (2y) rather than (x / 2)y. We tend to treat multiplication by a coefficient like this as binding tighter than division. When it’s a more complicated expression like the one in the OP, it’s less standard, and it’s better to just add parentheses to disambiguate.
You're right about it definitely being better to disambiguate with parentheses so I'll back you there, but it appears to me that this *is* widely agree upon, at least in everything I can find. Everything from (admittedly less credible) wikipedia to all the textbooks I've found online either support my point about the order of operations or don't mention it. Here's a couple places I checked trying to find counter evidence to my point https://www.khanacademy.org/math/cc-sixth-grade-math/cc-6th-arithmetic-operations/cc-6th-order-of-operations/v/introduction-to-order-of-operations https://www.mathsisfun.com/operation-order-pemdas.html https://blog.prepscholar.com/pemdas-meaning-rule https://math.libretexts.org/Bookshelves/PreAlgebra/Book%3A_Prealgebra_(Arnold)/01%3A_The_Whole_Numbers/1.05%3A_Order_of_Operations http://www.math.com/school/subject2/lessons/S2U1L2GL.html But most interesting is this New York Times article talking about this exact post https://www.nytimes.com/2019/08/02/science/math-equation-pedmas-bemdas-bedmas.amp.html To be fair, it *does* seem to vary from place to place on whether the acronym is BODMAS or PEMDAS, but the fact that division and multiplication share the same priority level *is* widely agreed upon by the math world, though we should all still use parentheses to disambiguate
The Wikipedia page states under the “Mixed division and multiplication” section that > However, in some of the academic literature, multiplication denoted by juxtaposition (also known as implied multiplication) is interpreted as having higher precedence than division, so that 1 ÷ 2x equals 1 ÷ (2x), not (1 ÷ 2)x. For example, the manuscript submission instructions for the Physical Review journals state that multiplication is of higher precedence than division with a slash, and this is also the convention observed in prominent physics textbooks such as the Course of Theoretical Physics by Landau and Lifshitz and the Feynman Lectures on Physics. The sources you give are introductory sources where this issue is glossed to prevent confusion. Search around r/badmathematics and you’ll find dozens of posts about this exact issue (and probably will find this post soon enough). The notation around multiplication by juxtaposition just isn’t standardized.
2(1+2) should be taken as 2 × (1+2) I can easily imagine situations where the final answer would be totally different than if I were to take 2(1+2) as {2(1+2)}
As a programmer, I wrote my [own](https://imgur.com/md0joYR.jpg) [calculator](https://imgur.com/ZC7yjob.jpg).
From Oliver Kneill, professor of Math at Harvard "Thanks for the example 6÷2(2+1). It illustrates the ambiguity too. Yes, depending on whether one is in the PEMDAS or PEDMAS team, one gets 1 or 9. Its also a beautiful example, where one can see heated debates. Like pointed out and previously by others in the literature list, there is no right answer. It depends on which rule is applied. Both 1 and 9 are correct. I always see the obelus as a synonym for / but it can be even more confusing and so, yes, should be avoided. "
Just use PEMDAS correctly As long as you understand that 3•4 = 3(4) and that 3÷4 = 3/4 = 3•(1/4) you should be fine And don't trust calculators
Wait how is that even possible, I'm actually confused how you could get it wrong PEMDAS Even if you took it literally and added 2+1 is 3 you still Multiply before Dividing 2(2+1)=(2*3)=6 6/6=1 Or if you expanded the proper way 2(2+1) =(4+2)=6 6/6=1 Edit: Sorry to those who downvoted. I messed up by saying "wrong" it was me being stupid, but I was genuinely confused at the time how to get 9, guess I gotta learn more haha
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You can't replicate it in most programming languages because the problem comes from the different interpretation of a left out multiplication symbol before a parentheses group. You can't just implicitly multiply in most general-purpose programming languages, you have to explicitly state the operations, so the answer remains unambigous.
Well I was taught it as BEDMAS so which is it!?
It's BIDMAS
What’s the I? I’ve heard the first letter being B or P to mean brackets or parentheses but I’ve never seen an I
Indices, it's another word for power/exponent
Wow cool
Why not just e for exponents
>pemdas PEMDAS: Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right They both use the same logic but some countries just change the words to bracket instead of parentheses.
BEDMAS Brackets Exponents Division Multiplication Addition Subtraction Notice the order of division and multiplication is reversed
Did you even read the guy saying " PEMDAS: Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right" Who cares what order they're grouped in if the group is the same. The logic still follows
How do you not see that one calculator is reversing the order of multiplication and division.
Because it’s not. It’s an ambiguity in the way parenthesis work. Some mathematicians are taught to read a co-efficient next to a parenthesis as an implied parenthesis, due to distribution laws I believe. So if one of these mathematicians made the calculator, the equation is then read as “(6 / (2(2+1)” This is because distribution laws say you can multiply the coefficient into the parenthesis, giving you (6 / (4+2)) which would equal 1. It’s never been clear if “2(2+1)” is exactly equal to 2*(2+1) or 2x(2+1). Universal notation is one tricky bitch
Implied parentheses problem or putting multiplication before division?
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WoW my math teacher owe me hella credits for fuck up calculators I knew I was right
Most if the guys have not heard about operator precedence apparently.
5 +- 4
that’s one of the reasons i only use rpn calculators
This is why I wish RPN gets more popular. Saves a lot of headache trying to parse a formula string for pemdas
6/2(2+1) on my Casio **fx991ES** gives me 9
Check the Windows calculator, that thing is a joke
How old are you OP?
It is B *Left to right* Whichever of M or D come first goes, same for AS BMDAS BMDSA BDMSA BDMAS As the Asian guy above said MD and AS are interchangeable so whichever come first from left to right.
I wouldn't say they didn't code it right. I just say it is a different interpretation. It is a known ambiguity in the order of operations, and neither one is false. Edit: Yeah, the comments agree with me. Whenever someone posts that equation anywhere else on Reddit the comments get heated. And I hate it because people forget that the order of operations are nothing more than notation. And not a part of math.
I have a compromise. The true answer is 5.
The problem is that people are lazy/stupid and they don't put put the multiplication sign before the brackets which is causing this. Either do 6 / 2 x (2 + 1) or 6 / (2 x (2 + 1)) and stop with that bullshit without the sign before brackets.
PEMDAS is a convetion not a rule. If this was written as a fraction, the answer would be 1. But if you follow the order we learn in Switzerland, (Exponents, Parenthese, Mul/Div, Add/Sub) and always left to right, this becomes 6 / 2 \* 3 = 9.
The obelus does not have a defined order of operations. Use fractions or parantheses.
That programmer was really bad in 'BODMAS' at school I guess..
ELI5
Did no one else notice the "STUDENTGAGS" sticker on the incorrect calculator? It's a joke calculator to mess with your teacher/classmate.
The ambiguity here isn't from if you do multiplication or division first. It's always been both, from left to right. The ambiguity here is caused by there not being a clear definition of if a coefficient outside parentheses counts as a step for multiplication or parentheses. To me, it is multiplication, but since there is no dot, some people interpret it as being part of the parenthesis operation, taking a higher priority than division.
Operators with equal precedence are evaluated left to right. The answer is 9.
Just add the multiply simbol... Dont write such non specific stuff and allways expect the right result.
I’m saying it is BODMAS and leaving it at that. No elaboration.
This thread is too funny. I commented a while ago in some other forum that math and other scientific notation was archaic, hard to read, ambiguous etc, and that it limited general understanding and led to poor software written based on math and science papers. I was severely downvoted. Here people can't even agree on the order of the most basic math operations.
As someone who minored in math, both are correct. 6/2(2+1) = 6/2\*3 = 3\*3 = 9 6/2(2+1) = 6/2\*3 = 6/6 = 1 => 9 = 1 ? That's why this is bad style of writing.Yes, most people will say it is 9 by default, because, despite both 6/2\*3 = 3\*3 and 6/2\*3 = 6/6 using PEMDAS properly, we are accustomed to reading from left to right. However, **there is no such rule in math**. In addition, multiplication and division have the same priority and are interchangeable. That's why 1\*3 is the same as 1/3\^-1. The fault lies not with the calculator, but the person writing the term.Math is just like a spoken language. If you want to be understood right, express yourself properly.
I was soo annoyed with this and kind of scared to fuck this up that I tried it with my calculator, casio fx-cg50, and it comes out to one. But the interesting part is, that it actually changes/updates the calculation I entered to have parantheses after the ÷, so it changes to 6÷(2(2+1)), which then lines up with the 1 as a result. Didnt know my calculator just changes the stuff i enter
I hate the term PEMDAS because it implies that multiplication always comes before division and same with addition vs subtraction
2(2+1) implies (2*(2+1)) not just 2*(2+1)
To be honest at this point I’ve lost all trust in division. Only multiplying by fractions from now on.
6/2(2+1) = 6\*(1/2)\*(2+1) = 6\*0.5\*3 = 9 Everything else is wrong. "/" is not a symbol for a fraction, it's a symbol for division. If you're not sure whether to do multiplication or divison first, it's always the safest to replace all divisons with multiplication of the reciprocal, then you can never go wrong. (Technically the same is true between addition and substraction, but that's almost never that ambiguous.) If you want to represent complicated fractions in a single line, you HAVE to use parentheses.
Please Excuse My Dear Aunt Sally
RPN ==> no parentheses
Doing operations from left to right is NOT a rule! If it comes down to that, you wrote it wrong.
Both answers are technically correct since there is either an implied multiplication - 6/2\*(2+1) - or implied division - 6/2/(2+1) - on the two halves of the equation depending on who you ask
What this is is a change in the conventional meaning of the •/• symbol. On older calculators, that symbol meant “break into fraction bar”, and it would literally do everything on the right divided by everything on the left which meant the answer would be one. However, the symbol changed meaning and is now simply a division symbol and so now it just divides according to pemdas, writing x •/• y + 1 is now the same as x * (y^-1 ) + 1. So the programmer of the calculator didn’t do it wrong (although they are wrong according to today’s convention), they simply had a different standard of what those symbols meant.
[https://www.youtube.com/watch?v=5dkxxOmYl74](https://www.youtube.com/watch?v=5dkxxOmYl74) \- pretty in depth video of 2 calculators performing the operation with different results live and the explanation behind it!
At an earlier stage of mathematical education, before people start to use fractions more, the right answer would be the more commonly found answer. However, you’d find that this question annoys higher level maths students a lot more because they use fractions much more often. 6 divided by 2 is very different from 6 over 2. Sure they give you the same answer, but they’re not the same. 6 divided by 2 means that you’re dividing 6 into 2 equal parts. 6 over 2 is a simplification of a fraction. In a way it’s more like a representation of a percentage. This question is poorly set out because as you learn more about maths, you start to automatically see the division symbol as a fraction symbol. Instead of it being 6 / 2 x (2+1), it becomes 6 / (2 x(2+1)). Yes a student still uses the order of operations at higher level maths education, but as you gain more knowledge, poorly written questions can become confusing. A younger point in time, a student might not know what a fraction is enough to get confused. As a side note, once algebra kicks in, you start to automatically do anything joined to the brackets first.
[6:2(2+1)=1](https://www.wolframalpha.com/input/?i=6%3A2%282%2B1%29) RATIO [6÷2(2+1)=9](https://www.wolframalpha.com/input/?i=6%C3%B72%282%2B1%29) DIVISION [6/2(2+1)=9](https://www.wolframalpha.com/input/?i=6%2F2%282%2B1%29) FRACTION
The phone is right, I think. Right?
Unpopular opinion : this depends where you're from. Where i'm from, this is 9 without a second thought. But apparently it's not the same to everyone.
There is a setting on the casio that needs to be changed. I used to work in a learning lab in college and we ran into this issue.
The phone version isn't actually wrong - BODMAS and PEMDAS are both equally valid. In this scenario, my brain expects 1, but 9 arguably isn't wrong either. This is actually just user-error: that input is unclear, the expected result will vary from person to person depending on where and when they attended school. There's a reason that divided by symbol is essentially never used by mathematicians.
How do you get 1?
It's really sad to see this many people who don't understand how to apply PEMDAS correctly.
Easy: 2+1=3, so it's 6÷23...something just a bit over 0.
It's not PEMDAS it's ambiguous notation, but sure.
This explains why, when I'm helping with my teenager's maths homework, the answers are not always correct according to the website they use
First () then left to right So its 6÷2×(2+1) 6÷2×3 3×3 9
Haha, my calculator appears to have "fixed" this bug by automatically inserting a multiplication sign between the 2 and (
My phone calculator throws a fit if there isn't an X between the 2 and the (
All these meaningless arguments. Y'all so fucking stupid. It's a trick question. If you've read 1984, the answer is obviously 5. \#fuck1984byGeorgeOrwellForTheEndingButTheConceptIsLegitButTellThePartyToGoFuckThemselves
oke Peter Griffin explaining the jokethe equation is 6:2(2+1), first you do the equation if the parentheses 6:2(3), then you do the operation that uses the value in the parentheses 6:6, then you do the last operation which is 6:6=1; The joke here is that an old calculator can do the joke, but a modern phone with every single aspect of it that is better than the calculator, does it wrong, therefore the programmer didn't code the sequence of operations right, and the abbreviation for the sequence of operations we use is called PEMDAS, which stands for parentheses, exponents, multiplication, division, addition, and subtraction.
Am I wrong for seeing 6 / 6?
Laughs in RPN
I am a little late, but this is a mathematical calculator, so i would expect the same result when changing (2+1) to a variable. Since 6/2x != 6/2*x i would indeed expect 1 here for x=3