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this post was submitted on 31 May 2025
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I'm sure we're all geniuses here, but just in case...
Please excuse my dear aunt Sally.
Parenthesis, exponents, multiplication, division, addition, subtraction.
Why? Because a bunch of dead Greeks say so!
3x3-3÷3+3
(3x3)-(3÷3)+3
9-1+3
8+3
11
I guess remembering grade school order of operation means you're a guinus now? Bar has gotten pretty low...
That's the point.
Set the bar low, but just high enough that tons of people still trip over it.
Sit back and enjoy the comment wars.
The people who are confident but wrong are too proud to admit they were wrong even if they realize it, and comment angrily.
The people who are right and know why, comment for corrections and some to show off how S-M-R-T they are.
The people who are wrong but willing to accept that just have their realization and probably don't think about it again. So do the people who don't know and/or care.
But those first two groups will keep the post going in both shares and comments, because "look at all these wrong people"
It's all designed to boost engagement.
And it will go even lower as people start relying mpre on AI...
The "why" goes a little further than that.
In actuality, it's because of fundamental properties of operations
a + b = b + a
a×b = b×a
(a + b) + c = a + (b + c)
(a×b)×c = a×(b×c)
a + 0 = a
a×1 = a
If you know that, then PEMDAS and such are useless because they're derived from those properties but do not fully encompass them.
Eg.
3×2×(2+2) = 3×(4+4) = 12+12 = 24
This is a correct solution that is improper if you're strictly adhering to PEMDAS rule as I've done multiplication before parenthesis from right to left.
I could even go completely out of order by doing 3×2×(2+2) = 2×(6+6) and it will still be correct
Not a genius. But if subtraction is last, why isn't it 9-4?
Because its not really "1 plus 3", its negative 1 plus 3 which is two. I know it seems a little weird but the minus sign is " tied" to the thing following it.
Addition/subtraction work out the same regardless of how you order the operations. If you do subtraction last you start with the original:
9-1+3
and you are adding 3 to the result of (9-1). Since you are trying to perform it before the (9-1) operation is carried out, you can add 3 to the 9:
12-1 = 11
or you can add three to the -1 and get:
9+2 = 11
You only end up with 9-4 if you were subtracting 3 rather than adding three. It all becomes more obvious if you read the original as:
9 + (-1) + 3
should actually be
Addition and subtraction are given the same priority, and are done in the same step, from left to right.
It's not a great system of notation, it could be made far clearer (and parenthesis allow you to make it as clear as you like), but it's essentially the universal standard now and it's what we're stuck with.
No, it should simply be "Parenthesis, exponents, multiplication, addition."
A division is defined as a multiplication, and a substraction is defined as an addition.
I am so confused everytime I see people arguing about this, as this is basic real number arithmetics that every kid in my country learns at 12 yo, when moving on from the simplified version you learn in elementary school.
No it isn't. Multiplication is defined as repeated addition. Division isn't repeated subtraction. They just happen to have opposite effects if you treat the quotient as being the result of dividing.
Yes, it is. The division of a by b in the set of real numbers and the set of rational numbers (which are, de facto, the default sets used in most professions) is defined as the multiplication of a by the multiplicative inverse of b. Alternative definitions are also based on a multiplication.
That's why divisions are called an auxilliary operation.
No it isn't.
No it isn't. The Quotient is defined as the number obtained when you divide the Dividend by the Divisor. Here it is straight out of Euler...
Emphasis on "alternative", not actual.
You want PEMA with knowledge of what is defined, when people can't even understand PEMDAS. You wish for too much.
I hate most math eduction because it's all about memorizing formulas and rules, and then memorizing exceptions. The user above's system is easier to learn, because there's no exceptions or weirdness. You just learn the rule that division is multiplication and subtraction is addition. They're just written in a different notation. It's simpler, not more difficult. It just requires being educated on it. Yes, it's harder if you weren't obviously, as is everything you weren't educated on.
That's because (strictly speaking) they aren't teaching math. They're teaching "tricks" to solve equations easier, which can lead to more confusion.
Like the PEMDAS thing that's being discussed here. There's no such thing as "order of operations" in math, but it's easier to teach by assuming that there is.
Edit: To the people downvoting: I want to hear your opinions. Do you think I'm wrong? If so, why?
Yes we are. Adults forgetting it is another matter altogether.
Yes there is! 😂
No, I know you're wrong.
If you don't solve binary operators before unary operators you get wrong answers. 2+3x4=14, not 20. 3x4=3+3+3+3 by definition
Yes and no. You teach how to solve equations, but not the fundamentals (and if you do then kudos to you, as it's not a trivial accomplishment). Fundamentals, most of the time, are taught in universities. It's so much easier that way, but doesn't mean it's right. People call it math, which is fair enough, but it's not really math in a sense that you don't understand the underlying principles.
Nope.
There's only commutation, association, distribution, and identity. It doesn't matter in which order you apply any of those properties, the result will stay correct.
2×2×(2-1)/2 = 2×(4-2)/2 = 1×(4-2) = 4-2 = 2
As you can see, I didn't follow any particular order and still got the correct result. Because no basic principle was broken.
Or I could also go
2×2×(2-1)/2 = 4×(2-1)/2 = 4×(1-0.5) = 4×0.5 = 2
Same result. Completely different order, yet still correct.
My response to the rest goes back to the aforementioned.
Nope. We teach the fundamentals. Adults not remembering them doesn't mean they weren't taught. Just pick up a Maths textbook. It's all in there. Always has been.
No they're not. They only teach order of operations from a remedial point of view. Most of them forget about The Distributive Law. I've seen multiple Professors be told by their students that they were wrong.
The Constructivist learners have no trouble at all understanding it.
Yep!
And many proofs of other rules, which you've decided to omit mentioning.
But the order you apply the operations does matter, hence the proven rules to be followed.
Notably you picked an example that has no addition, subtraction, or distribution in it. That's called cherry-picking.
Yep, because you cherry-picked a simple example where it doesn't matter. It's never going to matter when you only pick operations which have the same precedence.
...cherry-picking.
Sure. They are, however, not the focus. At least that's not how I've been taught in school. You're not teaching kids how to prove the quadratic formula, do you? No, you teach them how to use it instead. The goal here is different.
Again, with the order of operations. It's not a thing. I've given you two examples that don't follow any.
That's kinda random, but sure?
They all derive from each other. Even those fundamental properties are. For example, commutation is used to prove identity.
2+2-2 = 4-2 = 2+0 = 0
2 operators, no order followed.
If we take your example
2+3×4 then it's not an order of operation that plays the role here. You have no property that would allow for (2+3)×4 to be equal 2+3×4
Look, 2+3×4 = 1+3×(2+2)+1 = 1+(6+6)+1 = 7+7 = 14
Is that not correct?
It literally has subtraction and distribution. I thought you taught math, no?
2-2 is 2 being, hear me out, subtracted from 2
Same with 2×(2-2), I can distribute the value so it becomes 4-4
No addition? Who cares, subtraction literally works the same, but in opposite direction. Same properties apply. Would you feel better if I wrote (2-2) as (1+1-2)? I think not.
Also, can you explain how is that cherry-picking? You only need one equation that is solvable out of order to prove order of operation not existing. One is conclusive enough. If I give you two or more, it doesn't add anything meaningful.
If you had a bad teacher that doesn't mean everyone else had a bad teacher.
We teach them how to do proofs, including several specific ones.
We teach them how to use everything, and how to do proofs as well. Your whole argument is just one big strawman.
Happens to be the topic of the post.
Yes it is! 😂
So you could not do the brackets first and still get the right answer? Nope!
2×2×(2-2)/2=0
2×2×2-2/2=7
Not random at all, given you were talking about students understanding how Maths works.
Yes it is! 😂 If I have 1 2-litre bottle of milk, and 4 3-litre bottles of milk, there's only 1 correct answer for how many litres of milk of have, and it ain't 20! 😂 Even elementary school kids know how to work it out just by counting up.
No they don't. The proof of order of operations has got nothing to do with any of the properties you mentioned.
And neither is used to prove the order of operations.
Again with a cherry-picked example that only includes operators of the same precedence.
And yet we have a proof of why 14 is the only correct answer to 2+3x4, why you have to do the multiplication first.
Of course it is. So what?
No it didn't. It had Brackets (with subtraction inside) and Multiplication and Division.
Yep, and I just pointed out that what you just said is wrong. 2-2(1+2) has Subtraction and Distribution.
Which was done first because you had it inside Brackets, therefore not done in the Subtraction step in order of operations, but the Brackets step.
You already know - you know which operations to pick to make it look like there's no such thing as order of operations. If I tell you to look up at the sky at midnight and say "look - there's no such thing as the sun", that doesn't mean there's no such thing as the sun.
The Greeks certainly didn't come up with PEMDAS. US teachers too lazy to teach kids actual maths did. And that's before taking into account that the Greeks didn't come up with Algebra.
What’s lazy about learning PEMDAS? And what’s the non-lazy/superior way?
Nothing. Only people who don't know what they're talking about say that.
Learning the actual algebraic laws, instead of an order of operations to mechanically replicate. PEMDAS might get you through a standardised test but does not convey any understanding, it's like knowing that you need to press a button to call the elevator but not understand what elevators are for.
Though "lazy teachers" might actually be a bit too charitable a take given the literacy rates of US college graduates mastering in English. US maths teachers very well might not understand basic maths themselves, thinking it's all about a set of mechanical operations.
You might be smart, but you’re still wrong about the importance of order of operations; especially in algebra.
As far as teachers go, you’re being a dick by generalizing all (US) teachers are lazy and do not understand math.
Pro tip: opinions are like assholes; you too have one, and yes it too stinks.
Smart-arse more like. A serial troll who doesn't actually know what they're talking about.
Is it also lazy to learn Roy G. Biv to know the color spectrum instead of learning all the physics and optical properties behind that?
Or what about My Very Elderly Mother Just Served Us Nine Pickles to know the planets instead of learning orbital dynamics and astrophysics?
Christ man, it's a mnemonic device for elementary schoolers.
Those two things are memorisation tasks. Maths is not about memorisation.
You are not supposed to remember that the area of a triangle is
a * h / 2, you're supposed to understand why it's the case. You're supposed to be able to show that any triangle that can possibly exist is half the area of the rectangle it's stuck in: Start with the trivial case (right-angled triangle), then move on to more complicated cases. If you've understood that once, there is no reason to remember anything because you can derive the formula at a moment's notice.All maths can be understood and derived like that. The names of the colours, their ordering, the names of the planets and how they're ordered, they're arbitrary, they have no rhyme or reason, they need to be memorised if you want to recall them. Maths doesn't, instead it dies when you apply memorisation.
Ein Anfänger (der) Gitarre Hat Elan. There, that's the Guitar strings in German. Why do I know that? Because my music theory knowledge sucks. I can't apply it, music is all vibes to me but I still need a way to match the strings to what the tuner is displaying. You should never learn music theory from me, just as you shouldn't learn maths from a teacher who can't prove
a * h / 2, or thinks it's unimportant whether you can prove it.It is for ROTE learners.
Yes you are. A lot of students get the wrong answer when they forget the half.
Constructivist learners can do so, ROTE learners it doesn't matter. As long as they all know how to do Maths it doesn't matter if they understand it or not.
No they're not.
And if you haven't understood it then there is a reason to remember it.
Students aren't expected to be able to do that.
It can be by Constructivist learners, not ROTE learners.
No they're not. Colours are in spectrum order, the planets are in order from the sun.
A very substantial chunk of the population does just fine with having memorised Maths.
What fundamental property of the universe says that
6 + 4 / 2 is 8 instead of 5?
The fundamental property of Maths that you have to solve binary operators before unary operators or you end up with wrong answers.
But +, -, *, and / are all binary operators.
As far as I know, the only reason multiplication and division come first is that we've all agreed to it. But it can't be derived in a vacuum as that other dude contends it should be.
No, only multiply and divide are. 2+3 is really +2+3, but we don't write the first plus usually (on the other hand we do always write the minus if it starts with one).
No, they come first because you get wrong answers if you don't do them first. e.g. 2+3x4=14, not 20. All the rules of Maths exist to make sure you get correct answers. Multiplication is defined as repeated addition - 3x4=3+3+3+3 - hence wrong answers if you do the addition first (just changed the multiplicand, and hence the answer). Ditto for exponents, which are defined as repeated multiplication, a^2=(axa). Order of operations is the process of reducing everything down to adds and subtracts on a number line. 3^2=3x3=3+3+3
Typical examples of binary operations are the addition ( + {\displaystyle +}) and multiplication ( × {\displaystyle \times }) of numbers and matrices
Very confidently getting basic facts wrong doesn't inspire confidence in the rest of your comments.
Your example still doesn't give a reason why 2 + 3 * 4 is 2 + 3 + 3 + 3 +3 instead of 2 + 3 + 2 + 3 + 2 + 3 + 2 + 3 other than that we all agree to it.
...says person quoting Wikipedia and NOT a Maths textbook! 😂
Yes it does., need to work on your comprehension..
You can disagree as much as you want and 3x4 will still be defined as 3+3+3+3. It's been that way ever since Multiplication was invented.
The arithmetic operations, addition + , subtraction − , multiplication × , and division ÷
That better? Or you can find one you like all by yourself: https://duckduckgo.com/?q=binary+operator&ko=-1&ia=web
And you can shove the condescension up your ass until you understand the difference between unary and binary operators.
But to original point. I'm not disagreeing with anything and you're proving my point for me. There is no fundamental law of the universe that says multiplication comes first. It's defined by man and agreed to. If we encounter aliens someday, the area of their triangles are still going to be half the width times the height, the ratios of their circles circumference to diameter are still going to be pi, regardless of how they represent those values. But they could very well prioritize addition and subtraction over multiplication and division.
Is it a Maths textbook?
I already have dozens of Maths textbooks thanks.
It's not me who doesn't understand the difference.
Still need to work on your comprehension then. I did nothing of the sort.
Yes there is. The fact that it's defined as repeated addition. You don't do it first, you get wrong answers.
It's been defined and man has no choice but to agree with the consequences of the definition, or you get wrong answers.
No they couldn't. It gives wrong answers.
Actually, it is. Written by a PhD and used in a college course. It just happens to be distributed for free because Canada is cool like that.
May want to work on your own reading comprehension.
The facts disagree.
You can keep saying defined all you want, it doesn't change the underlying issue that it's defined by man. In the absence of all your books (which you clearly don't understand anyway based on our discussion of unary vs binary) order of operations only exists because we all agree to it.
Yeah there's an issue with them having forgotten the basic rules, since they don't actually teach them (except in a remedial way). Why do you think I keep trying to bring you back to actual Maths textbooks?
Nope. It's still not a textbook. Sounds more like a higher education version of Wikipedia.
With you, yes.
The notation is, the rules aren't.
Says person who doesn't understand the difference between unary and binary. Apparently EVERYTHING is binary according to you (and your website). 😂
It exists whether we agree with it or not. Don't obey it, get wrong answers.
It is though. Here's a link to buy a printed copy: https://libretexts.org/bookstore/order?math-7309
You keep mentioning textbooks but haven't actually shown any that support you. I have. I'll trust the PhD teaching a university course on the subject over the nobody on the internet who just keeps saying "trust me bro" and then being condescending while also being embarrassingly wrong.
And because I can't help it, I'll also trust Wolfram over you: Examples of binary operation on A from A×A to A include addition (+), subtraction (-), multiplication (×) and division (÷). Here, you can buy a copy of this too: https://www.amazon.com/exec/obidos/ASIN/1420072218/weisstein-20
Talking about yourself in the third person is weird. Even your nonsense about a silent "+" is really just leaving off the leading 0 in the equation 0+2. Because addition is a binary operator.
Only the ones that operate on two inputs. Some examples of unary operators are factorial, absolute value, and trig functions. The laughing face when you make a fool of yourself isn't really as effective as you think it is.
But we're getting off topic again. I can't keep trying to explain the same thing to you, so I would say this has been fun, but it's been more like talking to an unusually obnoxious brick wall. Next time you want to engage with someone try being less of a prick, or at least less wrong. You're not nearly as smart as you seem to think you are.
BWAHAHAHAHAHAHA! They print it out when someone places an order! 😂
No you haven't. You've shown 2 websites, both updated by random people.
I already pointed out to you that they DON'T teach order of operations at University. It's taught in high school. Dude on page you referred to was teaching Set theory, not order of operations.
Don't know who you're referring to. I'm a high school Maths teacher, hence the dozens of textbooks on the topic.
Proves I'm not weird then doesn't it.
You call what's in textbooks nonsense? That explains a lot! 😂
And yet the textbook says nothing of the kind. If I had 2+3, which is really +2+3 (see above textbook), do I, according to you, have to write 0+2+0+3? Enquiring minds want to know. And do I have to put another plus in front of the zero, as per the textbook, +0+2+0+3
No it isn't 😂
Now you're getting it. Multiply and divide take 2 inputs, add and subtract take 1.
Actually none of those are operators. The first 2 are grouping symbols (like brackets, exponents, and vinculums), the last is a function (it was right there in the name). The only unary operators are plus and minus.
You very nearly got it that time though! 😂
Again, it's not me who's wrong.
Welcome to the 21st century. We have this thing called the internet so people can share information without killing trees. It's the resource material for a college course. That's like the definition of a text book without costing the students a month's rent.
One is a PhD teaching a college course on the subject, the other is Wolfram. Neither of those are "random people" and their credentials beat "claims to be a high school math teacher but had trouble counting to 2" pretty soundly.
This portion of the discussion wasn't about order of operations, it was about the number of inputs an operator (+, and - in this case) has. Try to keep up.
Dear God if that's true I feel sorry for your students and embarrassed for whatever school is paying you. But this is the internet and with any luck that's a flat out lie. At least your repeated use of the plural maths means you're not anywhere near my kids.
Oh, I see the problem. We're back to reading comprehension. That section you highlighted specifically refers to when those symbols are being used as a "sign of the quality" of the number it's referring to, not when it's being used to indicate an operation like addition or subtraction. Hopefully that clears it up. This is ignoring the fact that a random screen shot could be anything. For all I know you wrote that yourself.
No. You also don't need to write +2+3 because the first "+" isn't an operator. It's, as your own picture says, a sign of the quality of 2.
I would love to know how you get to a sum or difference with only one input. Here, I'll try to spell it out using your own example so that even you can understand.
The inputs to 2 + 3 = 5 are 2 and 3. Let's count them together. 2 is the first, and 3 is the second. 1, 2. Two inputs for addition. Did you get it this time? Was that too fast? You can go back and read it again if you need to
Fine, operation then. The fact that you think "!" is the same thing as brackets doesn't do anything to help your bona fides. And I don't have the energy to write up a word doc and screen shot it since that's apparently what it takes for you to consider something valid.
Maybe you're just being weirdly pedantic about operator vs operation. Which would be a strange hill to die on since the original topic was operations.
If by "it" you mean through your thick skull, then you're more optimistic than I am.
Again, according to literally everyone, it is. I could keep providing sources, but I still don't have the time to screen shot some random crap with no supporting evidence. And as much as I enjoy dunking on dipshits, I've got other things to do.
Welcome to it's not a textbook (and it wasn't about order of operations anyway).
We also have this thing called textbooks, that schools order so that Maths classes don't have to be held in computer labs.
And the college doesn't teach order of operations.
by someone who can't back up their statements with actual textbooks.
Yep, exactly what I said - a random person as far as order of operations is concerned, since he teaches Set Theory and not order of operations.
Yeah, their programmers didn't know The Distributive Law either.
Happy to take that bet. Guarantee you neither of them has studied order of operations since they were in high school.
Yes it is. I said that order of operations dictates that you have to solve binary operators before unary operators, then you started trying to argue about unary operators.
Yep, the ones with more inputs, binary operators, have to be solved first.
Says person who's forgotten why we were talking about it to begin with! 😂
Well that outs yourself as living in a country which has fallen behind the rest of the world in Maths, where high school teachers don't even have to have Maths qualifications to teach Maths.
which is always. As usual, the comprehension issue is at your end.
Yes it is 😂
That you still have comprehension issues? I knew that already
The name of the book is in the top left. Not very observant either.
You don't care how much you embarrass yourself do you, given the name of the book is in the top left and anyone can find and download it. 😂
Yes it is! 😂
and a sign of the quality of the 3 too. There are 2 of them, one for each Term, since it's a 1:1 relationship.
You don't. Both need 2 Terms with signs. In this case +2 and +3.
Yep, corresponding to the 2 plus signs, +2 and +3. 1 unary operator, 1 Term, 2 of each.
2 jumps on the number line, starting from 0, +2, then +3, ends up at +5 on the number line. This is how it's taught in elementary school.
The real question is did you?
No, you just forgot one of the plus signs in your counting, the one we usually omit by convention if at the start of the expression (whereas we never omit a minus sign if it's at the start of the expression).
I'm not the one who doesn't know how unary operators work. Try it again, this time not leaving out the first plus sign.
Nope, not an operation either.
I see you don't know how grouping symbols work either.
Grouping symbols are neither.
You were the one who incorrectly brought grouping symbols into it, not me.
You haven't provided any yet! 😂
Glad you finally admitted you have no supporting evidence. Bye then! 😂
Nothing. And that's why people don't write equations like that: You either see
or
If you wrote
6 + 4 / 2in a paper you'd get reviewers complaining that it's ambiguous, if you want it to be on one line write(6+4) / 2or6 + (4/2)or6 + ⁴⁄₂or even½(6 + 4)Working mathematicians never came up with PEMDAS, which disambiguates it without parenthesis, US teachers did. Noone else does it that way because it does not, in the slightest, aid readability.Says someone who clearly hasn't looked in any Maths textbooks
Only if their Maths was very poor. #MathsIsNeverAmbiguous
Yes they did.
It was never ambiguous to begin with.
Says someone who has never looked in a non-U.S. Maths textbooks - BIDMAS, BODMAS, BEDMAS, all textbooks have one variation or another.
just say you like the smell of your own farts, it would be less text for us to read for the same result
This guy is the the guy posting the answer and then spending hours fighting the idiots who got it wrong on Facebook.
Nerd.
x/0 is the set {+inf,-inf}, fite me IRL.