6÷2(1+2) = ?

"And the correct answer from a mathematician is “you need to write this better so it’s not ambiguous”."

This is correct and is essentially what I said before.

Math is a language. A simple equation like this should give a reader a single unambiguous answer in about 2 seconds.

If you read a sentence and have to spend 10 minutes diagramming it and remembering exact grammatical rules to interpret what the author is saying, the writer failed. Unless the writer's purpose was to intentionally write an ambiguous sentence and get people to argue about it.

This equation failed to communicate. It succeeded in generating discussion.

Just Another Hobby Jogger wrote:

2600 bro wrote:

Except there are no "objective facts" within the conventions of symbolic notation. It's just what people have agreed on over the years, constantly changing, new symbols invented, new syntax becomes standard, etc.

Much like speaking and writing. You and I could barely make sense of Middle English but that doesn't mean they were speaking and writing counter to "objective fact" back in the day. Much like the assertions in this thread that their understanding of order of operations is solely correct...

The same is true about the symbols themselves. People have just agreed what signs they use for addition, subtraction, multiplication and division. Nothing objective there. For that matter, numbers are also what people agreed. Why is 2+3=5? Because that's what people agreed.

So does 2+3 =6 become the standard if enough people insist it is?

This is a big philosophical question.

I disagree. Numbers as concepts exist independent of the symbols used to write them. Cavemen could see that two rocks + two rocks = four rocks. Countable things exist and so do the operations used to manipulate them. Order of operations is another level of abstraction away from the "reality" of math and helps with unambiguous communication, thus if you fail to communicate unambiguously it's your problem, not reality's.

Again, you and many others here are adding parenthesis and changing the argument. I gave you a situation where you are taking half of six and then multiplying by (2+1). I don't need to alter the expression to model the situation.

Syntax is important. If you type 6/2x into your calculator, it assumes you are doing (6)/(2)*(x). You have to create a fraction or use parenthesis otherwise. Just like the famous "Let's eat Grandma," if you don't put in the comma after eat, sure you can argue that the sentence makes more sense that you are inviting Grandma to dinner rather than serving her for it, but syntactically that's not what you wrote.

If you want 6/(2x), write that, or actually create a standard fraction with 6 in the numerator and 2x in the denominator. The slash is not equivalent.

I teach grammar. This is not the same. "Let's eat Grandma" has one meaning. "Let's eat, Grandma" has another. Neither is ambiguous. They're just different things. It's only ambiguous if you don't know how to read.

The math problem is only ambiguous if you don't know how to read.

The slash is equivalent. No one is going to take the time to return-key down to create a horizontal dividing line.

6

___

2x

That is unnecessary and cumbersome. Just use '/' to symbolize the numerator/denominator line.

2600 bro wrote:

Just Another Hobby Jogger wrote:

The same is true about the symbols themselves. People have just agreed what signs they use for addition, subtraction, multiplication and division. Nothing objective there. For that matter, numbers are also what people agreed. Why is 2+3=5? Because that's what people agreed.

So does 2+3 =6 become the standard if enough people insist it is?

This is a big philosophical question.

I disagree. Numbers as concepts exist independent of the symbols used to write them. Cavemen could see that two rocks + two rocks = four rocks. Countable things exist and so do the operations used to manipulate them. Order of operations is another level of abstraction away from the "reality" of math and helps with unambiguous communication, thus if you fail to communicate unambiguously it's your problem, not reality's.

Platonism - except your argument isn't quite correct.

Anyways, just checking in to see if anyone has pointed out that "order of operations" isn't really a construct even belonging to math (unless you count like, abstract syntax trees or something as math). You're basically arguing rules of infix notation.

There is no Peano axiom for this, so it really is a language question.

a) It's the same because the intention is not equivalent to the written meaning. You may intend 6/(2(2+1)), but you didn't write it.

b) That it is cumbersome doesn't mean it doesn't have significance. Also, if that's what you mean, just add the parentheses. Much easier than wasting time with a denominator.

c) 6/2+4....does that equal 1? That is, am I to infer that 2+4 is the entire denominator? Of course not. So when you take a shortcut by using a slash, it requires you to clarify when you are terminator your division step. If no clarification, then we assume you are not including further digits like the 4 in your denominator.

d) Glad you teach grammar...I teach math.

2600 bro wrote:

I disagree. Numbers as concepts exist independent of the symbols used to write them. Cavemen could see that two rocks + two rocks = four rocks. Countable things exist and so do the operations used to manipulate them. Order of operations is another level of abstraction away from the "reality" of math and helps with unambiguous communication, thus if you fail to communicate unambiguously it's your problem, not reality's.

Numbers as concept exist independent of symbols. But symbols are human creation. Why do we write 2 instead of II or 二? Because we have agreed to use Arabic numerals. Why is it 2 and not 10? Because we have agreed to use decimal system instead of binary.

So while two rocks plus two rocks are four rocks, we write it as 2+2=4 instead of II + II = IV or 二 + 二 = 四, or 10 + 10 = 100 as a matter of mutual agreement.

Pre 1917 = 1.

Post 1917 = 9.

They had order of operation wrong back then.

Just Another Hobby Jogger wrote:

2600 bro wrote:

I disagree. Numbers as concepts exist independent of the symbols used to write them. Cavemen could see that two rocks + two rocks = four rocks. Countable things exist and so do the operations used to manipulate them. Order of operations is another level of abstraction away from the "reality" of math and helps with unambiguous communication, thus if you fail to communicate unambiguously it's your problem, not reality's.

Numbers as concept exist independent of symbols. But symbols are human creation. Why do we write 2 instead of II or 二? Because we have agreed to use Arabic numerals. Why is it 2 and not 10? Because we have agreed to use decimal system instead of binary.

So while two rocks plus two rocks are four rocks, we write it as 2+2=4 instead of II + II = IV or 二 + 二 = 四, or 10 + 10 = 100 as a matter of mutual agreement.

Yes everyone agrees on using Indian numerals but there's not one agreed upon way to parse the expression 6÷2(1+2) as has been pointed out in lots of ways.

Prof. Racket wrote:

2600 bro wrote:

This is a big philosophical question.

I disagree. Numbers as concepts exist independent of the symbols used to write them. Cavemen could see that two rocks + two rocks = four rocks. Countable things exist and so do the operations used to manipulate them. Order of operations is another level of abstraction away from the "reality" of math and helps with unambiguous communication, thus if you fail to communicate unambiguously it's your problem, not reality's.

Platonism - except your argument isn't quite correct.

Anyways, just checking in to see if anyone has pointed out that "order of operations" isn't really a construct even belonging to math (unless you count like, abstract syntax trees or something as math). You're basically arguing rules of infix notation.

There is no Peano axiom for this, so it really is a language question.

Well notation and communication is certainly a part of math. A mathematician that cannot express their thoughts doesn’t get much done.

You’re being a bit try-hard.

As far as I know there’s nothing that “does math” without acting through a symbolic intermediate.

Even action potentials in the brain use sodium and potassium flux as a symbolic languages to do their signal integration.

Maybe I disagree with myself now.

Of course it's one.

How about this:

Prove x^n +y^n does not equal z^n for any integer value of greater than 2 for n and any integers for x,y, and z.

Input this to any modern ( 1950<) calculator and you get the CORRECT ANSWER which is 9. The answer 1 came from older historical usage of order of operations. SIMPLE RULE: PEMDAS ( If same precedence - left to right)!!

Everyone who thinks 9 is not an answer should go to school or fuKK himself

hill_runner wrote:

Input this to any modern ( 1950<) calculator and you get the CORRECT ANSWER which is 9. The answer 1 came from older historical usage of order of operations. SIMPLE RULE: PEMDAS ( If same precedence - left to right)!!

Everyone who thinks 9 is not an answer should go to school or fuKK himself

That was my take-away as well, after listening to an explanation on you-tube , which was very specific about the historical interpretation once being prevalent, but now no longer generally accepted.

I also very much agree with the posters who say that the formula is ambiguous, almost as if it were purposefully written to be so.

Rippling Hysteresis wrote:

b) That it is cumbersome doesn't mean it doesn't have significance. Also, if that's what you mean, just add the parentheses. Much easier than wasting time with a denominator.

c) 6/2+4....does that equal 1? That is, am I to infer that 2+4 is the entire denominator? Of course not. So when you take a shortcut by using a slash, it requires you to clarify when you are terminator your division step. If no clarification, then we assume you are not including further digits like the 4 in your denominator.

Why waste time using the parentheses when they are unnecessary?

6/2+4 is not the equation we are talking about. Keep on task. 2+4 is not the entire denominator because you didn't write it that way.

The question we are talking about is this:

6 / 2x.

If x = (2+1), then it becomes: 6 / 6 = 1

You could also simply 6 / 2x to 3 / x.

If x = (2+1) then it becomes 3 /3 = 1

You are saying that we need to write it as 6 / (2x) or else no one will understand it. I am saying that you're suggesting something unnecessary.

2600 bro wrote:

Prof. Racket wrote:

Platonism - except your argument isn't quite correct.

Anyways, just checking in to see if anyone has pointed out that "order of operations" isn't really a construct even belonging to math (unless you count like, abstract syntax trees or something as math). You're basically arguing rules of infix notation.

There is no Peano axiom for this, so it really is a language question.

Well notation and communication is certainly a part of math. A mathematician that cannot express their thoughts doesn’t get much done.

You’re being a bit try-hard.

I have a PhD in math. It was literally my job at one point in my life

seattle prattle wrote:

hill_runner wrote:

Input this to any modern ( 1950<) calculator and you get the CORRECT ANSWER which is 9. The answer 1 came from older historical usage of order of operations. SIMPLE RULE: PEMDAS ( If same precedence - left to right)!!

Everyone who thinks 9 is not an answer should go to school or fuKK himself

That was my take-away as well, after listening to an explanation on you-tube , which was very specific about the historical interpretation once being prevalent, but now no longer generally accepted.

I also very much agree with the posters who say that the formula is ambiguous, almost as if it were purposefully written to be so.

This is because most calculators use infix notation (and postfix too I think).

No one like reverse Polish notation

Prof. Racket wrote:

2600 bro wrote:

Well notation and communication is certainly a part of math. A mathematician that cannot express their thoughts doesn’t get much done.

You’re being a bit try-hard.

I have a PhD in math. It was literally my job at one point in my life

Ok then I’m curious.

Don’t math types invent notation all the time to express themselves? Is this actually not considered “math?”

Are Feynman Diagrams not “physics?”