• HereIAm@lemmy.world
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    27 days ago

    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 issue normally with these “trick” questions

      There’s no “trick” - it’s a straight-out test of Maths knowledge.

      the ambiguous nature of that division sign

      Nothing ambiguous about it. The Term of the left divided by the Term on the right.

      A common mistake is to think division is prioritised above multiplication

      It’s not a mistake. You can do them in any order you want.

      when it actually has the same priority

      Which means you can do them in any order

      • HereIAm@lemmy.world
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        2 hours ago

        “A common mistake is to think division is prioritised above multiplication”

        That is what I said. I said it’s a mistake to think one of them has a precedence over the other. You’re arguing the same point I’m making?

        • I said it’s a mistake to think one of them has a precedence over the other

          And I said it’s not a mistake. You still get the right answer.

          You’re arguing the same point I’m making?

          No, I’m telling you that prioritising either isn’t a mistake. Mistakes give wrong answers. Prioritising either doesn’t give wrong answers.

    • vithigar@lemmy.ca
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      26 days ago

      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?

      • HereIAm@lemmy.world
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        26 days ago

        So let’s try out some different prioritization systems.

        Left to right:

        (((6 * 4) / 2) * 3) / 9
        ((24 / 2) * 3) / 9
        (12 * 3) / 9
        36 / 9 = 4
        

        Right to left:

        6 * (4 / (2 * (3 / 9)))  
        6 * (4 / (2 * 0.333...))  
        6 * (4 / 0.666...)  
        6 * 6 = 36
        

        Multiplication first:

        (6 * 4) / (2 * 3) / 9  
        24 / 6 / 9
        

        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... = 36
        

        And finally division first:

        6 * (4 / 2) * (3 / 9)  
        6 * 2 * 0.333...  
        12 * 0.333.. = 4 
        

        It’s ambiguous which one of these is correct. Hence the best method we have for “correct” is left to right.

        • Right to left:

          6 * (4 / (2 * (3 / 9)))

          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.

        • Melvin_Ferd@lemmy.world
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          26 days ago

          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

          • until the ambiguity is removed

            There isn’t any ambiguity.

            all those answers are correct

            No, only 1 answer is correct, and all the others are wrong.

            Until the author gives me clarity then that sentence has multiple meanings. With math

            Maths isn’t English and doesn’t have multiple meanings. It has rules. Obey the rules and you always get the right answer.

            it doesn’t click for people that the equation is incomplete.

            It isn’t incomplete.

            • Melvin_Ferd@lemmy.world
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              12 hours ago

              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.

              • Can you explain how that is? Like with an example?

                I’m not sure what you’re asking about. Explain what with an example?

                Math is exactly like English. It’s a language

                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.

                It’s an abstraction to describe something

                And that something is the Laws of the Universe. 1+1=2, F=ma, etc.

                Hell the word statement is used in math and English for a reason

                You won’t find the word “statement” used in Maths textbooks. I’m guessing you’re referring to Expressions.

                • Melvin_Ferd@lemmy.world
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                  5 hours ago

                  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”

                  • Those rules are based on axioms

                    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.

          • HereIAm@lemmy.world
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            26 days ago

            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.

            • Robust Mirror@aussie.zone
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              26 days ago

              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.

              • It’s so we don’t have to spam brackets everywhere

                No it isn’t. The order of operations rules were around for several centuries before we even started using Brackets in Maths.

                ((((((9+2)-1)+6)-4)+7)-3)+5

                It was literally never written like that

                we only need parentheses when we want to deviate from the norm

                That has always been the case

                • Robust Mirror@aussie.zone
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                  53 minutes ago

                  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.

                  • THEY took the position we should have brackets defining the order in every single equation or otherwise have them as undefined TODAY

                    Who’s this mysterious “THEY” you are referring to, because I can assure you that the history of Maths tells you that is wrong. e.g. look in Cajori and you’ll find the order of operations rules are at least 2 centuries older than the use of Brackets in Maths.,

                    It doesn’t matter when they were invented

                    The rules haven’t changed since then.

                    They are the one arguing it SHOULD BE

                    …and watch Physicists and Mathematicians promptly run out of room on blackboards if they did.

                    You’re getting caught up in the semantics of the wording

                    No, you’re making up things that never happened.

                    they’re saying brackets were always around and we chose left to right to avoid bracket mess

                    and that’s wrong. Left to right was around before Brackets were.

                    we chose and continue to choose to keep using the left to right convention over brackets everywhere

                    and you’re wrong, because that choice was made before we’d even started using Brackets in Maths, by at least a couple of centuries.

                    it would be unnecessary and make things more cluttered

                    They’ve always been un-necessary, unless you want to deviate from the normal order of operations.

                    They could have decided we should use them in every equation for absolute clarity of order

                    But they didn’t, because we already had clarity over order, and had done for several centuries.

                    Saying we should not do that based on tradition alone is a bad reason.

                    Got nothing to do with tradition. Got no idea where you got that idea from.

                    Things DO change.

                    The order of operations rules don’t, and the last change to the notation was in the 19th Century.

                    I could go on

                    and you’d still be wrong. You’re heading off into completely unrelated topics now.

                    you should argue more than “it’s tradition” or “we’ve done fine without it so far”

                    I never said either of those things.

                    Because they did fine with many things in mathematics until they decided they needed to change or expand it

                    And they changed the meaning of the Division symbol sometime in the 19th Century or earlier, and everything has been settled for centuries now.

        • barsoap@lemm.ee
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          26 days ago

          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.

          • 💡𝚂𝗆𝖺𝗋𝗍𝗆𝖺𝗇 𝙰𝗉𝗉𝗌📱@programming.dev
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            15 hours ago

            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.

            Yes we do, and it’s what we teach students to do.

          • HereIAm@lemmy.world
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            26 days ago

            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.

      • Robust Mirror@aussie.zone
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        26 days ago

        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

      • troistigrestristes@lemmy.eco.br
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        26 days ago

        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

        • HereIAm@lemmy.world
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          26 days ago

          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.

          • Except it does matter

            No it doesn’t. You disobeying the rules and getting lots of wrong answers in your examples doesn’t change that.

            I left some examples for another post with multiplication and division

            Which you did wrong.

            I’ll give you some addition and subtraction to see order matter with those operations as well

            And I’ll show you it doesn’t matter when you do it correctly

            Subtraction first: 1 + (2 - 3) + 4 1 + (-1) + 4 = 4

            Nope. Right answer for wrong reason - you only co-incidentally got the answer right. -3+1+2+4=-3+7=4

            Right to left: 1 + (2 - (3 + 4)) 1 + (2 - 7) 1 + (-5) = -4

            Nope. 4-3+2+1=1+2+1=3+1=4

            Edit: You can argue that, for example, the addition first could be (1 + 2) + (-3 + 4)

            Or you could just do it correctly in the first place, always obeying Left Associativity and never adding Brackets

            in my opinion that’s another ambiguous case

            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.

          • troistigrestristes@lemmy.eco.br
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            25 days ago

            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.

    • AnotherPenguin@programming.dev
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      25 days ago

      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.

      • 💡𝚂𝗆𝖺𝗋𝗍𝗆𝖺𝗇 𝙰𝗉𝗉𝗌📱@programming.dev
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        14 hours ago

        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).