What is the answer to 6/2(1+2)

[Superfluous J]

A=3
B=2

1 / AB = ?

You're not going to tell me anybody thinks it should be 2/3.

[Pixophir]

@Vanamonde: I think pembas ist just nonsensical (Edit: ninja'd and clarified by @RKunze, and the list given in their post below is what I recall. But pemdas should really correct this on the entry page). Btw., for those who need more multitude, there's also bodmas, which puts division first :-)

Looking in the first chapter of one of my math books (German) it starts with logic stuff (what's true and false), sets and operations on them, we get commutativity, associativity and distributivity and these are used to explain the rules of doing calculations with the numbers. That's the most basic I have here. I recall that last I saw an offspring dabbling with this they had parentheses, then mult/div, then add/sub. Exponents came later and slid in between parentheses and mult/div.

What's important is that div is the inverse of mult, and sub the inverse of add. That makes them interchangeable in their "level", thus there can't be a precedence. The whole thing is this causes the ambiguity of the expression in the OP. I have actually never seen that before, but when searching I immediately found a vast base of such things in the social networks.

It may also have such a wide spread because of social networks, which tend to share and multiply things like "look what I found", and when looking there actually nothing to see.

Edited July 25, 2022 by Pixophir

[RKunze]

13 minutes ago, Vanamonde said:

The version I learned ordered the operations by strict priority and did not have a left-to-right rule, and at least some of the sources I checked also explain it this way. 

The problem with "strict priority" is that multiplication and division (as well as addition and subtraction) actually have the same priority, so you need an evaluation order to disambiguate

And I just had a look at https://pemdas.info/ to check if they actually teach it differently (would have been a surprise, actually, because that would actually be wrong from a mathematical point of view), but they don't:

Quote

PEMDAS Rank and Priority
Multiplication and Division (MD in PEMDAS) have equal rank in the order of operations and do not need to be in order. Addition and Subtraction (AS in PEMDAS) also have equal rank and do not need to be in order. You could, for example, subtract before you add. The reason they have equal rank or priority is because the operations are inverse or are compliments of each other. Thus, the order of operations could also be interpreted as follows:

1 - Parentheses
2 - Exponents & Roots
3 - Multiplication & Division
5 - Addition & Subtraction

[Dman979]

6/2(1+2) = (6/1)*(1/(2*(1+2))) = 6*(1/(2*3)) = 6*(1/6) = (6/1)*(1/6) = 1

 

The problem is confusingly written if you have trouble remembering the order of operations but the answer to it is 1.

If you disagree, try plugging it into Desmos.

[JoeSchmuckatelli]

49 minutes ago, Vanamonde said:

Are there differences in the way this is taught? The version I learned ordered the operations by strict priority and did not have a left-to-right rule, and at least some of the sources I checked also explain it this way. 

I don't remember Left to Right being part of it when I went to school... but I was also a poor student.

I did hear my kid's teacher say that LtoR is a thing.

[Pixophir]

57 minutes ago, Superfluous J said:

A=3
B=2

1 / AB = ?

You're not going to tell me anybody thinks it should be 2/3.

Exactly that because the fraction bar only spreads over the A, and if there was a multiplication sign between A and B. Because it equals to 1/3*2, which is 2/3. One can confuse that with 1/(3*2), which is 1/6.

@Dman979

https://www.wolframalpha.com/input?i=6%2F2*(1%2B2)

 

Edited July 25, 2022 by Pixophir

[Vanamonde]

About things being the same order of precedence, that shouldn't matter. The whole thing is a set of arbitrary rules merely meant to ensure that everybody reading the same equation comes to the same answer. And the way I was taught, multiplication comes before division and addition comes before subtraction. If another set of rules is being taught, that defeats the purpose of having the rules in the first place. 

[Dman979]

 

[Pixophir]

16 minutes ago, Dman979 said:

6/2(1+2) = (6/1)*(1/(2*(1+2)))

That's wrong, there is no equivalence there. You would have to show that 6/2*3 = 6/(2*3). Good luck :--)

Edited July 25, 2022 by Pixophir

[JoeSchmuckatelli]

1 minute ago, Vanamonde said:

About things being the same order of precedence, that shouldn't matter. The whole thing is a set of arbitrary rules merely meant to ensure that everybody reading the same equation comes to the same answer. And the way I was taught, multiplication comes before division and addition comes before subtraction. If another set of rules is being taught, that defeats the purpose of having the rules in the first place. 

I think the crux of the problem is 'sentence case'.

If it were written

     6

_______

2(1+2)

We'd all get '1'.

 

Thus the sentence case is what's driving the discussion.

Part of my brain is also wanting to get to the part where you have 6/2(3) to get a result that asks 'what operation applies to the information in the parentheses?'... meaning you'd have to do the 6/2 first.  Part of my mind still thinks the way I did it first was correct.

 

I'd have to bug my kid's math teacher to discover the reasoning... but I do recall LtoR.  (Which, as stated, was new to me).

[Pixophir]

@Dman979:

I was curious and plugged 6/2*(1+2) in desmos and it has 9 as the result, just like wolfram alpha.

One must press arrow right to get the cursor out of the denominator, else one types 6/(2*(1+2)), which is another expression.

Try it out ;-)

Edited July 25, 2022 by Pixophir

[JoeSchmuckatelli]

1 hour ago, RKunze said:

The problem with "strict priority" is that multiplication and division (as well as addition and subtraction) actually have the same priority, so you need an evaluation order to disambiguate

... Multiplication and Division (MD in PEMDAS) have equal rank in the order of operations and do not need to be in order... 

And I just had a look at https://pemdas.info/ to check if they actually teach it differently (would have been a surprise, actually, because that would actually be wrong from a mathematical point of view), but they don't:

. 

I read your response after writing what I did above... 

The explanation quote you provided does not seem to answer the Left to Right question... Unless I'm missing something. 

(because if you don't use LtoR you can get different answers??) 

[RKunze]

1 hour ago, JoeSchmuckatelli said:

The explanation quote you provided does not seem to answer the Left to Right question... Unless I'm missing something. 

(because if you don't use LtoR you can get different answers??) 

Exactly. I'm not really sure if "left to right" is just cultural bias on my side, though (because I'm used to reading from left to right).

If you do real math, it's moot anyway, because you would either display that equation with a fraction bar and get either 

6
— ⋅ (1+2) =  9
2

or 

   6
——————— = 1
2⋅(1+2)

 And if the system you are using can't render math properly (like this forum ) and you have to resort to plain text, just use enough parentheses to make it unambiguous: Either
 

(6/2) * (1+2) = 9

or

6/(2*(1+3)) = 1

 

[razark]

Every time I see something like this brought up, I'm amazed.  Were people taught that "pemdas" (or other variants) was a rule or method, rather than a mnemonic device to remember order of operations?

 

Or do people think the solar system is made up of the sun and My Very Educated Mother Just Served Us Nachos, too?

 

 

[Pixophir]

Yeah, its a simplification and partly also a misunderstanding: pemdas are not rules, their are just a help for the untrained. Real world is ludicrously more complicated. Imagine they wouldn't get elementary school math right.

I too think left-to-right is cultural bias. When looking at a properly written and well formed expression one does that sorting into patterns automatically, almost subconsciously, like one would try to understand a written clause or sentence. Of course it depends on the complexity, and there may be a limit for everyone, specifically when machines start to take over in data science or so.

A long fraction bar, a function, an integral, a sum, these are all basic patterns that spring out, like punctuation in a well formed sentence. Sure one can obfuscate the meaning of anything, which can be fun like here. But in serious writing one would make the meaning clear. Look at the first two examples @RKunzewrote 2 posts above. Aren't they neat and clear and leave no question ?

Edited July 26, 2022 by Pixophir

[Nuke]

8 hours ago, Vanamonde said:

Are there differences in the way this is taught? The version I learned ordered the operations by strict priority and did not have a left-to-right rule, and at least some of the sources I checked also explain it this way. 

i don't know but i used to have this "reversed polish notation" calculator i got from an old engineer i knew. it used to confuse the living hell out of me because it worked completely different from what a normal calculator does.  iirc you had to enter your values first before your operator.  i don't know if this has any real advantages or is just a regional thing. 

[Pixophir]

36 minutes ago, Nuke said:

reversed polish notation

Has simple roots from back in the day: it reduces memory usage and uses a very simple data structure (a stack) to represent all its operations. It also speeds operations up because there is no parsing of parentheses like in algebraic notation. There are just operations as they come. Which also puts more "load" on the operator, who has to sort their input prior to typing.

https://www.hpmuseum.org/rpn.htm

https://en.wikipedia.org/wiki/HP-65

Edited July 26, 2022 by Pixophir

[kerbiloid]

49 minutes ago, Nuke said:

i don't know if this has any real advantages or is just a regional thing. 

It isn't.

Spoiler

And a whole PC emulator of them.

http://www.emulator3000.org/c3.htm

A (at least locally) popular programming language for built-in systems.

https://en.wikipedia.org/wiki/Forth_(programming_language)

The "reverse Polish" or "reverse bracketless" notation make it much easier to organize the stack of operations, so was/is very popular to make devices and compilers cheap and simple.

Actually, it was hard and weird for me to get from the "reverse notation" programming to the "normal one" in Basic, so much sticky it is.

"Is " 6/2(1+2)"  
6^2/1^2^+*
or
6^2/1^2^+/

, that's the question.

 

[cubinator]

In MATLAB, "6/2*(1+2)" yields 9. "6/2(1+2)" yields an error.

[Ryaja]

So what do I trust?

Spoiler

https://drive.google.com/file/d/1MejJNkuwlBBgizdb-HU_hhu7NXez17hH/view?usp=drivesdk

Spoiler

https://drive.google.com/file/d/1zKrOhryBdvR3mmJzc-SXn2kKUQKuOeFy/view?usp=drivesdk

Spoiler

If the second link is broken it is a screenshot of the desmos scientific calculator with the same question and it's answer was 1

 

Edited August 5, 2022 by Ryaja

[Rutabaga22]

Isn't this easily solvable with PEMDAS?

Parentheses: 1+2 = 3
The equation is now 6/2(3)
No exponent
Multiplication: 2(3) = 6
The equation is now 6/6
Division: 6/6 = 1
No addition
No Subtraction

Edited August 6, 2022 by Rutabaga22

[kerbiloid]

Summary.

Desmos and PEMDAS can't into operation priority.

[razark]

4 hours ago, Rutabaga22 said:

Isn't this easily solvable with PEMDAS?

No, because PEMDAS isn't a thing, it's simply a series of initials to help you remember the order of operations.

Why does your list of steps include multiplication before division, and addition before subtraction?  Multiplication IS division, so they occur at the same level of priority.  The same as subtraction being addition.

[OHara]

I learned arithmetic in the US in the 1970s, never saw the acronym 'PEMDAS', but was taught effectively P, E, MD, AS where ×÷ share the same priority, computed left-to-right.

Nevertheless, when I see the implied multiplication 1/AB,  I understand it to mean 1/(AB).  In compound units with a product dot in the denominator, like Wb/A·m or W/m·K,  I understand Weber per (Amp-metre) and Watt per (metre-Kelvin), as if multiplication was higher priority than division in that situation.

I suppose humanity could decide on a convention to take P,E,M,D,A,S literally, but I don't recommend it because:
 Calculators and programming languages tend to group ×÷ in the same priority, done left to right, and likewise +− .
 P,E,M,D,A,S  taken literally would give  1 − 2 + 3  =  −4

[Rutabaga22]

1 hour ago, OHara said:

I learned arithmetic in the US in the 1970s, never saw the acronym 'PEMDAS', but was taught effectively P, E, MD, AS where ×÷ share the same priority, computed left-to-right.

Nevertheless, when I see the implied multiplication 1/AB,  I understand it to mean 1/(AB).  In compound units with a product dot in the denominator, like Wb/A·m or W/m·K,  I understand Weber per (Amp-metre) and Watt per (metre-Kelvin), as if multiplication was higher priority than division in that situation.

I suppose humanity could decide on a convention to take P,E,M,D,A,S literally, but I don't recommend it because:
 Calculators and programming languages tend to group ×÷ in the same priority, done left to right, and likewise +− .
 P,E,M,D,A,S  taken literally would give  1 − 2 + 3  =  −4

I am now questioning everything I have ever been taught.