This is technically a repost, but the [old post](https://www.reddit.com/r/mathmemes/comments/r3hqj8/these_wouldnt_be_viral_if_people_remembered_order/?utm_source=share&utm_medium=ios_app&utm_name=iossmf) was not popular. I will let the other mods decide on this.
No, no one is out here trying to correct people. Use this comment as proof that no one ever corrects people. You disagree and straight to name calling.
Well first of all, you're wrong. Now i shall correct you by explaining that there is no better feeling than correcting people on the internet, as it asserts your intellectual superiority over them.
3rd you can’t be above me, do you know how many people I have put down today? 18 and it’s 6 am. That’s 3 an hour. You can’t keep up with these numbers. Clearly I’m the bigger man as I have built my self up on putting other people down. Look at how good I’m, I’m currently taking the high road you stupid mofo.
P.s. love the use of first off with no follow up points. That is a real Reddit move.
It's intentionally confusing. Properly written the implied parenthesis before the 2 would be included to clarify the order of operations. Or the first part of the equation would be placed in parenthesis if they wanted that portion solved first the rule is parenthesis, exponents, multiply/divide starting from the left, add/subtract starting from the left.
It’s like someone forgetting to add a comma or putting an apostrophe in a word that’s plural instead of a possessive or contraction. It’s a poor way of writing this problem, and it leads to ambiguity.
Yep. Notation is a form of communication. Intentionally writing your expressions in a confusing way isn't being clever, it's just being deliberately obtuse, and sabotaging your own message. Plus notation conventions are not really maths themselves, just a necessary part of the human practice of mathematics, so it's both funny and disappointing seeing people flex these posts as proof of their intellectual superiority
PS. IIRC this type of ambiguity is one that even calculators don't agree on, so there's that too
There was a subtle change in order of operations sometime in the second half of the 20th century and there are lots of people on both sides who were taught opposing methods.
Anything that manufactures outrage will get popular. The fact of the matter is this problem is poorly written and ambiguous. You need to tell us if the 6 is over the whole equation or not. If the 6 is over the entire equation then the answer is 1. This is why in higher math the division sign is never ever used. It is not accurate enough and writing the number over the other as a fraction will always be more specific.
For someone who doesn’t remember PEMDAS, it’s actually a pretty interesting puzzle. For someone who remembers PEMDAS, it’s annoying as shit to see. Just depends on relative knowledge.
AKA postfix for us CS types.
We typically learn it as an introduction to the idea of changing the human element as opposed to coding around the human. Since infix notation (what is most often used) requires non-sequential read operations to properly tokenize and calculate inputs it is much more demanding on a calculator or computer to process; HP and others developed calculators around the prefix and postfix notations to lessen this load on early calculators.
That’s how I was taught it in my maths course as well, cause the subject was predominantly CS majors
We were given two different equations and told to write them both with infix notation; they came out the same, we got told that infix creates ambiguity, and we got told that computers like pre- or postfix to avoid that problem
There's an insane amount of kanji with very specific meanings. Like in the latin alphabet you read a jargon word and don't know what it means but for every word you're even slightly unsure about, a japanese person wouldn't even be able to pronounce it. I'm not sure on the numbers but I think adults know ~30-40% of the kanji that make up almost all their daily language and just guess at the rest based on shapes and associations.
I will be messaging you in 350 years on [**2372-06-12 01:55:53 UTC**](http://www.wolframalpha.com/input/?i=2372-06-12%2001:55:53%20UTC%20To%20Local%20Time) to remind you of [**this link**](https://www.reddit.com/r/mathmemes/comments/v9qn8d/i_think_its_9/ic1hlaa/?context=3)
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Why? I think it’s fine personally. I switch between notations depending on the question and whatever feels right at the time.
It’s a matter of preference really.
In higher level maths it would be written out in a rich text to actually be something like
6
---------
2(2+1)
This helps clear up any confusion on order of operations. Or it would have descrete parens
(6)/(2(2+1)
Even worse: 6 / 2(1 + 2)
The slash operator is really ambiguous. Any programmer knows it has equal precedence to multiplication, but a mathematician may treat it like a large fraction bar and give it lower precedence
An actual mathematician would realize the ambiguity and use additional parentheses, or they would write an actual fraction. Most mathematicians appreciate that it isn't enough to be technically correct, they need to be understood, and so they avoid ambiguous or confusing notation if they have a simple way around it.
I believe ei283 was saying if a mathematician was _handed_ this ambiguous problem, not if they wrote it themselves.
Maybe the correct choice is to refuse to answer.
I can just imagine a 9 year old kid writing on an arithmetic test in pencil, overflowing out of the slot where the answer goes to fit the entire answer that he came up with:
```
:1: SyntaxWarning: 'int' object is not callable; perhaps you missed a comma?
Traceback (most recent call last):
File "", line 1, in
TypeError: 'int' object is not callable
```
the *real* ambiguity comes from the multiplication via juxtaposition [2(1+2) rather than 2*(1+2)]; since that's sometimes conventionally higher precedence than other multiplication/division.
There's a rule in many places that if the multiplication symbol is omitted, you treat the entire product as a part of a fraction, e.g. 4π/2π = 2 but 4π/2*π = 2π^2
a programmer may also see that function call has greater precedence than division (disregarding that it would currently be an error as you can't start a function name with a digit)
mult2 = ( x => 2*x );
answer = 6 / mult2( 1 + 2 );
No. Not at all. I mean more precedence. Noting that "precedence" doesn't have a mathematical meaning, but just using the normal English definition.
>precedence
>/ˈprɛsɪd(ə)ns,prɪˈsiːd(ə)ns/
>noun
>the condition of being considered more important than someone or something else; priority in importance, order, or rank.
If I see an equation like (6/2)*(2+1) then that's just an equation. But if it was 6/[2(2+1)] then I would consider it a fraction. The fraction takes precedence in classifying the equation even though it isn't the first thing you calculate.
Or as another example if you have sin(2pi) then I would consider the sine is the important part not the multiplication which you do first if you're calculating the answer.
> "precedence" doesn't have a mathematical meaning
Wikipedia gives a synonym of "order of operations" as "operation precedence", which is a really common phrase in computer science.
https://en.wikipedia.org/wiki/Order_of_operations
It might not be commonly used by mathematicians, but it's defined.
Usually when I see the ÷ symbol, my brain enters a mode where I insert a × symbol everywhere a multiplication occurs, giving 6 ÷ 2 × (1 + 2). I feel like it's more obvious then that division and multiplication have positional precedence descending left to right.
But with the / symbol, it becomes more ambiguous to me because it could represent a large fraction bar, as in 4π/2π = 2.
as a mathematician and a programmer, i have always just followed pemdas and assumed that the value represented by
6 ÷ 2(1 + 2)
is 6 over 2, the fraction, multiplied by 1 + 2.
a better way to write this would be (6 + 12)/2
People need to realize that the order of operations is simply a convention within the field. There's no one true universal standard that every single author and publication is required by law to follow and so you get some edge cases. The one here is that some authors consider multiplication by juxtaposition to have higher priority than the ÷, /, ·, ×, and \* operators, some authors do not. Another example is where most authors consider *a*^*b*^*c* to be equal to *a*^*d* where *d* = (*b*^(*c*)) (reddit formatting is fucky) while some consider it to be equal to (*a*^(*b*))^(*c*).
Moreover, PEMDAS (or any equivalent alternative) a mnemonic taught to pre-teen students that's convenient for the content they're likely to encounter.
It's not exhaustive, and doesn't take into account nuance or intention. Math is a language, not an algorithm.
But most people haven't taken math class more advanced than pre-algebra or algebra, and their understanding of math is limited by teachers more worried about making grading assignments easier than teaching actual comprehension.
So we wind up with four groups:
- PEMDAS is PEMDAS
- 'But it's ambiguous!' which often appears with a side of 'You can't use a division symbol!'
- Well my calculator says...
- Obviously, the intent was...
I personally fall into the last category. The choice of notation used to write multiplication was intentional, as there are multiple other options they could have easily used. In this version, there's also the intentional choice that spaces only appear on either side of the division symbol and before the equal sign.
Both comments above you understood and used this. The reply stated that no one would write it as \(a ^ {b^c }\) to indicate \(a ^ {b*c}\)
I can’t format
Also notation is not even maths, just a tool used to express it to other humans. So if you intentionally confuse people, you're not being smart, you're just being a dickhead. Saying someone doesn't know maths because they fucked up OOE is like saying someone doesn't know literature because they can't spell "pneumonia"
That's not the problem as much as "is the entire right side past the division symbol together or not" most everyone does the parentheses first.
So then you're left with the argument of is it:
6 / (2 x 3) = 1 -- this is what you'll get if you view the parentheses right next to the 2 as being indicative of it all being under the divisor, or...
6 / 2 x 3 = 9 -- this is what you get if you expand things out and follow order of operations assuming that the parentheses is not under the divisor.
Unfortunately it's an ambiguously formatted problem that's designed to provoke an argument.
Some mathematicians use GEMDAS as a more descriptive acronym: Grouping (parentheses, brackets, etc.), Exponentiation, (Multiplication & Division, of equal precedence evaluated left-to-right), (Addition & Subtraction, also evaluated left-to-right).
One reason I've heard for this order is based on group theory, but I'm fuzzy on the specifics. In general, following this process is less ambiguous and has a level of applicability in computer science.
In any case, that means "true" order of operations is:
6 / 2 (2 + 1)
6 / 2 • 3
3 • 3 = 9
Side note: if these operations are being processed by an old-school calculator or on a software program run on stack memory, order of operations may reverse: these cases can invert input, since that's the cheapest way to process calculations on limited memory. As such, they may not follow order of operations properly.
I didn't find any posts that meet the matching requirements for r/mathmemes.
It might be OC, it might not. Things such as JPEG artifacts and cropping may impact the results.
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My dad always finds these on Facebook and then goes "oh it's one of these dumb things designed to trick you" but then he goes "oh yeah and the answer is ____"
Based.
When I was first learning about fractions they made no sense to me. But now I'm convinced that fractions are one of, if not the most important thing in mathematics. Just seeing an equation with the ÷ instead of a fraction makes my brain stutter for a second.
how much is 6/2x while x=3?
From algebraic point of view this is a fraction and should be looked at that way and answer would be 1...
From purely arithmetic point of view it is 9... So you could say both are correct but the notation is wrongly presented
Even as arithmetic this = 1, if you answer 9then you broke the question into two and made your own question without realising, instead of solving as a single question
Did we change the order of operations to Warhammer 40k math recently? I don’t remember division ever occurring before multiplication.
6 is a term. 2(1+2) is a term. No matter what symbol you use to denote division and multiplication, these two are still separate terms. The only way you can get 6/2 is if you also divide 2(1+2) by 2. This would imply you have to divide the unknown on the result side by 2, though, which you could, you’d just have to remember to multiply the whole equation by 2 again.
6 / 2(1+2) = x
-> 6/2 / 2(1+2)/2 = x/2
--> 3 / (1+2) = x/2
---> 3 / 3 = x/2
----> 2(3) / 2(3) = x
-----> 6 / 6 = x = 1
We could also factor the 2 into the parentheses, so it would become 6 / (2 + 4). It still ends up as 1.
This expression just shouldn’t ever look like (6/2) * (1+2), though.
It took me a while to figure out why I was having such a hard time understanding how people could make that mistake. Turns out, I did this in my head, then wrote it down. I hadn’t tried a digital system.
It’s because our digital systems are not coded to handle the understanding a “n(n1…)” is multiplication. For example, to properly write this equation in excel would require writing it as 6/(2*(1+2)). This is the only way to get it to understand the order of operations correctly between the two terms. Otherwise, the digital system interprets 6/2 as a term and (1+2) as a term, which is not how this equation is written in analog.
It makes sense coding cannot support “n(n1…)” means multiplication, though. Imagine if in every instance of a set of parentheses, a computer would need to make sure it was or wasn’t a mathematical statement by comparing it to everything else around the instance of parentheses. I don’t think computers could handle something like that in a reasonable span of time. So, it was probably simplified during the earliest days of coding. If () meant multiplication, it couldn’t mean anything else in those first days because logic systems for being able to define variables weren’t a thing in coding yet.
However, the problem of a computer inherently understanding what a mathematical term is without excessive intervention is unanswered. We know 6 is a term. We know 2(1+2) is a term. A computer does not know 2(1+2) is a term. Even wolframalpha does not know what you are saying. As the user, the only way to make it understand what you see is to type in (2*(1+2)).
I think this kind of required specificity has caused a lot of people a lot of problems. It’s situations like these that have led to software engineers/coders/students/etc. having entire programs come up broken or giving wrong outputs without anyone knowing why. So, I now understand how people could end up with 9; however, if you are getting 9 because you used wolframalpha, a calculator, excel, or any other digital system, you’ve been the mathematical equivalent to improperly autocorrected. Ducking autocorrect…
The lack of an operator between 2 and (1+2) makes 2(1+2) a single grouping and is evaluated together. This is the same as how 3/2x always means 3/(2x) and not 3x/2. If it was 6/2 * (1+2) it would be 9, but 6/2(1+2) is 1.
> This is the same as how 3/2x always means 3/(2x) and not 3x/2.
I'm sorry, [what??](https://freeimage.host/i/hlqYSs)
Edit: this is to say, the notation is absolutely not as unambiguous as the parent comment makes it look like (I'm not trying to argue it means any of the two interpretations)
it gets even worse. in wolframalpha, `1/2W` is interpreted as half a Watt, but `1J/2W` is interpreted as (1J)/(2W) = half a second.
two disparate ways of treating the W after the division, despite both being interpreted as Watts.
edit: changed Coulombs and Amps to Joules and Watts, so it doesn't need Coulomb spelled out to disambiguate from Celcius.
Ok.in may opinion 6÷2(1+2) should be 1 but 6÷2×(1+2) should be 9 because because implicit multiplication **should** come before division. The reason I think so is because if someone writes 1÷2x will almost always mean 1÷(2x) and not (1÷2)x. If someone did mean that they would just write x/2 (or write the original problem as 6(1+2)÷2
i say a(b) is a function call regardless of what a actually is - even if it's 2. Of course, noone would seriously call a function '2' but i hate the ambiguity here and parentheses with something in front are generally function calls so why not be consistent?
It is the vastly more popular one. CASIO Calculators had issues with it because at first they did juxtaposition before division, then they didn't, then they did again.
After running many surveys for teachers, CASIO found out only American and possibly Indian teachers interpreted 1/2x as (1/2)x whereas German, French, Chinese, and overall the vast majority interpreted it as 1/(2x). So that's why latest CASIO calculators returned to evaluating juxtaposition first.
Similarly in Harvard there was a famous test where the teacher wrote 3x/2y-1 and not a single student interpreted it as (3x/2)y-1, most students interpreted it as (3x)/(2y)-1
yes we analyze it by order it is 9, all by operation hierarchy, many will think that because it is in parentheses it has priority, but it only represents multiplication, but not priority
They meant that some people think that the implied multiplication with parentheses, for example 2(1+2), has precedence over normal multiplication/division.
Order of operations is just notation and procedure, not an inherent property of math.
In a lot of settings parenthesis are given priority as a notation tool; it's like treating the (2+1) as a variable like 6÷2x.
I don't know many who would interpret that as 3x.
Ok so wait... I actually do have an interesting question maybe...
Is brackets a separate operation or do they only work in conjunction with another operation?
Like (1+2) = (3) or does (1+2) = 3?
Because this changes how you see the equation. In the second statement the brackets are no longer an operation in and of themselves but must always be used in conjunction with another operation and in the first one the brackets become part of a term, right?
6÷2(1+2) in the first scenario is only 2 terms: the 6 and the 2(1+2) which implies you solve the second term first because it contains the brackets and simplifies to 2(3) and then you once again simplify the term containing the brackets to 6 which creates the 6÷6 = 1.
Whereas in the second scenario you remove the brackets by using the addition operation and replace the brackets by a multiplication operation giving you 3 terms: the 6, 2 and 3. Now that they are split into three terms each split by an operation of equal presidence, the answer becomes 9.
Of course the problem with this is the way it's written down but I would have instinctively seen this as 6 OVER 2(1+2) which makes it 1 but I am also widely considered an idiot... But anyway
TL;DR can brackets be considered a separate operation or must they always be used in conjunction with other operations?
>Like (1+2) = (3) or does (1+2) = 3?
(3) = 3
Brackets aren't really an operation, they're just saying hey look do this first. When we write something like 2(3) it's just shorthand for 2 \* (3) which is just 2 \* 3.
Terms are separated by + or - so after evaluating the parenthesis both scenarios only have one term.
And both are equivalent since
2(3) = 2 \* 3
> 2(3) it’s just shorthand for 2 * (3) which is just 2 * 3.
It is but it’s not. You’re supposed to distribute what’s preceding the () to the numerical values inside the ()
For example y=2(x+5) becomes y=2x+10 not y=2(5x)
While the above comment is a bit rude, I must say I at least partly agree. The problem is not what's inside the parentheses, but rather what's directly next to them. The result should be written as
6 / 2(3)
Not multiplication, but juxtaposition, which means there is still ambiguity. Do we consider juxtaposition as "more important"? If yes, the answer would be
6 / (2(3)) = 6 / (2 \* 3) = 6 / 6 = 1
If we don't, then the answer would be
6 / 2 \* 3
which some still consider ambiguous. I personally believe it makes the most sense if the division operator divides by what is followed only right after the symbol. Otherwise we don't have a clear way of "ending" the denominator, so if I'd want to multiply the entire result by 3, I'd actually have to add in another division symbol (1 / (1 / 3) = 3). I find this more confusing, which is why I would consider it equal to
6 / (2) \* 3 = (6/2) \* 3 = 3 \* 3 = 9
As such, we can safely conclude that if anyone considers 6 / 2 \* 3 to equal 1 that they must be a psychopath.
Shouldn't it be 9 regardless?
First brackets, so 1+2 = 3
Then division or multiplication, so 6/2=3 and then 3\*3=9
There is no way to get 1 result, unless someone deliberately do multiplication 2 \* (1+2), before the division. Which isn't either correct with order of operation, nor it's actual order of the numbers in this equation.
Exactly. Both multiplication and division have the same priority, and they are both left-hand associative. That's how compilers deal with stuff like this, for example.
I honestly believe the confusion comes because of how we teach order of operations. We give students the acronym PEMDAS that is easy to remember but everyone forgets the two "from left to right" rules. They do multiplication before division and that causes the mistake, and addition before subtraction can cause mistakes too.
If they were taught the acronym PEDMSA instead, they wouldn't have to remember the "left to right" rules and may make less mistakes. Division before multiplication is always correct, and subtraction before addition is always correct. I always show this to my students.
This is technically a repost, but the [old post](https://www.reddit.com/r/mathmemes/comments/r3hqj8/these_wouldnt_be_viral_if_people_remembered_order/?utm_source=share&utm_medium=ios_app&utm_name=iossmf) was not popular. I will let the other mods decide on this.
Why do these problems get popular
It's basically clickbait. By saying something really stupid or really controversial, people are temped to "correct" the poster or other people.
I think it's Cunningham's law but a more funny person would say it's Murphy's law.
I think it's Cole's law
[удалено]
I usually put it in the sandwich, but each to their own
Could be dunning kruger a little bit too, the people who dont understand it properly think they know the right answer
And then theres 2 sides arguing which gives the debate more attention
[relevant xkcd](https://xkcd.com/1531/)
I **am** the law
you missed out on the chance to just say its called murphys law and have someone correct you
The shitty mobile ads of social media
No, no one is out here trying to correct people. Use this comment as proof that no one ever corrects people. You disagree and straight to name calling.
Well first of all, you're wrong. Now i shall correct you by explaining that there is no better feeling than correcting people on the internet, as it asserts your intellectual superiority over them.
3rd you can’t be above me, do you know how many people I have put down today? 18 and it’s 6 am. That’s 3 an hour. You can’t keep up with these numbers. Clearly I’m the bigger man as I have built my self up on putting other people down. Look at how good I’m, I’m currently taking the high road you stupid mofo. P.s. love the use of first off with no follow up points. That is a real Reddit move.
This one though is genuinely confusing. Both answers feel right. Like I get one but there is no law supporting my decision.
It's intentionally confusing. Properly written the implied parenthesis before the 2 would be included to clarify the order of operations. Or the first part of the equation would be placed in parenthesis if they wanted that portion solved first the rule is parenthesis, exponents, multiply/divide starting from the left, add/subtract starting from the left.
It’s like someone forgetting to add a comma or putting an apostrophe in a word that’s plural instead of a possessive or contraction. It’s a poor way of writing this problem, and it leads to ambiguity.
Yep. Notation is a form of communication. Intentionally writing your expressions in a confusing way isn't being clever, it's just being deliberately obtuse, and sabotaging your own message. Plus notation conventions are not really maths themselves, just a necessary part of the human practice of mathematics, so it's both funny and disappointing seeing people flex these posts as proof of their intellectual superiority PS. IIRC this type of ambiguity is one that even calculators don't agree on, so there's that too
It gives people something to argue over which drives engagement.
There was a subtle change in order of operations sometime in the second half of the 20th century and there are lots of people on both sides who were taught opposing methods.
What subtle change
Anything that manufactures outrage will get popular. The fact of the matter is this problem is poorly written and ambiguous. You need to tell us if the 6 is over the whole equation or not. If the 6 is over the entire equation then the answer is 1. This is why in higher math the division sign is never ever used. It is not accurate enough and writing the number over the other as a fraction will always be more specific.
For someone who doesn’t remember PEMDAS, it’s actually a pretty interesting puzzle. For someone who remembers PEMDAS, it’s annoying as shit to see. Just depends on relative knowledge.
It’s relatable
Oh fuck you and this division notation. I will create my own party and outlaw this bullshit on EU level.
You mean something like 6 2 ÷ 1 2 + ×?
Reverse Polish my beloved
AKA postfix for us CS types. We typically learn it as an introduction to the idea of changing the human element as opposed to coding around the human. Since infix notation (what is most often used) requires non-sequential read operations to properly tokenize and calculate inputs it is much more demanding on a calculator or computer to process; HP and others developed calculators around the prefix and postfix notations to lessen this load on early calculators.
That’s how I was taught it in my maths course as well, cause the subject was predominantly CS majors We were given two different equations and told to write them both with infix notation; they came out the same, we got told that infix creates ambiguity, and we got told that computers like pre- or postfix to avoid that problem
Prefix is nicer to program with I think but postfix works. Obviously it's different for calculators.
That might be unambiguous but readability is terrible. Just use fractions. Always. I don't care they aren't commonly supported in digital space.
The readability is bad because you aren't used to it. Like japanese is unreadable for me who uses the latin alphabet
Your point is correct but ironically you chose a language that even adult native speakers can't comprehensively read or spell out.
Really?
There's an insane amount of kanji with very specific meanings. Like in the latin alphabet you read a jargon word and don't know what it means but for every word you're even slightly unsure about, a japanese person wouldn't even be able to pronounce it. I'm not sure on the numbers but I think adults know ~30-40% of the kanji that make up almost all their daily language and just guess at the rest based on shapes and associations.
Oh.. well I'm glad I didn't chose japanese in school then
Alphabet writing are the best. Also, just imagine looking in a dictionary for a kanji without knowing where it could be.
[удалено]
Different symbol solves nothing.
> 6 2 ÷ 1 2 + × SyntaxError: unexpected token :p
Ever heard of reversed polish notation? It's supposed to replace expression trees
Ever heard of programming ? / is used for division, not ÷ \* is used for multiplication, not ×
Oh that's what you meant
Yup!
Using the obelus ÷ for division is already outlawed.
Never heard about somebody going to prison for using it.
They don’t go to prison, they are sent to wyoming
order ambiguous; got 1.653 - 4.247i
how
i do have a proof, but it's too large to fit in the comments
Im sure someone 350 years from now will write a post with the proof
RemindMe! 350 years
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Send a pastebin
My antivirus says it's a zip-bomb and deletes it...
Absolute Giga Fermat
Just upload it to google drive and send us the link
Man if only Fermat had a Google account
I love you
It’s trivial and left as an exercise for the reader
This guy maths
hory shid
also that division notation is pretty dumb
I used it only in primary school
Me too, but mostly in the bathrooms and we drew it more like this: #:-
Sir that's a dog's bollocks, not a division symbol
I've been enlightened and unlocked a new perspective of how I view the world around me
I literally just used it in my final maths a level exam lol
Good luck with that.
Imagine using it now
Why? I think it’s fine personally. I switch between notations depending on the question and whatever feels right at the time. It’s a matter of preference really.
The division sign is literally just a fraction with dots as placeholders.
Tbf it fits one one line, which a fraction bar doesn't
/
That's a good option too. Personally I always preferred the look of ÷ but / is certainly way better suited to digital use
… Is this for real??? I have blissfully lived my life thinking it was just a quirky plus sign.
The fact that 90+% of these feature said division operator should be enough justification to throw it out.
What is it supposed to be? I've never seen it a different way and you all have me very confused.
In higher level maths it would be written out in a rich text to actually be something like 6 --------- 2(2+1) This helps clear up any confusion on order of operations. Or it would have descrete parens (6)/(2(2+1)
That expression was designed to make people argue about it and gain attention
Too right, it's not even specified if this calculation should be done in Hilbert space or Banach space.
Even then, separable or non-separable?
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How is that more dirty. Isn’t there only one answer to this? The other one is purposely vague because it’s written poorly. This one is just 1.
Just don't use the divide sign. In higher level maths you always use fractions
Even worse: 6 / 2(1 + 2) The slash operator is really ambiguous. Any programmer knows it has equal precedence to multiplication, but a mathematician may treat it like a large fraction bar and give it lower precedence
If it was a fraction (or a large fraction bar as you call it) then you would calculate 2(1+2) first, giving it lower precedence…
my b, fixed "higher precedence" to "lower precedence"
An actual mathematician would realize the ambiguity and use additional parentheses, or they would write an actual fraction. Most mathematicians appreciate that it isn't enough to be technically correct, they need to be understood, and so they avoid ambiguous or confusing notation if they have a simple way around it.
I believe ei283 was saying if a mathematician was _handed_ this ambiguous problem, not if they wrote it themselves. Maybe the correct choice is to refuse to answer.
A real mathematician would solve it both ways and fine the average. So the reason answer is 5. /s
Not only that, but a programmer would most likely get a syntax error
I can just imagine a 9 year old kid writing on an arithmetic test in pencil, overflowing out of the slot where the answer goes to fit the entire answer that he came up with: ```:1: SyntaxWarning: 'int' object is not callable; perhaps you missed a comma?
Traceback (most recent call last):
File "", line 1, in
TypeError: 'int' object is not callable
```
the *real* ambiguity comes from the multiplication via juxtaposition [2(1+2) rather than 2*(1+2)]; since that's sometimes conventionally higher precedence than other multiplication/division.
That's a good point. It does seem like the division slash doesn't have this ambiguity issue with anything except for juxtaposed multiplication.
Wouldnt a large fraction bar be written as 6 / (2(1 + 2))?
There's a rule in many places that if the multiplication symbol is omitted, you treat the entire product as a part of a fraction, e.g. 4π/2π = 2 but 4π/2*π = 2π^2
a programmer may also see that function call has greater precedence than division (disregarding that it would currently be an error as you can't start a function name with a digit) mult2 = ( x => 2*x ); answer = 6 / mult2( 1 + 2 );
I understand what you are saying, but philosophically if that was a large bar then I would mentally give it more precedence and do the fraction last.
You mean less precedence
No. Not at all. I mean more precedence. Noting that "precedence" doesn't have a mathematical meaning, but just using the normal English definition. >precedence >/ˈprɛsɪd(ə)ns,prɪˈsiːd(ə)ns/ >noun >the condition of being considered more important than someone or something else; priority in importance, order, or rank. If I see an equation like (6/2)*(2+1) then that's just an equation. But if it was 6/[2(2+1)] then I would consider it a fraction. The fraction takes precedence in classifying the equation even though it isn't the first thing you calculate. Or as another example if you have sin(2pi) then I would consider the sine is the important part not the multiplication which you do first if you're calculating the answer.
> "precedence" doesn't have a mathematical meaning Wikipedia gives a synonym of "order of operations" as "operation precedence", which is a really common phrase in computer science. https://en.wikipedia.org/wiki/Order_of_operations It might not be commonly used by mathematicians, but it's defined.
To me that looks to be the exact level of ambiguity as OP’s division symbol.
Usually when I see the ÷ symbol, my brain enters a mode where I insert a × symbol everywhere a multiplication occurs, giving 6 ÷ 2 × (1 + 2). I feel like it's more obvious then that division and multiplication have positional precedence descending left to right. But with the / symbol, it becomes more ambiguous to me because it could represent a large fraction bar, as in 4π/2π = 2.
as a mathematician and a programmer, i have always just followed pemdas and assumed that the value represented by 6 ÷ 2(1 + 2) is 6 over 2, the fraction, multiplied by 1 + 2. a better way to write this would be (6 + 12)/2
People need to realize that the order of operations is simply a convention within the field. There's no one true universal standard that every single author and publication is required by law to follow and so you get some edge cases. The one here is that some authors consider multiplication by juxtaposition to have higher priority than the ÷, /, ·, ×, and \* operators, some authors do not. Another example is where most authors consider *a*^*b*^*c* to be equal to *a*^*d* where *d* = (*b*^(*c*)) (reddit formatting is fucky) while some consider it to be equal to (*a*^(*b*))^(*c*).
Moreover, PEMDAS (or any equivalent alternative) a mnemonic taught to pre-teen students that's convenient for the content they're likely to encounter. It's not exhaustive, and doesn't take into account nuance or intention. Math is a language, not an algorithm. But most people haven't taken math class more advanced than pre-algebra or algebra, and their understanding of math is limited by teachers more worried about making grading assignments easier than teaching actual comprehension. So we wind up with four groups: - PEMDAS is PEMDAS - 'But it's ambiguous!' which often appears with a side of 'You can't use a division symbol!' - Well my calculator says... - Obviously, the intent was... I personally fall into the last category. The choice of notation used to write multiplication was intentional, as there are multiple other options they could have easily used. In this version, there's also the intentional choice that spaces only appear on either side of the division symbol and before the equal sign.
why would a^b^c ever mean (a^(b))^c though since the latter can always be expressed as a^(b*c).
I think they meant to write something like \(a^{b^c}\) which cannot be represented in reddit markdown, I think?
petition to add support for LaTeX math formatting to Reddit comments
Both comments above you understood and used this. The reply stated that no one would write it as \(a ^ {b^c }\) to indicate \(a ^ {b*c}\) I can’t format
Also notation is not even maths, just a tool used to express it to other humans. So if you intentionally confuse people, you're not being smart, you're just being a dickhead. Saying someone doesn't know maths because they fucked up OOE is like saying someone doesn't know literature because they can't spell "pneumonia"
6÷2(1+2) = 6÷2•(1+2) = 6÷2•1+2•2 = 3+4 = 7 The only correct answer.
Why are people taking this seriously
Not sure if you respond to the comment or the other replies to the comment.
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That's also incorrect. Here's how it is >6÷2×(1+2)= 6÷2×3= 9 Please know the difference, the one you were solving was >6÷(2(2+1))
It's equal to 6 ÷ 2(1+2) it's not really ambiguous.
Stop writing equations like 3rd graders.
this question is breaking the internet lmao
parentheses is first
That's not the problem as much as "is the entire right side past the division symbol together or not" most everyone does the parentheses first. So then you're left with the argument of is it: 6 / (2 x 3) = 1 -- this is what you'll get if you view the parentheses right next to the 2 as being indicative of it all being under the divisor, or... 6 / 2 x 3 = 9 -- this is what you get if you expand things out and follow order of operations assuming that the parentheses is not under the divisor. Unfortunately it's an ambiguously formatted problem that's designed to provoke an argument.
why do we have BODMAS? I still dont get why other than in algebra
Some mathematicians use GEMDAS as a more descriptive acronym: Grouping (parentheses, brackets, etc.), Exponentiation, (Multiplication & Division, of equal precedence evaluated left-to-right), (Addition & Subtraction, also evaluated left-to-right). One reason I've heard for this order is based on group theory, but I'm fuzzy on the specifics. In general, following this process is less ambiguous and has a level of applicability in computer science. In any case, that means "true" order of operations is: 6 / 2 (2 + 1) 6 / 2 • 3 3 • 3 = 9 Side note: if these operations are being processed by an old-school calculator or on a software program run on stack memory, order of operations may reverse: these cases can invert input, since that's the cheapest way to process calculations on limited memory. As such, they may not follow order of operations properly.
This is just a stupid expression that is intended to confuse people, you have to clarify either 6/(2a) or 6a/2 with a being (1+2).
Honestly, is the only point of "maths" to confuse people on Reddit?
I like how for you maths isn't a real thing, but a fake concept that needs to be surrounded by quotes
Oh look it’s the monthlyPEMDAS post. u/repostsleuthbot
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My dad always finds these on Facebook and then goes "oh it's one of these dumb things designed to trick you" but then he goes "oh yeah and the answer is ____"
Poland infix notation is better
The keyword here is not order, it's disambiguity
There’s just no excuse for poor notation
Opinon: We should completely remove the division symbol. It serves no purpose. No one uses it. Fraction bar notation only from now on. Change my mind.
Based. When I was first learning about fractions they made no sense to me. But now I'm convinced that fractions are one of, if not the most important thing in mathematics. Just seeing an equation with the ÷ instead of a fraction makes my brain stutter for a second.
how much is 6/2x while x=3? From algebraic point of view this is a fraction and should be looked at that way and answer would be 1... From purely arithmetic point of view it is 9... So you could say both are correct but the notation is wrongly presented
Even as arithmetic this = 1, if you answer 9then you broke the question into two and made your own question without realising, instead of solving as a single question
Clearly, this is how it's meant to be calculated: 6 / 2(1+2) break out the parenthesis 6 / 2 + 4 3 + 4 = 7 :D edit: woooosh-material below
I hate those problems They should just make a proper division line to know what exactly is above or below
Did we change the order of operations to Warhammer 40k math recently? I don’t remember division ever occurring before multiplication. 6 is a term. 2(1+2) is a term. No matter what symbol you use to denote division and multiplication, these two are still separate terms. The only way you can get 6/2 is if you also divide 2(1+2) by 2. This would imply you have to divide the unknown on the result side by 2, though, which you could, you’d just have to remember to multiply the whole equation by 2 again. 6 / 2(1+2) = x -> 6/2 / 2(1+2)/2 = x/2 --> 3 / (1+2) = x/2 ---> 3 / 3 = x/2 ----> 2(3) / 2(3) = x -----> 6 / 6 = x = 1 We could also factor the 2 into the parentheses, so it would become 6 / (2 + 4). It still ends up as 1. This expression just shouldn’t ever look like (6/2) * (1+2), though. It took me a while to figure out why I was having such a hard time understanding how people could make that mistake. Turns out, I did this in my head, then wrote it down. I hadn’t tried a digital system. It’s because our digital systems are not coded to handle the understanding a “n(n1…)” is multiplication. For example, to properly write this equation in excel would require writing it as 6/(2*(1+2)). This is the only way to get it to understand the order of operations correctly between the two terms. Otherwise, the digital system interprets 6/2 as a term and (1+2) as a term, which is not how this equation is written in analog. It makes sense coding cannot support “n(n1…)” means multiplication, though. Imagine if in every instance of a set of parentheses, a computer would need to make sure it was or wasn’t a mathematical statement by comparing it to everything else around the instance of parentheses. I don’t think computers could handle something like that in a reasonable span of time. So, it was probably simplified during the earliest days of coding. If () meant multiplication, it couldn’t mean anything else in those first days because logic systems for being able to define variables weren’t a thing in coding yet. However, the problem of a computer inherently understanding what a mathematical term is without excessive intervention is unanswered. We know 6 is a term. We know 2(1+2) is a term. A computer does not know 2(1+2) is a term. Even wolframalpha does not know what you are saying. As the user, the only way to make it understand what you see is to type in (2*(1+2)). I think this kind of required specificity has caused a lot of people a lot of problems. It’s situations like these that have led to software engineers/coders/students/etc. having entire programs come up broken or giving wrong outputs without anyone knowing why. So, I now understand how people could end up with 9; however, if you are getting 9 because you used wolframalpha, a calculator, excel, or any other digital system, you’ve been the mathematical equivalent to improperly autocorrected. Ducking autocorrect…
It's 9.
6/2(2+1) = 2(6/2) + 1(6/2) = 12/4 + 6/2 = 3 + 3 = 6😎
Almost correct. 2(6/2) = 12/2
Youre no fun :(
Sorry. :'(
The lack of an operator between 2 and (1+2) makes 2(1+2) a single grouping and is evaluated together. This is the same as how 3/2x always means 3/(2x) and not 3x/2. If it was 6/2 * (1+2) it would be 9, but 6/2(1+2) is 1.
> This is the same as how 3/2x always means 3/(2x) and not 3x/2. I'm sorry, [what??](https://freeimage.host/i/hlqYSs) Edit: this is to say, the notation is absolutely not as unambiguous as the parent comment makes it look like (I'm not trying to argue it means any of the two interpretations)
it gets even worse. in wolframalpha, `1/2W` is interpreted as half a Watt, but `1J/2W` is interpreted as (1J)/(2W) = half a second. two disparate ways of treating the W after the division, despite both being interpreted as Watts. edit: changed Coulombs and Amps to Joules and Watts, so it doesn't need Coulomb spelled out to disambiguate from Celcius.
The operator is * It's not lacking, just not written out. What the fuck else is it gonna be, square root?
Ok.in may opinion 6÷2(1+2) should be 1 but 6÷2×(1+2) should be 9 because because implicit multiplication **should** come before division. The reason I think so is because if someone writes 1÷2x will almost always mean 1÷(2x) and not (1÷2)x. If someone did mean that they would just write x/2 (or write the original problem as 6(1+2)÷2
This > 6÷2(1+2) And this > 6÷2×(1+2) Are two totally different equations because of the implication
i say a(b) is a function call regardless of what a actually is - even if it's 2. Of course, noone would seriously call a function '2' but i hate the ambiguity here and parentheses with something in front are generally function calls so why not be consistent?
Obvs it's 5. 6/2 = 3x1 = 3+2 = 5
Your both wrong it's 6
6÷2(1+2) = 6÷2(3)
It's 1
It’s 1. Juxtaposition takes precedence over symbols indicating multiplication/division. Brackets: 6/2(3) Exponents: 6/2(3) Juxtaposition: 6/6 MD: 1 AS: 1
“Juxtaposition takes precedence over symbols indicating multiplication/division” isn’t standard convention.
It is the vastly more popular one. CASIO Calculators had issues with it because at first they did juxtaposition before division, then they didn't, then they did again. After running many surveys for teachers, CASIO found out only American and possibly Indian teachers interpreted 1/2x as (1/2)x whereas German, French, Chinese, and overall the vast majority interpreted it as 1/(2x). So that's why latest CASIO calculators returned to evaluating juxtaposition first. Similarly in Harvard there was a famous test where the teacher wrote 3x/2y-1 and not a single student interpreted it as (3x/2)y-1, most students interpreted it as (3x)/(2y)-1
It's standard enough to be programmed into most calculators
THIS
No one in their right mind would write this formula like that.
yes we analyze it by order it is 9, all by operation hierarchy, many will think that because it is in parentheses it has priority, but it only represents multiplication, but not priority
Parentheses 100% has priority, what are you on about
Yes and “2(3)” is not inside parentheses. The only thing in the parentheses at that point is “3,” which is already simplified.
2(3) by some convention operates to 6 by priority distribution though, so the point that 3 is "the only thing in the parentheses" is moot.
They meant that some people think that the implied multiplication with parentheses, for example 2(1+2), has precedence over normal multiplication/division.
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Order of operations is just notation and procedure, not an inherent property of math. In a lot of settings parenthesis are given priority as a notation tool; it's like treating the (2+1) as a variable like 6÷2x. I don't know many who would interpret that as 3x.
Yeah excactly
How was the order of operations decided? And how is it shown to be correct?
Ok so wait... I actually do have an interesting question maybe... Is brackets a separate operation or do they only work in conjunction with another operation? Like (1+2) = (3) or does (1+2) = 3? Because this changes how you see the equation. In the second statement the brackets are no longer an operation in and of themselves but must always be used in conjunction with another operation and in the first one the brackets become part of a term, right? 6÷2(1+2) in the first scenario is only 2 terms: the 6 and the 2(1+2) which implies you solve the second term first because it contains the brackets and simplifies to 2(3) and then you once again simplify the term containing the brackets to 6 which creates the 6÷6 = 1. Whereas in the second scenario you remove the brackets by using the addition operation and replace the brackets by a multiplication operation giving you 3 terms: the 6, 2 and 3. Now that they are split into three terms each split by an operation of equal presidence, the answer becomes 9. Of course the problem with this is the way it's written down but I would have instinctively seen this as 6 OVER 2(1+2) which makes it 1 but I am also widely considered an idiot... But anyway TL;DR can brackets be considered a separate operation or must they always be used in conjunction with other operations?
>Like (1+2) = (3) or does (1+2) = 3? (3) = 3 Brackets aren't really an operation, they're just saying hey look do this first. When we write something like 2(3) it's just shorthand for 2 \* (3) which is just 2 \* 3. Terms are separated by + or - so after evaluating the parenthesis both scenarios only have one term. And both are equivalent since 2(3) = 2 \* 3
> 2(3) it’s just shorthand for 2 * (3) which is just 2 * 3. It is but it’s not. You’re supposed to distribute what’s preceding the () to the numerical values inside the () For example y=2(x+5) becomes y=2x+10 not y=2(5x)
6/6, perfect meme.
Division symbols are ambiguous and that’s why real math professionals use fractions because the parenthesis are implied. 6 ——— 2(1+2)
BIDMAS (Do brackets/parenthesis first) so you end up with 6 / (2 * 3) -> 2 * 3 is 6 and 6/6 is 1.
You have to do the paranthesis first, not pull completely new ones out of your ass! 6 / 2 * (1 + 2) becomes 6 / 2 * 3 NOT 6 / (2 * 3)
While the above comment is a bit rude, I must say I at least partly agree. The problem is not what's inside the parentheses, but rather what's directly next to them. The result should be written as 6 / 2(3) Not multiplication, but juxtaposition, which means there is still ambiguity. Do we consider juxtaposition as "more important"? If yes, the answer would be 6 / (2(3)) = 6 / (2 \* 3) = 6 / 6 = 1 If we don't, then the answer would be 6 / 2 \* 3 which some still consider ambiguous. I personally believe it makes the most sense if the division operator divides by what is followed only right after the symbol. Otherwise we don't have a clear way of "ending" the denominator, so if I'd want to multiply the entire result by 3, I'd actually have to add in another division symbol (1 / (1 / 3) = 3). I find this more confusing, which is why I would consider it equal to 6 / (2) \* 3 = (6/2) \* 3 = 3 \* 3 = 9 As such, we can safely conclude that if anyone considers 6 / 2 \* 3 to equal 1 that they must be a psychopath.
Agreed 100% A sociopath at least.
Bro, just put it on a calculator, and if you dont want to, know the answer is 9
Shouldn't it be 9 regardless? First brackets, so 1+2 = 3 Then division or multiplication, so 6/2=3 and then 3\*3=9 There is no way to get 1 result, unless someone deliberately do multiplication 2 \* (1+2), before the division. Which isn't either correct with order of operation, nor it's actual order of the numbers in this equation.
PEMDAS Arent multiplication and division given equal order? And if not, isn't multiplication listed before division?
Multiplication and division have equal priority, so then the order is from left to right
Exactly. Both multiplication and division have the same priority, and they are both left-hand associative. That's how compilers deal with stuff like this, for example.
Almost everyone uses the convention that implicit multiplication takes priority over ordinary multiplication and division, though.
I honestly believe the confusion comes because of how we teach order of operations. We give students the acronym PEMDAS that is easy to remember but everyone forgets the two "from left to right" rules. They do multiplication before division and that causes the mistake, and addition before subtraction can cause mistakes too. If they were taught the acronym PEDMSA instead, they wouldn't have to remember the "left to right" rules and may make less mistakes. Division before multiplication is always correct, and subtraction before addition is always correct. I always show this to my students.
Thats one point of confusion, another is the priority of implicit multiplication
7
1
It’s clearly 9. Let me break it down for you 6%2(3)=6%6=9 duh. People are fucking dumb.
I can't tell if this is sarcastic, especially when they use the modulo operator (%) instead of, y'know, a slash (/)
6%6 = 0 just fyi, look up that operator
You need a life.
They’re right, though. % is a well-defined operator, and any keyboard with % will have a /.
1 you guys forget Bodmas