The issue normally with these “trick” questions is the ambiguous nature of that division sign (not so much a problem here) or people not knowing to just go left to right when all operators are of the same priority. A common mistake is to think division is prioritised above multiplication, when it actually has the same priority. Someone should have included some parenthesis in PEDMAS aka. PE(DM)(AS) 😄
The same priority operations can be done in any order without affecting the result, that’s why they can be same priority and don’t need an explicit order.
6 × 4 ÷ 2 × 3 ÷ 9 evaluates the same regardless of order. Can you provide a counter example?
Oh my god now this is going to be Lemmy’s top thread for 6 months, isn’t it?
Btw, yeah I’m with you on this, you just need to know the priorities and you’re good, because the order doesn’t matter for operations with the same priority
Except it does matter. I left some examples for another post with multiplication and division, I’ll give you some addition and subtraction to see order matter with those operations as well.
Right to left:
1 + (2 - (3 + 4))
1 + (2 - 7)
1 + (-5) = -4
Left to right:
((1 + 2) - 3) + 4
(3 - 3) + 4 = 4
Edit:
You can argue that, for example, the addition first could be (1 + 2) + (-3 + 4) in which case it does end up as 4, but in my opinion that’s another ambiguous case.
Oh, but of course the statement changes if you add parentheses. Basically, you’re changing the effective numbers that are being used, because the parentheses act as containers with a given value (you even showed the effective numbers in your examples).
Get this : + 1 - 1 + 1 - 1 + 1 - 1 + 1
You can change the result several times by choosing where you want to put the parentheses. However, the order of operations of same priority inside a container (parentheses) does not change the resulting value of the container.
In the example, there were no parentheses, so no ambiguity (there wouldn’t be any ambiguity with parentheses either, the correct way of calculating would just change), and I don’t think you can add “ambiguity” by adding parentheses — you’re just changing the effective expression to be evaluated.
By the way, this is the reason why I absolutely overuse parentheses in my engineering code. It can be redundant, but at least I am SURE that it is going to follow the order that I wanted.
Nope! 6 × 4 ÷ 2 × 3 ÷ 9 =4 right to left is 6 ÷ 9 x 3 ÷ 2 × 4 =4. You disobeyed the rule of Left Associativity, and your answer is wrong
Multiplication first: (6 * 4) / (2 * 3) / 9
Also nope. Multiplication first is 6 x 4 x 3 ÷ 2 ÷ 9 =4
Left first: (24 / 6) / 9
Still nope. 6 × 4 x 3 ÷ 2 ÷ 9 =4
Right side first: 24 / (6 / 9)
Still nope. 6 × 4 x 3 ÷ 9 ÷ 2 =4
And finally division first: 6 * (4 / 2) * (3 / 9)
And finally still nope. 6 ÷ 9 ÷ 2 x 4 x 3 =4
Hint: note that I never once added any brackets. You did, hence your multiple wrong answers.
It’s ambiguous which one of these is correct
No it isn’t. Only 4 is correct, as I have just shown repeatedly.
Hence the best method we have for “correct” is left to right
It’s because students don’t make mistakes with signs if you don’t change the order. I just showed you can still get the correct answer with different orders, but you have to make sure you obey Left Associativity at every step.
It’s ambiguous which one of these is correct. Hence the best method we have for “correct” is left to right.
The solution accepted anywhere but in the US school system range from “Bloody use parenthesis, then” over “Why is there more than one division in this formula why didn’t you re-arrange everything to be less confusing” to “50 Hertz, in base units, are 50s-1”.
More practically speaking: Ultimately, you’ll want to do algebra with these things. If you rely on “left to right” type of precedence rules re-arranging formulas becomes way harder because now you have to contend with that kind of implicit constraint. It makes everything harder for no reason whatsoever so no actual mathematician, or other people using maths in earnest, use that kind of notation.
The solution accepted anywhere but in the US school system range from “Bloody use parenthesis, then” over “Why is there more than one division in this formula why didn’t you re-arrange everything to be less confusing” to “50 Hertz, in base units, are 50s-1”.
No, the solution is learn the rules of Maths. You can find them in Maths textbooks, even in U.S. Maths textbooks.
so no actual mathematician, or other people using maths in earnest, use that kind of notation.
I fully agree that if it comes down to “left to right” the problem really needs to be rewritten to be more clear. But I’ve just shown why that “rule” is a common part of these meme problems because it is so weird and quite esoteric.
No, you weren’t. Most of their answers were wrong. You were right. See my reply. 4 is the only correct answer, and if you don’t get 4 then you did something wrong, as they did repeatedly (kept adding brackets and thus changing the Associativity).
Maybe I’m wrong but the way I explain it is until the ambiguity is removed by adding in extra information to make it more specific then all those answers are correct.
“I saw her duck”
Until the author gives me clarity then that sentence has multiple meanings. With math, it doesn’t click for people that the equation is incomplete. In an English sentence, ambiguity makes more sense and the common sense approach would be to clarify what the meaning is
Can you explain how that is? Like with an example?
Math is exactly like English. It’s a language. It’s an abstraction to describe something. Ambiguity exists in math and in English. It impacts the validity of a statement. Hell the word statement is used in math and English for a reason.
Those rules are based on axioms which are used to create statements which are used within proofs. As far as I know statements are pretty common and are a foundational part of all math.
Defining math as a language though is also going to be pointless here. It’s not really a yes or no thing. I’ll say it is a language but sure it’s arguable.
And again laws are created using statements. I have plenty of textbooks that contain “statements”
100% with you. “Left to right” as far as I can tell only exists to make otherwise “unsolvable” problems a kind of official solution. I personally feel like it is a bodge, and I would rather the correct solution for such a problem to be undefined.
100% with you. “Left to right” as far as I can tell only exists to make otherwise “unsolvable” problems a kind of official solution
It’s not a rule, it’s a convention, and it exists so as to avoid making mistakes with signs, mistakes you made in almost every example you gave where you disobeyed left to right.
That’s just clutter for no good reason when we can just say if it doesn’t have parentheses it’s left to right. Having a default evaluation order makes sense and means we only need parentheses when we want to deviate from the norm.
Another common issue is thinking “parentheses go first” and then beginning by solving the operation beside them (mostly multiplication). The point being that what’s inside the parentheses goes first, not what’s beside them.
Another common issue is thinking “parentheses go first”
There’s no “think” - it’s an absolute rule.
then beginning by solving the operation beside them
a(b) isn’t an operation - it’s a Product. a(b)=(axb) per The Distributive Law.
(mostly multiplication)
NOT Multiplication, a Product/Term.
The point being that what’s inside the parentheses goes first, not what’s beside them
Nope, it’s the WHOLE Bracketed Term. a/bxc=ac/b, but a/b( c )=a/(bxc). Inside is only a “rule” in Elementary School, when there isn’t ANYTHING next to them (students aren’t taught this until High School, in Algebra), and it’s not even really a rule then, it’s just that there isn’t anything ELSE involved in the Brackets step than what is inside (since they’re never given anything on the outside).
So order of operations is hard?
Not for students it isn’t. Adults who’ve forgotten the rules on the other hand…
The issue normally with these “trick” questions is the ambiguous nature of that division sign (not so much a problem here) or people not knowing to just go left to right when all operators are of the same priority. A common mistake is to think division is prioritised above multiplication, when it actually has the same priority. Someone should have included some parenthesis in PEDMAS aka. PE(DM)(AS) 😄
There’s no “trick” - it’s a straight-out test of Maths knowledge.
Nothing ambiguous about it. The Term of the left divided by the Term on the right.
It’s not a mistake. You can do them in any order you want.
Which means you can do them in any order
The same priority operations can be done in any order without affecting the result, that’s why they can be same priority and don’t need an explicit order.
6 × 4 ÷ 2 × 3 ÷ 9 evaluates the same regardless of order. Can you provide a counter example?
Oh my god now this is going to be Lemmy’s top thread for 6 months, isn’t it?
Btw, yeah I’m with you on this, you just need to know the priorities and you’re good, because the order doesn’t matter for operations with the same priority
Except it does matter. I left some examples for another post with multiplication and division, I’ll give you some addition and subtraction to see order matter with those operations as well.
Let’s take:
1 + 2 - 3 + 4
Addition first:
(1 + 2) - (3 + 4)
3 - 7 = -4
Subtraction first:
1 + (2 - 3) + 4
1 + (-1) + 4 = 4
Right to left:
1 + (2 - (3 + 4))
1 + (2 - 7)
1 + (-5) = -4
Left to right:
((1 + 2) - 3) + 4
(3 - 3) + 4 = 4
Edit: You can argue that, for example, the addition first could be
(1 + 2) + (-3 + 4)in which case it does end up as 4, but in my opinion that’s another ambiguous case.No it doesn’t. You disobeying the rules and getting lots of wrong answers in your examples doesn’t change that.
Which you did wrong.
And I’ll show you it doesn’t matter when you do it correctly
Nope. Right answer for wrong reason - you only co-incidentally got the answer right. -3+1+2+4=-3+7=4
Nope. 4-3+2+1=1+2+1=3+1=4
Or you could just do it correctly in the first place, always obeying Left Associativity and never adding Brackets
There aren’t ANY ambiguous cases. In every case it’s equal to 4. If you didn’t get 4, then you made a mistake and got a wrong answer.
Oh, but of course the statement changes if you add parentheses. Basically, you’re changing the effective numbers that are being used, because the parentheses act as containers with a given value (you even showed the effective numbers in your examples).
Get this : + 1 - 1 + 1 - 1 + 1 - 1 + 1
You can change the result several times by choosing where you want to put the parentheses. However, the order of operations of same priority inside a container (parentheses) does not change the resulting value of the container.
In the example, there were no parentheses, so no ambiguity (there wouldn’t be any ambiguity with parentheses either, the correct way of calculating would just change), and I don’t think you can add “ambiguity” by adding parentheses — you’re just changing the effective expression to be evaluated.
By the way, this is the reason why I absolutely overuse parentheses in my engineering code. It can be redundant, but at least I am SURE that it is going to follow the order that I wanted.
It sure does, but they don’t seem to understand that.
Another person already replied using your equation, but I felt the need to reply with a simpler one as well that shows it:
9-1+3=?
Subtraction first:
8+3=11
Addition first:
9-4=5
Nope. Addition first is 9+3-1=12-1=11. You did 9-(1+3), incorrectly adding brackets and changing the answer (thus a wrong answer)
So let’s try out some different prioritization systems.
Left to right:
Right to left:
Multiplication first:
Here the path divides again, we can do the left division or right division first.
Left first: (24 / 6) / 9 4 / 9 = 0.444... Right side first: 24 / (6 / 9) 24 / 0.666... = 36And finally division first:
It’s ambiguous which one of these is correct. Hence the best method we have for “correct” is left to right.
Nope! 6 × 4 ÷ 2 × 3 ÷ 9 =4 right to left is 6 ÷ 9 x 3 ÷ 2 × 4 =4. You disobeyed the rule of Left Associativity, and your answer is wrong
Also nope. Multiplication first is 6 x 4 x 3 ÷ 2 ÷ 9 =4
Still nope. 6 × 4 x 3 ÷ 2 ÷ 9 =4
Still nope. 6 × 4 x 3 ÷ 9 ÷ 2 =4
And finally still nope. 6 ÷ 9 ÷ 2 x 4 x 3 =4
Hint: note that I never once added any brackets. You did, hence your multiple wrong answers.
No it isn’t. Only 4 is correct, as I have just shown repeatedly.
It’s because students don’t make mistakes with signs if you don’t change the order. I just showed you can still get the correct answer with different orders, but you have to make sure you obey Left Associativity at every step.
The solution accepted anywhere but in the US school system range from “Bloody use parenthesis, then” over “Why is there more than one division in this formula why didn’t you re-arrange everything to be less confusing” to “50 Hertz, in base units, are 50s-1”.
More practically speaking: Ultimately, you’ll want to do algebra with these things. If you rely on “left to right” type of precedence rules re-arranging formulas becomes way harder because now you have to contend with that kind of implicit constraint. It makes everything harder for no reason whatsoever so no actual mathematician, or other people using maths in earnest, use that kind of notation.
No, the solution is learn the rules of Maths. You can find them in Maths textbooks, even in U.S. Maths textbooks.
Yes we do, and it’s what we teach students to do.
I fully agree that if it comes down to “left to right” the problem really needs to be rewritten to be more clear. But I’ve just shown why that “rule” is a common part of these meme problems because it is so weird and quite esoteric.
I stand corrected
No, you weren’t. Most of their answers were wrong. You were right. See my reply. 4 is the only correct answer, and if you don’t get 4 then you did something wrong, as they did repeatedly (kept adding brackets and thus changing the Associativity).
Maybe I’m wrong but the way I explain it is until the ambiguity is removed by adding in extra information to make it more specific then all those answers are correct.
“I saw her duck”
Until the author gives me clarity then that sentence has multiple meanings. With math, it doesn’t click for people that the equation is incomplete. In an English sentence, ambiguity makes more sense and the common sense approach would be to clarify what the meaning is
There isn’t any ambiguity.
No, only 1 answer is correct, and all the others are wrong.
Maths isn’t English and doesn’t have multiple meanings. It has rules. Obey the rules and you always get the right answer.
It isn’t incomplete.
Can you explain how that is? Like with an example?
Math is exactly like English. It’s a language. It’s an abstraction to describe something. Ambiguity exists in math and in English. It impacts the validity of a statement. Hell the word statement is used in math and English for a reason.
I’m not sure what you’re asking about. Explain what with an example?
No it isn’t. It’s a tool for calculating things, with syntax rules. We even have rules around how to say it when speaking.
And that something is the Laws of the Universe. 1+1=2, F=ma, etc.
You won’t find the word “statement” used in Maths textbooks. I’m guessing you’re referring to Expressions.
Those rules are based on axioms which are used to create statements which are used within proofs. As far as I know statements are pretty common and are a foundational part of all math.
Defining math as a language though is also going to be pointless here. It’s not really a yes or no thing. I’ll say it is a language but sure it’s arguable.
And again laws are created using statements. I have plenty of textbooks that contain “statements”
100% with you. “Left to right” as far as I can tell only exists to make otherwise “unsolvable” problems a kind of official solution. I personally feel like it is a bodge, and I would rather the correct solution for such a problem to be undefined.
It’s not a rule, it’s a convention, and it exists so as to avoid making mistakes with signs, mistakes you made in almost every example you gave where you disobeyed left to right.
It’s so we don’t have to spam brackets everywhere
9+2-1+6-4+7-3+5=
Becomes
((((((9+2)-1)+6)-4)+7)-3)+5=
That’s just clutter for no good reason when we can just say if it doesn’t have parentheses it’s left to right. Having a default evaluation order makes sense and means we only need parentheses when we want to deviate from the norm.
No it isn’t. The order of operations rules were around for several centuries before we even started using Brackets in Maths.
It was literally never written like that
That has always been the case
Another common issue is thinking “parentheses go first” and then beginning by solving the operation beside them (mostly multiplication). The point being that what’s inside the parentheses goes first, not what’s beside them.
There’s no “think” - it’s an absolute rule.
a(b) isn’t an operation - it’s a Product. a(b)=(axb) per The Distributive Law.
NOT Multiplication, a Product/Term.
Nope, it’s the WHOLE Bracketed Term. a/bxc=ac/b, but a/b( c )=a/(bxc). Inside is only a “rule” in Elementary School, when there isn’t ANYTHING next to them (students aren’t taught this until High School, in Algebra), and it’s not even really a rule then, it’s just that there isn’t anything ELSE involved in the Brackets step than what is inside (since they’re never given anything on the outside).