# multiplication by juxtaposition and order of operations

#### Ioannes

##### Oh man, I shot Marvin in the face.
It is always multiplication before dividing
Wrong! It's from left to right! (once brackets and exponents are completed)

#### Darthfett

##### Aerospace/Cybersecurity Software Engineer
Xienoph, you are correct so far so that it is "2 * 3", but then you are doing the dividing before your multiplying which is a big no.
It is always multiplication before dividing! And most certainly in this case since you have this:
Code:
``````2 ( 3 ) they belong together, you can't just take away the two and divide it with the other crap.
the two has to be multiplied into the parenthesis before you can remove it, thus making it like this:
6 / 2 * ( 1 + 2 )
= 6 / 2 * ( 3 )
= 6 / 6
= 1

Do you see it? You can't just remove the parenthesis without putting the number two there. Even my damn calculator did not get it and it was really damn expensive, that is probably what is so funny with the equation to begin with, the fact that all calculators get it wrong I mean.``````
You need to read my post, as well as tooltiperror's. Multiplication and Division both have the exact same precedence. You do not pick one or the other, you merely work them left to right. Read also the wikipedia article. Parenthesis only state that operations INSIDE them are done first. 1 * (2) has the EXACT same meaning as 1 * 2.

#### Samael88

##### Evil always finds a way
Wrong! It's from left to right! (once brackets and exponents are completed)
How can I be wrong when I have said nothing different? The 2 before the brackets belong to the brackets, it has to be multiplied in before the brackets can be removed at all.
You can also see the equation like this:
Code:
``````___[U]6[/U]___
2*(1+2)``````

#### Darthfett

##### Aerospace/Cybersecurity Software Engineer
How can I be wrong when I have said nothing different? The 2 before the brackets belong to the brackets
Try adding brackets around a single pair of equations in the correct order. Your mind is playing tricks on you, the rest of that equation is not automatically bracketed together.

6 / 2 * (1 + 2) -> 6 / 2 * 3

Here, if it is ambiguous to you, you can make it clearer by changing all x / y into x * (1 / y), or by grouping parenthesis around the first pair of operands, of the highest precedence (according to the order of operations).

Therefore, after adding 1 + 2, since they are inside parenthesis, you can take the result of 6 / 2 * 3, and either change it to 6 * (1 / 2) * 3, or group the first multiplication or division operands inside parenthesis: (6 / 2) * 3. Either result gives you the correct answer.

#### monoVertex

##### I'm back!
The answer is 9. Multiplication and Division have the same priority, as already stated. You compute the calculations inside brackets then you go from left to right.

6 / 2 * (1 + 2) = 6 / 2 * 3 = 9

I fail to see why would you multiply 2 with 3 first and then do the division. It makes no sense.

Anyway, want to make sure? Put this into code and run it. There's nothing that can bring more proof than this.

EDIT: As the original poster already did. Three different programs gave the same result, how can you guys still argue that? This isn't even a debate, how can you debate solid mathematic laws?

EDIT 2: I read the end of the OP again and noted the actual question. The system is already clear enough. You can't go any clearer than that. Any source that may bring an individual to believe that the answer to that equation is 1 is written by someone that didn't quite get his math right.

#### tooltiperror

##### Super Moderator
Smith, you are wrong, we are right.

Code:
``````6/2(1+2) =

6
----------
2(1+2)

=

6
----------
6``````
The stem of the debate is notation. In math there is only one right answer, but many ways of writing it.

1 + 1 = 2
1 + 1 + 1 - 1 = 2
1 + 1 + 1 + 1 - 1 - 1 = 2

#### Darthfett

##### Aerospace/Cybersecurity Software Engineer
Smith, you are wrong, we are right.

Code:
``````6/2(1+2) =

6
----------
2(1+2)

=

6
----------
6``````
The stem of the debate is notation. In math there is only one right answer, but many ways of writing it.

1 + 1 = 2
1 + 1 + 1 - 1 = 2
1 + 1 + 1 + 1 - 1 - 1 = 2
I may have misspoken when I told Samael88 to reread your post. Yes, PEMDAS is leading, but not because you have to group Division before multiplication, or addition before subtraction. PEMDAS is misleading because for precedence, Parenthesis > Exponents > Multiplication = Division > Addition = Subtraction. This is the way it's stated everywhere, even in Wikipedia.

Also, you realize that your post said absolutely nothing, right? The way you read 6 / 2(1+2) to be not confused is "six halves times the quantity one plus two". Do some research and give some examples/proofs before you respond next.

#### tooltiperror

##### Super Moderator
I don't have to do any research. My math teachers (of whom have degrees in Mathematics) have taught me to treat division as grouping, and I will. What you say to my on these forums does not matter, and regardless of your qualification, I will do math the way my teachers have taught me.

3 / 4 * 2 is not 1.5, it is 0.375, because that is how I've learned grouping is done. That is how I get problems right on my quizzes, that is how I'm graded, and that's how I'll do my math.

Code:
``````6/2(1+2) =

6
----------
2(1+2)

=

6
----------
2(3)

=

6
----------
6

=

1``````
If you are using the standard I have been taught with, that division indicates fractions and therefore they must be treated specially, then the answer is 1.

If you have been taught with the standard that Multiplication and Division are done left to right with no precedence given to either, then it is 9.

(is it worth noting my calculator spits out "6/2(1+2) = 1"?)

#### Darthfett

##### Aerospace/Cybersecurity Software Engineer
I don't have to do any research. My math teachers (of whom have degrees in Mathematics) have taught me to treat division as grouping, and I will. What you say to my on these forums does not matter, and regardless of your qualification, I will do math the way my teachers have taught me.

3 / 4 * 2 is not 1.5, it is 0.375, because that is how I've learned grouping is done. That is how I get problems right on my quizzes, that is how I'm graded, and that's how I'll do my math.

If you are using the standard I have been taught with, that division indicates fractions and therefore they must be treated specially, then the answer is 1.

If you have been taught with the standard that Multiplication and Division are done left to right with no precedence given to either, then it is 9.

(is it worth noting my calculator spits out "6/2(1+2) = 1"?)
It depends on what calculator you are using. According to the wikipedia article I linked earlier, Windows Calculator will work in different ways, depending on if you have scientific mode on or not. It also states that with physical calculators, you will get different answers, depending on if the calculator has a stack or not (e.g. if it is implemented using a stack, you will get the answer following order of operations, if it does not, you might get a different answer, depending on the equation).

Also by your rules, the equation 1 - 2 + 3 might be ambiguous. I interpret this to have the answer 2, but by your definition of grouping, it would be 1 - (2 + 3) = -4.

My question is, why would you go against all of academics because of what you believe one of your teachers taught you? You won't go far in life if you only believe what you're first taught. I'm not going to argue the point, but my personal belief is that you misunderstood, or weren't clear on the topic with one of your teachers, and that your version of the order of operations has not been addressed by any of your other teachers.

#### monoVertex

##### I'm back!
I don't have to do any research. My math teachers (of whom have degrees in Mathematics) have taught me to treat division as grouping, and I will. What you say to my on these forums does not matter, and regardless of your qualification, I will do math the way my teachers have taught me.

3 / 4 * 2 is not 1.5, it is 0.375, because that is how I've learned grouping is done. That is how I get problems right on my quizzes, that is how I'm graded, and that's how I'll do my math.

Code:
``````6/2(1+2) =

6
----------
2(1+2)

=

6
----------
2(3)

=

6
----------
6

=

1``````
If you are using the standard I have been taught with, that division indicates fractions and therefore they must be treated specially, then the answer is 1.

If you have been taught with the standard that Multiplication and Division are done left to right with no precedence given to either, then it is 9.

(is it worth noting my calculator spits out "6/2(1+2) = 1"?)
You are imagining a parenthesis, where it does not exist. In my entire education, which has been largely based on Math, no teacher (including College professors) ever said something like what you just said. Multiplication and Division are done left to right, with no precedence.

And as I said, the OP found various sources that state that the way I did it is correct. Because there is no parenthesis enclosing 2*(1+2), the division is done first.

#### Romek

##### Super Moderator
Staff member
I think this isn't as clear cut as some of you make it out to be. Everyone's well aware that multiplication and division take the same precedence, and work from left to right if both operations are adjacent to each other.
So 6/2*3 is 9, not 1. Your calculators can prove it, blah, blah, yeah, etc.

What about using a variable/constant in place of the parenthesis in a calculator or google? There were some examples earlier in this thread with 2y/2y being y^2, not 1.
What about trying 3/2pi? Interestingly enough, multiplication by juxtaposition takes priority there. My scientific calculator confirms this. Funnily enough, 3/2(pi) seems to go back to throwing in a *. I think multiplication by juxtaposition takes priority, just calculators and stuff haven't quite adjusted to it. We need some clearer defined rules on the matter.

2y implies "two y's", not necessarily "2 times y". As similar as that may seem, when solving equations and the sort, there's a difference, and using parenthesis around any such variables is an unnecessary nuisance (especially in the case of 2(2+1), which would become (2(2 + 1)) - ugly!), and It seems that rules that were made to avoid confusion are being taken too literally, and actually causing it.

WolframAlpha on 2y/2y

It actually seems that multiplication by juxtaposition with an unknown/constant involved is stronger than multiplication by juxtaposition otherwise. I think that's just a flaw, and they should both take precedence over regular multiplication.

Edit: I have an old and new calculator, and just tried 6/2(2+1) with both of them. The new one actually gives 1, and the old one gives 9. They both give 9 when I enter 6/2*(2+1).

#### Darthfett

##### Aerospace/Cybersecurity Software Engineer
I think that WolframAlpha is trying to interpret the input as a human would. If you add spaces between everything, it changes the answer. Interesting that the newer calculator gives a different answer with the * missing or added. I'd like to do some more research. #### perkeyone

##### something clever
im editing my op to have the question at the top instead of at the bottom lol
perkeyone said:
my question to you all is...

since this method of representing multiplication can be ambiguous, what changes, if any, do you think could be made to either the syntax of multiplication or (the wording of) the order of operations to avoid said ambiguity.

i am not asking whether you think the answer is 1 or 9. i am not asking for your math professor's expert opinion. i am asking for a set of rules for which such an equation would have only one valid interpretation.
if we could get some examples or some links which support a certain order or that describe a set of rules which are unambiguous, especially if specifically describe the case of juxtaposition, that would be great. (although if they specifically mention it, it is likely they have special precedence for it)

there are at least 6 possible cases of jux:
constant - constant (thats 2*3. doesnt make sense, is not used)
constant - variable [2b](commonly understood to be one logical unit)
constant - parenthesis [2(b)]
variable - variable [ab](commonly understood to be one logical unit)
variable - parenthesis [a(3)]
parenthesis - parenthesis [(2)(3)]

so, the set of rules could describe how to handle each of those cases specifically or describe how to handle jux in general.

#### Samael88

##### Evil always finds a way
If you are using the standard I have been taught with, that division indicates fractions and therefore they must be treated specially, then the answer is 1.

If you have been taught with the standard that Multiplication and Division are done left to right with no precedence given to either, then it is 9.
Exactly, that is what makes that particular equation so funny.

(is it worth noting my calculator spits out "6/2(1+2) = 1"?)
Thanks, that is the only calculator I have heard of yet that have the correct way of handling this.

I would like to add that the only reason why this damn equation is so damn misinterpreted all the time is because of the slash. A division was at first shown in the same way as fractions where, that made it much clearer what should be where in the equation, that is the reason why we have parentheses in the first place.

We might as well start arguing about the pronunciation of the word "tomato" here and come no further.

BUT however I am going to stop writing here because I get so pissed of because there are so many ways to calculate things in math, there should be strict guidelines like we had when I learned it(we where not allowed to use the slash without using parentheses).

My last word on this will be that tooltiperror and I are right on this one from our point of views, and honestly it is the most logical way to calculate it as well since if we where to put it in our way with the answer being nine it would look like this:
Code:
``````_[U]6[/U]_
2  * (1+2)``````
and to me that just look downright weird.

#### ertaboy356b

##### Old School Gamer
I had this kind of problem once.. The problem was the calculator.. I tried solving it the MDAS way.. then when I tried using the calculator, It gave me another answer.. Since the professor relies on the calculator, I got the wrong answer XD..

#### Prometheus

##### Everything is mutable; nothing is sacred
Staff member
6/2(1+2) == 6/2*(1+2)
PE(MD)(AS)
6/2*(3)
3*3
9

There seems to be some mis-teaching which said multiplication took priority over division, but in actuality they have the same priority.

• 1 person

#### Samael88

##### Evil always finds a way
PE(MD)(AS)
Please explain those for us who does not have English as our first language.

And no, it is not mis-teaching, you have learned to calculate in a "modern" way, we who see it as the answer being one calculates with the old ways only.
If I am wrong then every formula I have learned in my occupation is wrong and many people are going to die because of it. When it comes to calculations involving power then a nine instead of a one is a big difference.

##### Super Moderator
Staff member
Please explain those for us who does not have English as our first language.
P: Parenthesis (Brackets)
E: Exponential (Indices)
(MD): Multiplication/Division - They both have the same priority so are read from left to right.
(AS): Addition/Subtraction - The same as (MD).

#### Samael88

##### Evil always finds a way
P: Parenthesis (Brackets)
E: Exponential (Indices)
(MD): Multiplication/Division - They both have the same priority so are read from left to right.
(AS): Addition/Subtraction - The same as (MD).
Ah, then it is incorrect since Exponentials are prioritized before brackets.

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