A fake Facebook event disguised as a math problem has been one of its top posts for 6 months
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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 = 4Right to left:
6 * (4 / (2 * (3 / 9))) 6 * (4 / (2 * 0.333...)) 6 * (4 / 0.666...) 6 * 6 = 36Multiplication first:
(6 * 4) / (2 * 3) / 9 24 / 6 / 9Here 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:
6 * (4 / 2) * (3 / 9) 6 * 2 * 0.333... 12 * 0.333.. = 4It'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.
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I stand corrected
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).
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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.
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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.
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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
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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.
Yes we do, and it's what we teach students to do.
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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=11Addition first:
9-4=5Addition first:
9-4=5Nope. 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)
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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 + 4Addition first:
(1 + 2) - (3 + 4)
3 - 7 = -4Subtraction first:
1 + (2 - 3) + 4
1 + (-1) + 4 = 4Right to left:
1 + (2 - (3 + 4))
1 + (2 - 7)
1 + (-5) = -4Left to right:
((1 + 2) - 3) + 4
(3 - 3) + 4 = 4Edit:
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 = 4Nope. 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) = -4Nope. 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.
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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.
Oh, but of course the statement changes if you add parentheses
It sure does, but they don't seem to understand that.
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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).
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÷ could be a minus sign, see: https://en.wikipedia.org/wiki/Division_sign?wprov=sfla1
÷ could be a minus sign
No it couldn't.
Did you check the reference? It says % can be used as a minus sign, not the obelus. Welcome to what happens when you're next-door neighbour Joe Blow can edit Wikipedia.
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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.
Yes, it is
No it isn't.
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
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...
Alternative definitions are also based on a multiplication
Emphasis on "alternative", not actual.
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Hey, this is Presh Talwalkar
Person who has forgotten about The Distributive Law and lied about 1917.
Discussion of a brief history of this viral math problem
Including lying about 1917
Ultimately followed by brief discussion on the order of operations
But forgets about Terms and The Distributive Law.
And that’s the answer
Now watch his other ones, where he screws it up royally. Dude has no idea how to handle brackets. Should be avoided at all costs.
I've seen many of his videos and haven't noticed any obvious errors. Could you please link to the specific video(s) that you are referencing in regards to errors he has made, especially those related to the distributive law and what you reference to as "1917," as well as any explanation as to what is incorrect/misleading/lying?
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Yes, it is
No it isn't.
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
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...
Alternative definitions are also based on a multiplication
Emphasis on "alternative", not actual.
No it isn't.
Yes, it is.
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...
I'm defining the division operation, not the quotient. Yes, the quotient is obtained by dividing... Now define dividing.
Emphasis on "alternative", not actual.
The actual is the one I gave. I did not give the alternative definitions. That's why I said they are also defined based on a multiplication, implying the non-alternative one (understand, the actual one) was the one I gave.
Feel free to send your entire Euler document rather than screenshotting the one part you thought makes you right.
Note, by the way, that Euler isn't the only mathematician who contributed to the modern definitions in algebra and arithmetics.
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No it isn't.
Yes, it is.
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...
I'm defining the division operation, not the quotient. Yes, the quotient is obtained by dividing... Now define dividing.
Emphasis on "alternative", not actual.
The actual is the one I gave. I did not give the alternative definitions. That's why I said they are also defined based on a multiplication, implying the non-alternative one (understand, the actual one) was the one I gave.
Feel free to send your entire Euler document rather than screenshotting the one part you thought makes you right.
Note, by the way, that Euler isn't the only mathematician who contributed to the modern definitions in algebra and arithmetics.
I’m defining the division operation, not the quotient
Yep, the quotient is the result of Division. It's right there in the definition in Euler. Dividend / Divisor = Quotient <= no reference to multiplication anywhere
Yes, the quotient is obtained by dividing… Now define dividing.
You not able to read the direct quote from Euler defining Division? Doesn't mention Multiplication at all.
The actual is the one I gave
No, you gave an alternative (and also you gave no citation for it anyway - just something you made up by the look of it). The actual definition is in Euler.
That’s why I said they are also defined based on a multiplication
Again, emphasis on "alternative", not actual.
implying the non-alternative one (understand, the actual one) was the one I gave
The one you gave bears no resemblance at all to what is in Euler, nor was given with a citation.
Feel free to send your entire Euler document rather than screenshotting the one part
The name of the PDF is in the top-left. Not too observant I see
you thought makes you right
That's the one and only actual definition of Division. Not sure what you think is in the rest of the book, but he doesn't spend the whole time talking about Division, but feel free to go ahead and download the whole thing and read it from cover to cover to be sure!
Note, by the way, that Euler isn’t the only mathematician who contributed to the modern definitions in algebra and arithmetics.
And none of the definitions you have given have come from a Mathematician. Saying "most professions", and the lack of a citation, was a dead giveaway!
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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.
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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.
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.
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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.
But +, -, *, and / are all binary operators?
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).
As far as I know, the only reason multiplication and division come first is that we’ve all agreed to it.
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
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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.
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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.
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"
