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.
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”
Nope! The order of operations rules come from the proof of the definitions in the first place. 3x4=3+3+3+3 by definition, therefore if you don’t do the multiplication first in 2+3x4 you get a wrong answer (having changed the multiplicand).
As far as I know statements are pretty common
And yet you’ve not been able to quote a Maths textbook using that word.
are a foundational part of all math
Expressions are.
It’s not really a yes or no thing
It’s really a no thing.
And again laws are created using statements
Not the Laws of Maths. e.g. The Distributive Law is expressed with the identity a(b+c)=(ab+ac). An identity is a special type of equation. We have…
Numerals
Pronumerals
Expressions
Equations (or Formula)
Identities
No statements. Everything is precisely defined in Maths, everything has one meaning only.
Left to right is a convention. Left Associativity is a hard rule. Left to right is a convention which obeys the rule of Left Associativity.
It’s something agreed upon
It’s something that is a natural consequence of the definitions of the operators in the first place. As soon as Multiplication was defined in terms of Addition, that guaranteed we would always have to do Multiplication before Addition to get right answers.
is it not something that is universally true
Yes it is! All of Maths is universally true! 😂
Solve for X X^2=4
You know that’s no longer an order of operations problem, right?
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.
You’re literally arguing nothing right now. THEY took the position we should have brackets defining the order in every single equation or otherwise have them as undefined TODAY. It doesn’t matter when they were invented. Obviously it’s never been written like that. They are the one arguing it SHOULD BE. I said that would be stupid vs following the left to right convention already established. You’re getting caught up in the semantics of the wording.
What you inferred: they’re saying brackets were always around and we chose left to right to avoid bracket mess.
What I was actually saying: we chose and continue to choose to keep using the left to right convention over brackets everywhere because it would be unnecessary and make things more cluttered.
And yes, that IS a position mathematicians COULD have chosen once brackets WERE invented. They could have decided we should use them in every equation for absolute clarity of order. Saying we should not do that based on tradition alone is a bad reason.
The “always been the case” argument could justify any legacy system. We don’t still use Roman numerals for arithmetic just because they were traditional. Things DO change.
Ancient Greeks and Romans strongly resisted zero as a concept, viewing it as philosophically problematic. Negative numbers were even more controversial with many mathematicians into the Renaissance calling them “fictitious” or “absurd numbers.” It took centuries for these to become accepted as legitimate mathematical objects.
Before Robert Recorde introduced “=” in 1557, mathematicians wrote out “is equal to” in words. Even after its introduction, many resisted it for decades, preferring verbal descriptions or other symbols.
I could go on but if you’re going to argue why something shouldn’t be the case, you should argue more than “it’s tradition” or “we’ve done fine without it so far”. Because they did fine with many things in mathematics until they decided they needed to change or expand it.
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.
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.
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”
Nope! The order of operations rules come from the proof of the definitions in the first place. 3x4=3+3+3+3 by definition, therefore if you don’t do the multiplication first in 2+3x4 you get a wrong answer (having changed the multiplicand).
And yet you’ve not been able to quote a Maths textbook using that word.
Expressions are.
It’s really a no thing.
Not the Laws of Maths. e.g. The Distributive Law is expressed with the identity a(b+c)=(ab+ac). An identity is a special type of equation. We have…
Numerals
Pronumerals
Expressions
Equations (or Formula)
Identities
No statements. Everything is precisely defined in Maths, everything has one meaning only.
Order of operations is not a hard rule. It is a convention. It’s something agreed upon but is it not something that is universally true.
Solve for X
X^2=4
Yes it is.
Left to right is a convention. Left Associativity is a hard rule. Left to right is a convention which obeys the rule of Left Associativity.
It’s something that is a natural consequence of the definitions of the operators in the first place. As soon as Multiplication was defined in terms of Addition, that guaranteed we would always have to do Multiplication before Addition to get right answers.
Yes it is! All of Maths is universally true! 😂
You know that’s no longer an order of operations problem, right?
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
You’re literally arguing nothing right now. THEY took the position we should have brackets defining the order in every single equation or otherwise have them as undefined TODAY. It doesn’t matter when they were invented. Obviously it’s never been written like that. They are the one arguing it SHOULD BE. I said that would be stupid vs following the left to right convention already established. You’re getting caught up in the semantics of the wording.
What you inferred: they’re saying brackets were always around and we chose left to right to avoid bracket mess.
What I was actually saying: we chose and continue to choose to keep using the left to right convention over brackets everywhere because it would be unnecessary and make things more cluttered.
And yes, that IS a position mathematicians COULD have chosen once brackets WERE invented. They could have decided we should use them in every equation for absolute clarity of order. Saying we should not do that based on tradition alone is a bad reason.
The “always been the case” argument could justify any legacy system. We don’t still use Roman numerals for arithmetic just because they were traditional. Things DO change.
Ancient Greeks and Romans strongly resisted zero as a concept, viewing it as philosophically problematic. Negative numbers were even more controversial with many mathematicians into the Renaissance calling them “fictitious” or “absurd numbers.” It took centuries for these to become accepted as legitimate mathematical objects.
Before Robert Recorde introduced “=” in 1557, mathematicians wrote out “is equal to” in words. Even after its introduction, many resisted it for decades, preferring verbal descriptions or other symbols.
I could go on but if you’re going to argue why something shouldn’t be the case, you should argue more than “it’s tradition” or “we’ve done fine without it so far”. Because they did fine with many things in mathematics until they decided they needed to change or expand it.
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.