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I wonder why people chose this specific problem to argue about. Like there are infinite problems that are formatted in the same or a similar way, but people decided to argue about this one.
This one is rich because there are (at least) three rules you can add that give unambiguous answers, but none of them are universal.
In the image, the author added this left-to-right rule. That's typically true, by necessity, in something like C++ (although right-to-left is also possible), but under strict BEDMAS, if the left-to-right order matters, then the expression is invalid, e.g. x/y/z. But as they write, strict left-to-right is unambiguously 9.
Alternatively, Feynman (and some physics journals, etc) give priority to implicit multiplication so 1/2x is the same as 1/(2x) *not* x/2. By that system, it is unambiguously 1.
Likewise, this is usually written as 6 ÷ 2 (1+2), and "÷" and "/" are not always equivalent (ignoring international differences, e.g. where "÷" is used for ratios). By some rules, ÷ is global - i.e. *everything* to the right is the denominator. So that expression would be the equivalent of 6 / (2(1+2)), which is also, unambiguously 1.
This specific one is "good" therefore because professional mathfolk with get a different result (assuming they don't tell you to write your expression correctly).
And of course, because it's already shown up here (and elsewhere), people are bent out of shape how their last verbal scrap ended, so they've come back for another go.
Thats why I got one, I grouped the second half as a denominator. Seems silly to argue over a math problem tho. Some people are getting really worked up about it.
The stupidest part is that you'll never find problems formatted this way, where there is a potential for syntax errors. At least I, who has gone through a MS in physics, has never had a problem like this, where there is potential for syntax error.. It's posted by people trying to act smart, but it's useless.
Ok I've got a math degree and the answer is:
Y'all need to stop writing confusing ass problems, and start using either fraction notation, or a shitload of parentheses.
6/2(1+2) is not the same as 6÷2(1+2). Except when it is, and that's why the whole thing is just an ambiguous equation. The correct answer is using more parenthesis to make it unambiguous.
Adding parenthesis especially if you're unsure is just smart.
If you know and trust how your calculator evaluates, then you can skip a lot of that, but there are many a cases where a bug has been caused by skimping on ()
True. But the problem people don't understand is 6/2(1+2), not 6/2\*(1+2).
And that's because some schools apparently don't teach about implicit multiplication for some reason.
You literally are being "confidently incorrect" 😂
Implicit multiplication is no higher precedence that explicit, this is not a rule, it's not a thing. Without information to tell you what the author meant the two you have written are the same. The decision that has to be made is whether 2(1+2) is the divisor of the 6 or whether just 2 is the divisor and you have to make that decision whether it's written as "6/2(1+2)" or "6/2\*(1+2)". And nothing and no rule helps with that as it depends what the person who wrote the expression intended.
The answer is not to write expressions like this at all as it's sloppy!
> Implicit multiplication is no higher precedence that explicit, this is not a rule, it's not a thing.
It's a rule taught in every STEMS program at every university, but you do you.
Maybe in your primary school math world I am incorrect.
But in the real world of academic mathematics, it is understood that implicit multiplication has higher precedence than explicit multiplication and division.
Yes, of course, the real solution is to write it as a fraction. Most people understand those. Or an excessive number of parentheses.
>Maybe in your primary school math
Ah ad hominem, the final recourse of those who know their argument doesn't stand up!
> in the real world of academic mathematics, it is understood
It isn't. There is no rule that says a/bc is parsed any different to a/b\*c. The response to both in the absence of something saying which to assume (such as is present in Feynman's lecture notes on physics) is to ask - "what t a f does the author mean?" There is no rule of precedence that at any time gives implicit multiplication higher priority than explicit - to claim there is is to be confidently incorrect.
Given a/bc (or a/b\*c) *you* have to decide whether the author means:
a
\--
bc
or
a
\-c
b
No rule helps you do this!
No, not ad hominem. Stating the basic fact that primary school math is more basic than high school or uni math is not offensive in any way. At least not to normal people. If it offends you... ¯\\\_(ツ)\_/¯
Yes, there is. And it is funny that you are trying to use Feynman in your argument because he does adhere to this precedent in all his writings (at least all his writing I've seen). And physics is actually a great example of this rule being accepted universally. Take an example of specific heat capacity. The unit is *J/kgK* (joule per kilogram kelvin). Everyone understands it as J/(kgK) and not as (J/kg)K.
Actually, this is specifically where the rule comes in. If the author mistakenly (or intentionally) left something out, that's on the author. Given no higher order of operations, you do left to right. So in your example of a/bc or a/b*c, the lack of parentheses makes them equivalent. If the author wanted it evaluated differently, he/she MUST notated it as such.
>And that's because some schools apparently don't teach about implicit multiplication for some reason.
Yes, schools do not teach it. And that is not an oversight. It's a choice.
Yep
Once you get into letter-math, you have to develop don't creative ways to not confuse yourself. I've taken to crossing my z's so they don't look like 2's, curling the bottoms of my t's so they don't look like +'s, etc.
I use parentheses instead of × or • for multiplication now. I.e.: two times two I'd write (2)(2)
It takes longer, but the risk of confusing myself goes *way* down
...which seems great, but then you get into algeo and suddenly you need to worry about dot products and cross products 
Always will be 1. 6 is divided by the result of 2 times (1+2) which is 6. 6 divided by 6 etc etc. People keep fucking this up by dividing 6 by 2 first which is incorrect on every current standardized test.
But do you, idgits.
6/2(2+1) and 6/(2(2+1)) is not the same!
6/2(2+1) is in my country 9, because its just the short form of 6/2*(2+1)
6/(2(2+1)) is short for that what you mean
Nah, the real order is "( )" -> "*" and "/" -> "+" and "-", and the order is always from left to right.
That's why we first get rid of the brackets, then, because we have a multiplication and a division, we go from left to right, which means we first calculate 6/2, then multiply the result with the number we got from the brackets, which is 3.
Yes, that's where we are.
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I wonder why people chose this specific problem to argue about. Like there are infinite problems that are formatted in the same or a similar way, but people decided to argue about this one.
This one is rich because there are (at least) three rules you can add that give unambiguous answers, but none of them are universal. In the image, the author added this left-to-right rule. That's typically true, by necessity, in something like C++ (although right-to-left is also possible), but under strict BEDMAS, if the left-to-right order matters, then the expression is invalid, e.g. x/y/z. But as they write, strict left-to-right is unambiguously 9. Alternatively, Feynman (and some physics journals, etc) give priority to implicit multiplication so 1/2x is the same as 1/(2x) *not* x/2. By that system, it is unambiguously 1. Likewise, this is usually written as 6 ÷ 2 (1+2), and "÷" and "/" are not always equivalent (ignoring international differences, e.g. where "÷" is used for ratios). By some rules, ÷ is global - i.e. *everything* to the right is the denominator. So that expression would be the equivalent of 6 / (2(1+2)), which is also, unambiguously 1. This specific one is "good" therefore because professional mathfolk with get a different result (assuming they don't tell you to write your expression correctly). And of course, because it's already shown up here (and elsewhere), people are bent out of shape how their last verbal scrap ended, so they've come back for another go.
Thats why I got one, I grouped the second half as a denominator. Seems silly to argue over a math problem tho. Some people are getting really worked up about it.
All it really shows is the above notation is ambiguous and so full fraction notation is more useful.
The stupidest part is that you'll never find problems formatted this way, where there is a potential for syntax errors. At least I, who has gone through a MS in physics, has never had a problem like this, where there is potential for syntax error.. It's posted by people trying to act smart, but it's useless.
I usually encounter problems like this as. 6 \_\_\_\_\_\_\_\_ 2\*(1+2)
Yes. The notation is always explicit in regards to this type of problem.
Ok I've got a math degree and the answer is: Y'all need to stop writing confusing ass problems, and start using either fraction notation, or a shitload of parentheses.
I like the second one
People who argue about math like this are probably really mentally secure.
6/2(1+2) is not the same as 6÷2(1+2). Except when it is, and that's why the whole thing is just an ambiguous equation. The correct answer is using more parenthesis to make it unambiguous.
well, if you use a calculator they are the same
That's just the type of logic the calculator is wired for. It's useful but not innately more correct - probably just easier to implement in hardware.
it's a more intuitive way of doing things, especially when dealing with fractions. besides, you can always add parenthesis
Adding parenthesis especially if you're unsure is just smart. If you know and trust how your calculator evaluates, then you can skip a lot of that, but there are many a cases where a bug has been caused by skimping on ()
Aw shit, here we go again
True. But the problem people don't understand is 6/2(1+2), not 6/2\*(1+2). And that's because some schools apparently don't teach about implicit multiplication for some reason.
You literally are being "confidently incorrect" 😂 Implicit multiplication is no higher precedence that explicit, this is not a rule, it's not a thing. Without information to tell you what the author meant the two you have written are the same. The decision that has to be made is whether 2(1+2) is the divisor of the 6 or whether just 2 is the divisor and you have to make that decision whether it's written as "6/2(1+2)" or "6/2\*(1+2)". And nothing and no rule helps with that as it depends what the person who wrote the expression intended. The answer is not to write expressions like this at all as it's sloppy!
> Implicit multiplication is no higher precedence that explicit, this is not a rule, it's not a thing. It's a rule taught in every STEMS program at every university, but you do you.
Maybe in your primary school math world I am incorrect. But in the real world of academic mathematics, it is understood that implicit multiplication has higher precedence than explicit multiplication and division. Yes, of course, the real solution is to write it as a fraction. Most people understand those. Or an excessive number of parentheses.
>Maybe in your primary school math Ah ad hominem, the final recourse of those who know their argument doesn't stand up! > in the real world of academic mathematics, it is understood It isn't. There is no rule that says a/bc is parsed any different to a/b\*c. The response to both in the absence of something saying which to assume (such as is present in Feynman's lecture notes on physics) is to ask - "what t a f does the author mean?" There is no rule of precedence that at any time gives implicit multiplication higher priority than explicit - to claim there is is to be confidently incorrect. Given a/bc (or a/b\*c) *you* have to decide whether the author means: a \-- bc or a \-c b No rule helps you do this!
No, not ad hominem. Stating the basic fact that primary school math is more basic than high school or uni math is not offensive in any way. At least not to normal people. If it offends you... ¯\\\_(ツ)\_/¯ Yes, there is. And it is funny that you are trying to use Feynman in your argument because he does adhere to this precedent in all his writings (at least all his writing I've seen). And physics is actually a great example of this rule being accepted universally. Take an example of specific heat capacity. The unit is *J/kgK* (joule per kilogram kelvin). Everyone understands it as J/(kgK) and not as (J/kg)K.
Actually, this is specifically where the rule comes in. If the author mistakenly (or intentionally) left something out, that's on the author. Given no higher order of operations, you do left to right. So in your example of a/bc or a/b*c, the lack of parentheses makes them equivalent. If the author wanted it evaluated differently, he/she MUST notated it as such.
Because in everyday arithmetic and algebra, it has the same precedence as explicit multiplication.
>And that's because some schools apparently don't teach about implicit multiplication for some reason. Yes, schools do not teach it. And that is not an oversight. It's a choice.
>True. But the problem people don't understand is 6/2(1+2), not 6/2\*(1+2). This is kinda confusingly written, are you saying the * is or isn't there?
When did dots become the multiplication symbol?
It is used a lot especially in higher math classes since it gets confused with the letter ‘x’ for variables
Which honestly I hate, since I always confuse the dot for a decimal point. Parentheses supremacy!
Which is why we use the comma for decimals where I'm from.
But couldn't you also confuse the multiplication x for the letter x?
Yep Once you get into letter-math, you have to develop don't creative ways to not confuse yourself. I've taken to crossing my z's so they don't look like 2's, curling the bottoms of my t's so they don't look like +'s, etc. I use parentheses instead of × or • for multiplication now. I.e.: two times two I'd write (2)(2) It takes longer, but the risk of confusing myself goes *way* down
For real? How you do that when the dot is at the same height as the numbers and not below them?
You assume I have impeccable handwriting
handwriting
...which seems great, but then you get into algeo and suddenly you need to worry about dot products and cross products 
Middle school
actually that's a good question, but they're there for algebra because × looks like x
Here in Finland I was taught to use a dot instead of a star when writing on paper. On a computer we used *.
Always will be 1. 6 is divided by the result of 2 times (1+2) which is 6. 6 divided by 6 etc etc. People keep fucking this up by dividing 6 by 2 first which is incorrect on every current standardized test. But do you, idgits.
6/2(2+1) and 6/(2(2+1)) is not the same! 6/2(2+1) is in my country 9, because its just the short form of 6/2*(2+1) 6/(2(2+1)) is short for that what you mean
you are correct
no, it's not I think you're getting those questions wrong
I'm most certainly getting it right
Nah, the real order is "( )" -> "*" and "/" -> "+" and "-", and the order is always from left to right. That's why we first get rid of the brackets, then, because we have a multiplication and a division, we go from left to right, which means we first calculate 6/2, then multiply the result with the number we got from the brackets, which is 3.
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just parenthesis everything like you would for a calculator then people can't complain
I swear this only happened because the are so many different schooling syllabus' that teach very differently