If it helps, I saw what you did there, and I exhaled slightly harder out of my nose while smiling wryly. It’s even better the op didn’t get it. So like, well done and stuff 😊
Your obviously is only a convention and not everyone agree with that. Not even all peogramming languages or calculators.
If you wanted obviously, it would have to have different order or parentheses or both. Of course everything in math is convention but I mean more obvious.
2+2*4 is obvious with PEDMAS, but hardy obvious to common people
2+(2*4) is more obvious to common people
2*4+2 is even more obvious to people not good with math. I would say this is the preferred form.
(2*4)+2 doesn’t really add more to it, it just emphasises it more, but unnecessarily.
Honestly that’s my pet peeve about this category of content. Over the years I’ve seen (at least) hundreds of these check-out-how-bad-at-math-everyone-is posts and it’s nearly always order of operations related. Apparently, a bunch of people forgot (or just never learned) PEMDAS.
Now, having an agreed-upon convention absolutely matters for arriving at expected computational outcomes, but we call it a convention for a reason: it’s not a “correct” vs “incorrect” principle of mathematics. It’s just a rule we agreed upon to allow consistent results.
So any good math educator will be clear on this. If you know the PEMDAS convention already, that’s good, since it’s by far the most common today. But if you don’t yet, don’t worry. It doesn’t mean you’re too dumb to math. With a bit of practice, you won’t even have to remember the acronym.
Most actual math people never have to think about pemdas here because no one would ever write a problem like this. The trick here is “when was the last time I saw an X to mean multiplication” so I would already be off about it
1 + 1/2 in my brain is clearly 1.5, but 1+1÷2 doesn’t even register in my brain properly.
Right, and that clue IMO unravels the more troubling aspect of why this content spreads so quickly:
It’s deliberately aimed at people with a rudimentary math education who can be made to feel far superior to others who, in spite of having roughly the same level of proficiency, are missing/forgetting a single fact that has a disproportionate effect on the result they expect.
That is, it’s blue-dress-level contentious engagement bait for anyone with low math skills, whether or not they remember PEMDAS.
“No one” in this context meant “no one who actually does maths professionally.”
In a Maths textbook
Right, and I have decades of maths experience outside of textbooks. So it’s probably been 20 years since I had a meaningful interaction with the × multiplication symbol.
You don’t know that the obelus means divide??
I clearly know what the symbol means, I demonstrated a use of it. But again, haven’t had a meaningful interaction with the symbol in 20 years, and yet I deal with / for division daily.
When I see 1+½ i can instantly say “one and a half”, but when I see 1 + 1 ÷ 2 i actually have to pause for a moment to think about order of operations. Same with 1+2x vs 1 + 2 × x … one I recognize the structure of the problem immediately, and one feels foreign.
The point is that people who do maths for a living, and are probably above average in maths, tend to write things differently than people who are stopped their maths education in high school (or lower), and these types of memes are designed around making people who know high school maths feel smart. People who actually know maths don’t need memes to justify being better at maths than the rest of the public.
I’m not sure what motivates you to so generously offer your various dyadic tokens of knowledge on this subject without qualification while ignoring my larger point, but will assume in good faith that your thirst for knowledge rivals that of your devotion to The Rules.
First, a question: what are conventions if not agreed upon rules? Second, here is a history of how we actually came to agree upon the aforementioned rules which you may find interesting:
I’m a Maths teacher with a Masters - thanks for asking - how about you?
while ignoring my larger point
You mean your invalid point, that I debunked?
what are conventions if not agreed upon rules?
Conventions are optional, rules aren’t.
here is a history of how we actually came to agree upon the aforementioned rules which you may find interesting
He’s well-known to be wrong about his “history”, and if you read through the comments you’ll find plenty of people telling him that, including references. Cajori wrote the definitive books about the history of Maths (notation). They’re available for free on the Internet Archive - no need to believe some random crank and his blog.
By qualification I meant explanation. My doctorate is irrelevant to the truth.
Since you asked, my larger point was about the unhelpful nature of this content, which makes students of math feel inordinately inferior or superior hinged entirely on a single point of familiarity. I don’t handle early math education, but many of my students arrive with baggage from it that hinders their progress, leading me to suspect that early math education sometimes discourages students unnecessarily. In particular, these gotcha-style math memes IMO deepen students’ belief that they’re just bad at math. Hence my dislike of them.
Re: Dave Peterson, I’ll need to read more about this debate regarding the history of notation and I’ll search for the “proven rules” you mentioned (proofs mean something very specific to me and I can’t yet imagine what that looks like WRT order of operations).
If what riled you up was my use of the word “conventions” I can use another, but note that conventions aren’t necessarily “optional” when being understood is essential. Where one places a comma in writing can radically change the meaning of a sentence, for example. My greater point however has nothing to do with that. Here I am only concerned about the next generation of maths student and how viral content like this can discourage them unnecessarily.
It sure is. I’ve seen a PhD who didn’t read the only textbook he had referenced in his thesis, which proved his idea that teachers were doing it wrong and he wasn’t, was wrong. 😂 Should’ve listened to the people who teach it (or actually read the textbook he referenced 🙄 ).
which makes students of math feel inordinately inferior
They don’t. All students get this correct. It’s only adults who have forgotten the rules that get it wrong.
these gotcha-style math memes IMO deepen students’ belief that they’re just bad at math
Nope. Students never get these wrong.
proofs mean something very specific to me and I can’t yet imagine what that looks like WRT order of operations
All you have to do is see which way gives wrong answers for 2+3x4 and you’ve proven which ways don’t work 😂
note that conventions aren’t necessarily “optional”
Yes they are.
when being understood is essential
You don’t understand how to do 2+3x4-5 without knowing which conventions people use for the order of the plus and minus?
Here I am only concerned about the next generation of maths student and how viral content like this can discourage them unnecessarily
I feel like people should at least remember math at a 4th grade level and be able to get 10. What is the point of making it obvious the universe will never ever arrange itself in such a fashion. The point is if you remember simple rules you applied for a 10-15 years.
You’ve completely not understood that order of operations is an arbitrary convention. How did you decide to expand the definition of multiplication before evaluating the addition? Convention.
You can’t write 2 + 2 ÷ 2 like this, so how are you gonna decide whether to decide to divide or add first?
How are you gonna write 2 + 2 ÷ 2 with repeated addition?
The definition of Multiplication as being repeated addition
That doesn’t mean it has to be expanded first. You could expand 2 + 2 × 3 as (2+2)+(2+2)+(2+2) and you are unable to tell me what mathematical law prohibits it.
If this were a universal law, reverse polish notation wouldn’t work as it does. In RPL, 2 2 + 3 × is 12 but 2 3 × 2 + is 8. If you had to expand multiplication first, how would it work? The same can be done with prefix notation, and the same can be done with “pre-school” order of operations.
Different programming languages have different orders of operations, and those languages work just fine.
Your argument amounts to saying that it makes the most sense to do multiplication before addition. Which is true, but that only gives you a convention, not a rule.
How are you gonna write 2 + 2 ÷ 2 with repeated addition?
You don’t, because the second 2 is associated with a Division that has to be done before the addition. Maybe go back to school and learn how to do Maths 🙄
That doesn’t mean it has to be expanded first.
Yes it does. Everything has to be expanded before you do the addition and subtraction, or you get wrong answers 🙄
2+3x4=2+3+3+3+3=2+12=14 correct
2+3x4=5x4=5+5+5+5=20 wrong
You could expand 2 + 2 × 3 as (2+2)+(2+2)+(2+2)
Says someone who can’t tell the difference between (2+2)x3=12 and 2+2x3=8 🙄
you are unable to tell me what mathematical law prohibits it
The order of operations rules 😂
reverse polish notation wouldn’t work as it does
It works because it treats every operation as bracketed without writing the brackets. Also that’s only a Maths notation, not the Maths itself.
In RPL, 2 2 + 3 × is 12
Because the way it calculates that is (2+2)x3, not complicated. Same order of operations rules as other Maths notations - just a different way of writing the same thing
If you had to expand multiplication first, how would it work?
It works because Brackets - 2 2 + = (2+2) - are before Multiplication
The same can be done with prefix notation
Another Maths notation, same rules of Maths
Different programming languages have different orders of operations
You don’t, because the second 2 is associated with a Division that has to be done before the addition. Maybe go back to school and learn how to do Maths 🙄
Right, so you cannot derive precedence order from the definition of the operations. Your argument based on the definition of multiplication as repeated addition is wrong.
or you get wrong answers
This is begging the question. We are discussing whether the answers are flat wrong or whether there is a layer of interpretation. Repeating that they are wrong does nothing for this discussion, so there’s no need to bother.
You have nothing to say that I can see about why the different interpretations are impossible, or contradictory, or why they ought to qualify as “wrong” even though maths works regardless; you’ve just heard a school-level maths teacher tell you it’s done one way and believe that’s the highest possible authority. I’m sorry, but lots of things we get taught in high school are wrong, or only partially right. I see from your profile that you are a maths teacher, so it’s actually your job to understand maths at a higher level than the level at which you teach it. It may be easier to to teach high school maths this way, but it’s not a good enough level of understanding for an educator (or for a mathematician).
Left to right is a convention (as is not writing the brackets in RPN). Brackets before Multiplication before Addition are rules.
OK, let’s try a different tack. When I hear the word “rules”, I think you’re talking either about a rule of inference in first order logic or an axiom in a first-order system. But there is no such rule or axiom in, for example, first order Peano arithmetic. So what are you talking about? Can you find somewhere an enumeration of all the rules you’re talking about? Because maybe we’re just talking at cross-purposes: if you deviate from the axioms of Peano arithmetic then we’re fundamentally not doing arithmetic any more. But I contend that you will not find included in any axiomatisation anything which specifies order of operations. This is because from the point of view of the “rules” (i.e. the axioms) the addition and multiplication operations are just function symbols with certain properties. Even the symbols themselves are not really part of the axiomatisation; you could just as well get rid of the + symbol and write A(x, y, z) instead of “x + y = z”; you’d have the exact same arithmetic, the exact same rules.
If you’re able to answer this, we can get away from these vague terms which you keep introducing like “notation definition”, and we can instead think about what it means to be a convention versus whatever it is you mean by “rule”. (For example, Peano arithmetic has a privileged position amongst candidates for arithmetic because it encompasses our intuition about how numbers work: you can’t just take an alternative arithmetic, like say arithmetic modulo 17, and say that’s an “alternative convention” because when you add an apple to a bowl of 16 apples, they don’t all disappear. But there’s no such intuition about how to write mathematics to express a certain thing. I contend that is all convention.)
It works because it treats every operation as bracketed without writing the brackets. Also that’s only a Maths notation, not the Maths itself.
So, you understand that a notation can evaluate things in a certain order with what you call “treating every operation as bracketed without writing brackets.” What does it mean to be “bracketed without writing brackets”? There are exactly two aspects to brackets:
the symbols themselves - but we’re not writing them! So this isn’t relevant.
the effect they have - the effect on the order of evaluation of operations
So what you’re admitting with these phantom brackets is that a notation can evaluate operations in a different order, even though there are no written brackets.
So I can specify these fake brackets to always wrap the left-most operation first: (2 + 3) x 5 and hey look, this notation now has left-to-right order of evaluation, not the usual multiplication first. If you prefer to think of there being invisible brackets there, go right ahead, but the effect is the same.
So, how do we decide whether our usual notation “has bogus brackets” or not? Convention. We could choose one way or the other. Nothing breaks if we choose one or the other. Symmetrically, we could say that left-to-right evaluation is the notation “without bogus brackets” and that BODMAS evaluation is the notation “with bogus brackets”. Which choice we make is entirely arbitrary. That is, unless you can find a compelling reason why one is right and the other wrong, rather than just saying it once again.
They don’t actually. Welcome to most e-calcs give wrong answers because the programmers failed to deal with it correctly.
What problems does it cause? Are the problems purely that they don’t have the order of operations you expect, and so get different answers if you don’t clarify with brackets? Because that, again, is begging the question.
To re-iterate, you are in a discussion where you’re trying to establish that it’s a fundamental law of maths that you must do multiplication before addition. The fact that you’ve written a post in which you document how some calculators don’t follow this convention and said that they’re wrong is not evidence of that. It’s just your opinion. Indeed, it’s really (weak) evidence that your opinion is wrong, because you’re less of an authority than the manufacturers of calculators.
On calculators, there’s something important you need to realise: basic, non-scientific, non-graphing calculators all have left-to-right order of operations. You can test this with e.g. windows calculator in “standard” mode by typing 2, +, 3, x, 5 (it will give you 25, not 17). Switch it to “scientific” mode and it will give you 17.
Why is it different? Because “standard” mode is emulating a basic calculator which has a single accumulator and performs operations on that accumulated value. When you type “x 2” you are multiplying the accumulator by 2; the calculator has already forgotten everything that you typed to get the accumulator. This was done in the early days of calculators because it was more practical when memory looked like this:
Now, you can go on about your bogus brackets until you’re blue in the face, but the fact is that this isn’t “wrong”. It has a different convention for a sensible reason and if you expect something different then it is you who are using the device wrong.
From your other comment, since having two threads seems pointless:
So if you have one “notation definition” as you call it which says that 2+2*3 means ”first add two to two, then multiply by three” and another which says “first multiply two by three, then add it to two”, why on earth do the “rules” have anything further to say about order of operations?
No we don’t. We have another notation which says to do paired operations (equivalent to being in brackets) first.
What do you mean “we don’t”? I just made the definition. It exists. This is why terms like “notation definition” are not actually helpful IMO, so let’s be precise and use terms that are either plain english (like “convention”) or mathematical (like “axiom”, “definition”, etc).
Right, so you cannot derive precedence order from the definition of the operations.
Yes you can. I’m not sure what you’re not understanding about Division before Addition 😂
Your argument based on the definition of multiplication as repeated addition is wrong
No it isn’t! 😂
We are discussing whether the answers are flat wrong or whether there is a layer of interpretation
Flat wrong, as per the rules Of Maths 🙄
Repeating that they are wrong does nothing for this discussion, so there’s no need to bother
So stop doing wrong things and I can stop saying you’re doing it wrong 😂
why they ought to qualify as “wrong” even though maths works regardless
If you have 1 2 litre bottle of milk, and 4 3 litre bottles of milk, even a 3rd grader can count up and tell you how many litres there are, and that any other answer is wrong. 🙄 2+3x4=2+3+3+3+3=14 correct 2+3x4=5x4=5+5+5+5=20 wrong See how the Maths doesn’t work regardless? 😂
you’ve just heard a school-level maths teacher tell you it’s done one way and believe that’s the highest possible authority
Nope, I’ve proven it myself - that’s the beauty of Maths, that anyone at all can try it for themselves and find out. I’m guessing that you didn’t try it yourself 😂
lots of things we get taught in high school are wrong
says person failing to give a single such example 🙄
it’s actually your job to understand maths at a higher level than the level at which you teach it
No it isn’t. I’m required to to get the Masters degree which is required to be a teacher here, and that’s the end of it.
It may be easier to to teach high school maths this way
The correct way, yes 😂
When I hear the word “rules”, I think you’re talking either about a rule of inference in first order logic or an axiom in a first-order system
Nope, neither.
So what are you talking about?
What don’t you understand about 20 being a wrong answer for 2+3x4??
whatever it is you mean by “rule”
Thing which results in wrong answers if disobeyed - like 2+3x4=20 - not complicated. This is what we teach to students - if you always obey all the rules then you will always get the correct answer.
arithmetic modulo 17, and say that’s an “alternative convention”
Of course not, just a different function of Maths, that doesn’t involve Arithmetic at all (other than the steps along the way in doing the long division), unlike 2+3x4 🙄
I contend that is all convention
Nope! Just a different rule to Arithmetic 🙄
What does it mean to be “bracketed without writing brackets”?
Same thing as we’re adding the 2 in 2+3 without writing a plus (or a zero) in front of the 2 - all Arithmetic starts from zero on the number-line. Maths textbooks explicitly teach this, that we can leave the sign omitted at the start if it’s a plus.
the symbols themselves - but we’re not writing them! So this isn’t relevant
Just like we aren’t writing the plus sign in 2+3 🙄
So what you’re admitting with these phantom brackets is that a notation can evaluate operations in a different order, even though there are no written brackets.
Nope. Same order as though we did write it in a notation using Brackets, same as we always start with adding the 2 even though we didn’t write a plus sign in 2+3.
So I can specify these fake brackets to always wrap the left-most operation first: (2 + 3) x 5
No you can’t, because you get a wrong answer 🙄
this notation now has left-to-right order of evaluation
No it doesn’t, Multiplication before Addition 🙄
If you prefer to think of there being invisible brackets there
You know we were writing this without brackets for several centuries before we started using brackets in Maths, right?? 😂
So, how do we decide whether our usual notation “has bogus brackets” or not? Convention
Nope. proven rules 🙄
We could choose one way or the other.
No we can’t. Even a 3rd grader who is counting up can tell you that 🙄
Nothing breaks if we choose one or the other.
Yes it does. Again ask the 3rd grader how many litres we have, and then try doing Addition first to get that answer 😂
we could say that left-to-right evaluation is the notation “without bogus brackets”
No we can’t. Ask the 3rd grader, or even try it yourself with Cuisenaire rods
Which choice we make is entirely arbitrary
Nope. proven rules 🙄
That is, unless you can find a compelling reason why one is right and the other wrong, rather than just saying it once again
Count up how many litres we have 🙄
What problems does it cause?
wrong answers 😂
you’re trying to establish that it’s a fundamental law of maths that you must do multiplication before addition
As per Maths textbooks 😂
you’ve written a post in which you document how some calculators don’t follow this
rule
said that they’re wrong is not evidence of that
says person ignoring the Maths textbooks I quoted and the actual calculators giving the correct answer 🙄
It’s just your opinion
Nope! proven rules as found in Maths textbooks 🙄
it’s really (weak) evidence that your opinion is wrong,
says person ignoring the Maths textbooks I quoted and the actual calculators giving the correct answer 🙄
you’re less of an authority than the manufacturers of calculators
Demonstrably not 😂
basic, non-scientific, non-graphing calculators all have left-to-right order of operations
No they don’t! 😂
e.g. windows calculator in “standard” mode
The Windows calculator is an e-calc which was written by a programmer who didn’t check that their Maths was correct. 🙄 Now try it with any actual calculator 🙄
Why is it different?
Written by a different programmer, but one who didn’t know The Distributive Law, so even in Scientific mode it gives wrong answers to 8/2(1+3) 🙄
Because “standard” mode is emulating a basic calculator
No it isn’t. All basic calculators obey Multiplication before Addition, 🙄 and if the programmer had tried it then they would’ve found that out
performs operations on that accumulated value
Instead of using the stack, to store the Multiplication first, like all actual calculators do 🙄
When you type “x 2” you are multiplying the accumulator by 2
No, the dumb programmer is. All actual calculators did the Multiplication first and put the result on the stack
the calculator has already forgotten everything that you typed to get the accumulator
But actual calculators have put that result on the stack 🙄
This was done in the early days of calculators
No it wasn’t. All calculators “in the early days” used the stack
It has a different convention for a sensible reason
Nope, it’s just disobeying the rules of Maths because dumb programmer didn’t check their Maths was correct 🙄
it was more practical when memory looked like this:
And even then the stack existed 🙄
the fact is that this isn’t “wrong”
Yes it is! 😂 Again, ask the 3rd grader to count up and tell you the correct answer
if you expect something different then it is you who
knows the rules of Maths 🙄
What do you mean “we don’t”?
What don’t you understand about “we don’t”?
I just made the definition
Of the notation, not the rules 🙄
We have another notation which says to do paired operations (equivalent to being in brackets) first
And this notation says to do paired operations first, same as if they were in Brackets. You so nearly had it 🙄
plain english (like “convention”)
says person who keeps calling the rules “convention” 🙄
mathematical (like “axiom”, “definition”, etc)
You know we have Mathematical definitions of the difference between conventions and rules, right??
PEMDAS isn’t obvious to “common people”? Why not? It doesn’t seem like an arbitrary convention to me…
If “×” means “groups of,” then “2+2×4” means “two plus two groups of four” which only makes sense, to me, to be read as “two plus two groups of four” rather than “two plus two groups of four”
Sure the order of operations could be arbitrarily different, but I feel like we settled on that order because it simply makes more sense intuitively.
I’m aware of the possibility that it only feels natural and intuitive to me because I was taught that way, but I at least don’t think that applies to this specific example
It didn’t occur to me that the poll may function that way. Does it? I thought this was engagement bait in which the poll’s author lists only wrong answers as options
it would have to have different order or parentheses or both.
Neither. Multiplication is always before Addition, hence “obviously”
Of course everything in math is convention
Nope. The vast majority of it is proven rules
2+(2*4) is more obvious to common people
Weird then how many people were able to get this right without brackets for centuries before we started using brackets in Maths (which we’ve only had for 300 years)
Tell that obvious to over half the population who get this wrong
Way less than half actually. No teachers or students ever get this wrong, only adults who have forgotten the rules, and poll after poll puts this down around 40-45% of adults.
I mean, obviously ten.
But I at least understand 16.
I deeply worry about the percentage just next to the other three numbers.
13 is probably the next most chosen because it’s closest to 10.
Not including the correct answer is also a form of engagement bait to get additional comments and such saying “wait the real answer is 10, wtf?”
Why worry? You can see them on the right side of the image
It not even remotely possible to make an odd number out of that.
The numbers on the right-hand side are what I’m actually working about.
I was trying yo make a shitty joke conflating you worrying (having concern) with you worrrying (wondering what).
If it helps, I saw what you did there, and I exhaled slightly harder out of my nose while smiling wryly. It’s even better the op didn’t get it. So like, well done and stuff 😊
sorry about that, completely wooshed me
Saul Goodman
Why worry about obviously fake bullshit?
Your obviously is only a convention and not everyone agree with that. Not even all peogramming languages or calculators.
If you wanted obviously, it would have to have different order or parentheses or both. Of course everything in math is convention but I mean more obvious.
2+2*4 is obvious with PEDMAS, but hardy obvious to common people
2+(2*4) is more obvious to common people
2*4+2 is even more obvious to people not good with math. I would say this is the preferred form.
(2*4)+2 doesn’t really add more to it, it just emphasises it more, but unnecessarily.
Honestly that’s my pet peeve about this category of content. Over the years I’ve seen (at least) hundreds of these check-out-how-bad-at-math-everyone-is posts and it’s nearly always order of operations related. Apparently, a bunch of people forgot (or just never learned) PEMDAS.
Now, having an agreed-upon convention absolutely matters for arriving at expected computational outcomes, but we call it a convention for a reason: it’s not a “correct” vs “incorrect” principle of mathematics. It’s just a rule we agreed upon to allow consistent results.
So any good math educator will be clear on this. If you know the PEMDAS convention already, that’s good, since it’s by far the most common today. But if you don’t yet, don’t worry. It doesn’t mean you’re too dumb to math. With a bit of practice, you won’t even have to remember the acronym.
Most actual math people never have to think about pemdas here because no one would ever write a problem like this. The trick here is “when was the last time I saw an X to mean multiplication” so I would already be off about it
1 + 1/2 in my brain is clearly 1.5, but 1+1÷2 doesn’t even register in my brain properly.
Right, and that clue IMO unravels the more troubling aspect of why this content spreads so quickly:
It’s deliberately aimed at people with a rudimentary math education who can be made to feel far superior to others who, in spite of having roughly the same level of proficiency, are missing/forgetting a single fact that has a disproportionate effect on the result they expect.
That is, it’s blue-dress-level contentious engagement bait for anyone with low math skills, whether or not they remember PEMDAS.
Blue-dress-level?
Old internet thing. Hotly debated at the time.
https://en.wikipedia.org/wiki/The_dress
I’ll add the contextual link above for others, since it’s been awhile.
And yet Maths textbooks do! 😂
In a Maths textbook
You don’t know that the obelus means divide??
“No one” in this context meant “no one who actually does maths professionally.”
Right, and I have decades of maths experience outside of textbooks. So it’s probably been 20 years since I had a meaningful interaction with the × multiplication symbol.
I clearly know what the symbol means, I demonstrated a use of it. But again, haven’t had a meaningful interaction with the symbol in 20 years, and yet I deal with
/for division daily.When I see
1+½i can instantly say “one and a half”, but when I see1 + 1 ÷ 2i actually have to pause for a moment to think about order of operations. Same with1+2xvs1 + 2 × x… one I recognize the structure of the problem immediately, and one feels foreign.The point is that people who do maths for a living, and are probably above average in maths, tend to write things differently than people who are stopped their maths education in high school (or lower), and these types of memes are designed around making people who know high school maths feel smart. People who actually know maths don’t need memes to justify being better at maths than the rest of the public.
No it doesn’t. Everyone who does Maths professionally does it the same way as in Maths textbooks 🙄
And that would be wrong. It’s 1 plus one half. 1½ is one and a half.
You don’t know to Divide before Adding??
Says person with “decades of maths experience outside of textbooks” 🙄
That would be me
Nope. We all write it the same way as we were taught, even those who have done Maths at University (also me).
No, they’re designed around getting those who have forgotten the rules to argue about it. i.e. engagement bait
I learned BEDMAS. Doesn’t really change your comment other than effectively “spelling” of a single term
Proven rules actually
No we don’t - the order of operations rules
The rules most definitely are
proven rules which are true whether you agree to it or not! 😂
Yep
No it isn’t.
As long as you know the rules then that’s all that matters
Dear Mr Rules,
I’m not sure what motivates you to so generously offer your various dyadic tokens of knowledge on this subject without qualification while ignoring my larger point, but will assume in good faith that your thirst for knowledge rivals that of your devotion to The Rules.
First, a question: what are conventions if not agreed upon rules? Second, here is a history of how we actually came to agree upon the aforementioned rules which you may find interesting:
https://www.themathdoctors.org/order-of-operations-historical-caveats/
Happy ruling to you.
I’m a Maths teacher with a Masters - thanks for asking - how about you?
You mean your invalid point, that I debunked?
Conventions are optional, rules aren’t.
He’s well-known to be wrong about his “history”, and if you read through the comments you’ll find plenty of people telling him that, including references. Cajori wrote the definitive books about the history of Maths (notation). They’re available for free on the Internet Archive - no need to believe some random crank and his blog.
Dear colleague,
By qualification I meant explanation. My doctorate is irrelevant to the truth.
Since you asked, my larger point was about the unhelpful nature of this content, which makes students of math feel inordinately inferior or superior hinged entirely on a single point of familiarity. I don’t handle early math education, but many of my students arrive with baggage from it that hinders their progress, leading me to suspect that early math education sometimes discourages students unnecessarily. In particular, these gotcha-style math memes IMO deepen students’ belief that they’re just bad at math. Hence my dislike of them.
Re: Dave Peterson, I’ll need to read more about this debate regarding the history of notation and I’ll search for the “proven rules” you mentioned (proofs mean something very specific to me and I can’t yet imagine what that looks like WRT order of operations).
If what riled you up was my use of the word “conventions” I can use another, but note that conventions aren’t necessarily “optional” when being understood is essential. Where one places a comma in writing can radically change the meaning of a sentence, for example. My greater point however has nothing to do with that. Here I am only concerned about the next generation of maths student and how viral content like this can discourage them unnecessarily.
It sure is. I’ve seen a PhD who didn’t read the only textbook he had referenced in his thesis, which proved his idea that teachers were doing it wrong and he wasn’t, was wrong. 😂 Should’ve listened to the people who teach it (or actually read the textbook he referenced 🙄 ).
They don’t. All students get this correct. It’s only adults who have forgotten the rules that get it wrong.
Nope. Students never get these wrong.
All you have to do is see which way gives wrong answers for 2+3x4 and you’ve proven which ways don’t work 😂
Yes they are.
You don’t understand how to do 2+3x4-5 without knowing which conventions people use for the order of the plus and minus?
It doesn’t. None of them get it wrong. 🙄
I feel like people should at least remember math at a 4th grade level and be able to get 10. What is the point of making it obvious the universe will never ever arrange itself in such a fashion. The point is if you remember simple rules you applied for a 10-15 years.
common people who are not good at math…
PEMDAS is in the 5th-grade curriculum.
My obviously is gated to people who can hadle 5th-grade math.
I would say we should not provide the mathematically illiterate any say in the matter. They need to spend 10 minutes on Youtube and learn it.
Try RPN for a whole different beast
I am familiar with RPN. At least RPN is always unambiguous
There’s just 5 lots of 2. If it’s hard then think of x being just a bunch of + smooshed together. So
2 + 2 x 4
expands to
2 + 2 + 2 + 2 + 2
or contracts to
5 x 2
You’ve completely not understood that order of operations is an arbitrary convention. How did you decide to expand the definition of multiplication before evaluating the addition? Convention.
You can’t write 2 + 2 ÷ 2 like this, so how are you gonna decide whether to decide to divide or add first?
No, you’ve completely not understood that they are universal rules of Maths
The definition of Multiplication as being repeated addition
Yes you can
The rules of Maths, which says Division must be before Addition
How are you gonna write 2 + 2 ÷ 2 with repeated addition?
That doesn’t mean it has to be expanded first. You could expand 2 + 2 × 3 as (2+2)+(2+2)+(2+2) and you are unable to tell me what mathematical law prohibits it.
If this were a universal law, reverse polish notation wouldn’t work as it does. In RPL, 2 2 + 3 × is 12 but 2 3 × 2 + is 8. If you had to expand multiplication first, how would it work? The same can be done with prefix notation, and the same can be done with “pre-school” order of operations.
Different programming languages have different orders of operations, and those languages work just fine.
Your argument amounts to saying that it makes the most sense to do multiplication before addition. Which is true, but that only gives you a convention, not a rule.
You don’t, because the second 2 is associated with a Division that has to be done before the addition. Maybe go back to school and learn how to do Maths 🙄
Yes it does. Everything has to be expanded before you do the addition and subtraction, or you get wrong answers 🙄
2+3x4=2+3+3+3+3=2+12=14 correct
2+3x4=5x4=5+5+5+5=20 wrong
Says someone who can’t tell the difference between (2+2)x3=12 and 2+2x3=8 🙄
The order of operations rules 😂
It works because it treats every operation as bracketed without writing the brackets. Also that’s only a Maths notation, not the Maths itself.
Because the way it calculates that is (2+2)x3, not complicated. Same order of operations rules as other Maths notations - just a different way of writing the same thing
It works because Brackets - 2 2 + = (2+2) - are before Multiplication
Another Maths notation, same rules of Maths
Maths doesn’t
They don’t actually. Welcome to most e-calcs give wrong answers because the programmers failed to deal with it correctly.
No, my argument is it’s a universal rule of Maths, as found in Maths textbooks 🙄
Left to right is a convention (as is not writing the brackets in RPN). Brackets before Multiplication before Addition are rules.
Right, so you cannot derive precedence order from the definition of the operations. Your argument based on the definition of multiplication as repeated addition is wrong.
This is begging the question. We are discussing whether the answers are flat wrong or whether there is a layer of interpretation. Repeating that they are wrong does nothing for this discussion, so there’s no need to bother.
You have nothing to say that I can see about why the different interpretations are impossible, or contradictory, or why they ought to qualify as “wrong” even though maths works regardless; you’ve just heard a school-level maths teacher tell you it’s done one way and believe that’s the highest possible authority. I’m sorry, but lots of things we get taught in high school are wrong, or only partially right. I see from your profile that you are a maths teacher, so it’s actually your job to understand maths at a higher level than the level at which you teach it. It may be easier to to teach high school maths this way, but it’s not a good enough level of understanding for an educator (or for a mathematician).
OK, let’s try a different tack. When I hear the word “rules”, I think you’re talking either about a rule of inference in first order logic or an axiom in a first-order system. But there is no such rule or axiom in, for example, first order Peano arithmetic. So what are you talking about? Can you find somewhere an enumeration of all the rules you’re talking about? Because maybe we’re just talking at cross-purposes: if you deviate from the axioms of Peano arithmetic then we’re fundamentally not doing arithmetic any more. But I contend that you will not find included in any axiomatisation anything which specifies order of operations. This is because from the point of view of the “rules” (i.e. the axioms) the addition and multiplication operations are just function symbols with certain properties. Even the symbols themselves are not really part of the axiomatisation; you could just as well get rid of the + symbol and write A(x, y, z) instead of “x + y = z”; you’d have the exact same arithmetic, the exact same rules.
If you’re able to answer this, we can get away from these vague terms which you keep introducing like “notation definition”, and we can instead think about what it means to be a convention versus whatever it is you mean by “rule”. (For example, Peano arithmetic has a privileged position amongst candidates for arithmetic because it encompasses our intuition about how numbers work: you can’t just take an alternative arithmetic, like say arithmetic modulo 17, and say that’s an “alternative convention” because when you add an apple to a bowl of 16 apples, they don’t all disappear. But there’s no such intuition about how to write mathematics to express a certain thing. I contend that is all convention.)
So, you understand that a notation can evaluate things in a certain order with what you call “treating every operation as bracketed without writing brackets.” What does it mean to be “bracketed without writing brackets”? There are exactly two aspects to brackets:
So what you’re admitting with these phantom brackets is that a notation can evaluate operations in a different order, even though there are no written brackets.
So I can specify these fake brackets to always wrap the left-most operation first:
(x 5 and hey look, this notation now has left-to-right order of evaluation, not the usual multiplication first. If you prefer to think of there being invisible brackets there, go right ahead, but the effect is the same.2 + 3)So, how do we decide whether our usual notation “has bogus brackets” or not? Convention. We could choose one way or the other. Nothing breaks if we choose one or the other. Symmetrically, we could say that left-to-right evaluation is the notation “without bogus brackets” and that BODMAS evaluation is the notation “with bogus brackets”. Which choice we make is entirely arbitrary. That is, unless you can find a compelling reason why one is right and the other wrong, rather than just saying it once again.
What problems does it cause? Are the problems purely that they don’t have the order of operations you expect, and so get different answers if you don’t clarify with brackets? Because that, again, is begging the question.
To re-iterate, you are in a discussion where you’re trying to establish that it’s a fundamental law of maths that you must do multiplication before addition. The fact that you’ve written a post in which you document how some calculators don’t follow this convention and said that they’re wrong is not evidence of that. It’s just your opinion. Indeed, it’s really (weak) evidence that your opinion is wrong, because you’re less of an authority than the manufacturers of calculators.
On calculators, there’s something important you need to realise: basic, non-scientific, non-graphing calculators all have left-to-right order of operations. You can test this with e.g. windows calculator in “standard” mode by typing 2, +, 3, x, 5 (it will give you 25, not 17). Switch it to “scientific” mode and it will give you 17.
Why is it different? Because “standard” mode is emulating a basic calculator which has a single accumulator and performs operations on that accumulated value. When you type “x 2” you are multiplying the accumulator by 2; the calculator has already forgotten everything that you typed to get the accumulator. This was done in the early days of calculators because it was more practical when memory looked like this:
Now, you can go on about your bogus brackets until you’re blue in the face, but the fact is that this isn’t “wrong”. It has a different convention for a sensible reason and if you expect something different then it is you who are using the device wrong.
From your other comment, since having two threads seems pointless:
What do you mean “we don’t”? I just made the definition. It exists. This is why terms like “notation definition” are not actually helpful IMO, so let’s be precise and use terms that are either plain english (like “convention”) or mathematical (like “axiom”, “definition”, etc).
Yes you can. I’m not sure what you’re not understanding about Division before Addition 😂
No it isn’t! 😂
Flat wrong, as per the rules Of Maths 🙄
So stop doing wrong things and I can stop saying you’re doing it wrong 😂
If you have 1 2 litre bottle of milk, and 4 3 litre bottles of milk, even a 3rd grader can count up and tell you how many litres there are, and that any other answer is wrong. 🙄 2+3x4=2+3+3+3+3=14 correct 2+3x4=5x4=5+5+5+5=20 wrong See how the Maths doesn’t work regardless? 😂
Nope, I’ve proven it myself - that’s the beauty of Maths, that anyone at all can try it for themselves and find out. I’m guessing that you didn’t try it yourself 😂
says person failing to give a single such example 🙄
No it isn’t. I’m required to to get the Masters degree which is required to be a teacher here, and that’s the end of it.
The correct way, yes 😂
Nope, neither.
What don’t you understand about 20 being a wrong answer for 2+3x4??
Thing which results in wrong answers if disobeyed - like 2+3x4=20 - not complicated. This is what we teach to students - if you always obey all the rules then you will always get the correct answer.
Of course not, just a different function of Maths, that doesn’t involve Arithmetic at all (other than the steps along the way in doing the long division), unlike 2+3x4 🙄
Nope! Just a different rule to Arithmetic 🙄
Same thing as we’re adding the 2 in 2+3 without writing a plus (or a zero) in front of the 2 - all Arithmetic starts from zero on the number-line. Maths textbooks explicitly teach this, that we can leave the sign omitted at the start if it’s a plus.
Just like we aren’t writing the plus sign in 2+3 🙄
Nope. Same order as though we did write it in a notation using Brackets, same as we always start with adding the 2 even though we didn’t write a plus sign in 2+3.
No you can’t, because you get a wrong answer 🙄
No it doesn’t, Multiplication before Addition 🙄
You know we were writing this without brackets for several centuries before we started using brackets in Maths, right?? 😂
Nope. proven rules 🙄
No we can’t. Even a 3rd grader who is counting up can tell you that 🙄
Yes it does. Again ask the 3rd grader how many litres we have, and then try doing Addition first to get that answer 😂
No we can’t. Ask the 3rd grader, or even try it yourself with Cuisenaire rods
Nope. proven rules 🙄
Count up how many litres we have 🙄
wrong answers 😂
As per Maths textbooks 😂
rule
says person ignoring the Maths textbooks I quoted and the actual calculators giving the correct answer 🙄
Nope! proven rules as found in Maths textbooks 🙄
says person ignoring the Maths textbooks I quoted and the actual calculators giving the correct answer 🙄
Demonstrably not 😂
No they don’t! 😂
The Windows calculator is an e-calc which was written by a programmer who didn’t check that their Maths was correct. 🙄 Now try it with any actual calculator 🙄
Written by a different programmer, but one who didn’t know The Distributive Law, so even in Scientific mode it gives wrong answers to 8/2(1+3) 🙄
No it isn’t. All basic calculators obey Multiplication before Addition, 🙄 and if the programmer had tried it then they would’ve found that out
Instead of using the stack, to store the Multiplication first, like all actual calculators do 🙄
No, the dumb programmer is. All actual calculators did the Multiplication first and put the result on the stack
But actual calculators have put that result on the stack 🙄
No it wasn’t. All calculators “in the early days” used the stack
Nope, it’s just disobeying the rules of Maths because dumb programmer didn’t check their Maths was correct 🙄
And even then the stack existed 🙄
Yes it is! 😂 Again, ask the 3rd grader to count up and tell you the correct answer
knows the rules of Maths 🙄
What don’t you understand about “we don’t”?
Of the notation, not the rules 🙄
And this notation says to do paired operations first, same as if they were in Brackets. You so nearly had it 🙄
says person who keeps calling the rules “convention” 🙄
You know we have Mathematical definitions of the difference between conventions and rules, right??
PEMDAS isn’t obvious to “common people”? Why not? It doesn’t seem like an arbitrary convention to me…
If “×” means “groups of,” then “2+2×4” means “two plus two groups of four” which only makes sense, to me, to be read as “two plus two groups of four” rather than “two plus two groups of four”
Sure the order of operations could be arbitrarily different, but I feel like we settled on that order because it simply makes more sense intuitively.
I’m aware of the possibility that it only feels natural and intuitive to me because I was taught that way, but I at least don’t think that applies to this specific example
Everyone is taught the rules of Maths
It means repeated addition actually
No, it means 2+2+2+2+2
No they can’t
It’s because Multiplication is defined as repeated addition, so if you don’t do it before addition you get wrong answers
Clearly not if most of these answers are incorrect. If it was obvious, there wouldn’t be as many answers as there are.
It didn’t occur to me that the poll may function that way. Does it? I thought this was engagement bait in which the poll’s author lists only wrong answers as options
Nope. Rules of Maths
Neither. Multiplication is always before Addition, hence “obviously”
Nope. The vast majority of it is proven rules
Weird then how many people were able to get this right without brackets for centuries before we started using brackets in Maths (which we’ve only had for 300 years)
Tell that obvious to over half the population who get this wrong
Way less than half actually. No teachers or students ever get this wrong, only adults who have forgotten the rules, and poll after poll puts this down around 40-45% of adults.