Mathematician: Now that the Physicist and I have answered questions like these ones at least 9 times via email, I figure we should get this horrible topic out of the way once and for all with a short post.
The “order of operations” tells us the sequence in which we should carry out mathematical operations. It is merely a matter of convention (not a matter of right or wrong), giving us a standardized way to decide whether 6/2*3 is (6/2)*3 or 6/(2*3). The convention has been accepted just so that when different people see an equation they can agree on its meaning, even if they hate each other’s guts. You could come up with a different convention, but why bother? Let other people waste their time coming up with arbitrary conventions so that you can merely waste time reading posts about them.
The rules to follow are:
1. Things in parenthesis get computed before things not in parenthesis (this is because of the (little known fact) that parentheses kick ass).
2. Then, exponents get dealt with (including roots). Hence a^b*c = (a^b)*c.
3. Next, multiplication and division get carried out from left to right. So a*b/c*d is ((a*b)/c)*d. Note that division can be thought of as still carrying out a multiplication since a/b = a*(1/b), so it makes a certain sense that neither division nor multiplication takes precedence over one another.
4. Finally, additions and subtractions get carried out from left to right. Subtraction can be thought of as being like addition since a – b = a + (-b) so it makes sense that subtraction is treated on an equal footing with addition.
People sometimes use the expressions PEMDAS, “Please Excuse My Dear Aunt Sally” or “Puke Exhaust Mud Dolls Autumn Sasquatch” to remember these rules. These all stand for “Parenthesis, Exponents, Multiplication and Division, Addition and Subtraction”. It is easy to get confused by these mnemonics though because they don’t capture the fact that multiplication is treated the same way as division (carried out left to right), and addition is treated the same way as subtraction (carried out left to right). That’s why I prefer: “Puke Exhaust Mud And Dolls Autumn And Sasquatch”
One interesting thing to note about the order of operations is that there is nothing interesting about them whatsoever. But that implies there is something interesting about them. This is a contradiction, and hence the order of operations form a paradox. They are also about as fun as getting to watch a fish crawl up the stairs as a reward for answering math questions.
But, that won’t stop me from giving an example. Consider:
a^b*c/d+e-f.
Without a convention, it has many possible interpretations, such as
a^(b*(c/(d+e)))-f
or
(a^b)*((c/(d+e))-f)
which could be horribly confusing for people trying to communicate with each other. But, the correct interpretation according to the rules mentioned above, is:
(((((a^b)*c)/d)+e)-f).
When in doubt about what the orders of operation are for an expression, there is a simple solution which works even better than tattooing PEMDAS to your face. Just paste the expression into the google search bar and hit “search”. For instance, if I paste in:
4^2*3/6+1-5
it gives back
Not only is this the correct answer, but it shows you explicitely the order of operations that were used, so it’s useful for learning.
Google even gets this one right:
4^2*3/6 + 1 – 5/14*3 + 6/10 – 4*2/14*6^3/18 + 14
which is the kind of question little sisters give you when they are old enough to finally sort of get what a mathematician is.
It’s important to note that not all calculators handle the order of operations in the standard way. Why? Because people who make calculators hate children. The preceding sentence was not intended to be a factual statement.
Also, a lot of programming languages use the standard order of operation rules, but then you have exceptions like Smalltalk.
Oh, and in case you were wondering:
a^b^c = 
Regarding that last rule, that a^b^c = a^(b^c). Is that somehow implied by the pemdas convention?
I don’t believe so. Seems like it’s just more made up stuff to memorize.
Does anything interesting follow if we were to change the convention on the order of operations? Is it even valid to consider that?
Order of operations is like syntax in language. If you screw up your parentheses (or forget order of operations), that’s like screwing up your word order or your punctuation. It can drastically change your meaning.
For example:
The panda eats shoots and leaves.
The panda eats, shoots, and leaves.
You can use other conventions if you like, such as: carry out all operations from right to left with no concern for which operation is which. That would give, for example:
a^b*c/d+e-f = a^(b*(c/(d+(e-f))))
But, no, nothing interesting that I know of happens when you do this, besides the fact that you confuse other people.
I always thought this was whole business was just silly, until I came across this article by Compher: http://www.matthewcompher.com/?p=9 You’ve got to read that to see where the real subtle problem comes in with order-of-operations and division.
The best convention is that one that makes the problem easier to solve. A few scientific calculators and programming languages out there, for instance, employ the Reverse Polish Notation, in which all that matters is the left to right sequence of entries, so much so that even the operators themselves come after the numbers they’re operating over.
The pemdas example provided, “4^2*3/6+1-5”, is easily rendered in RPN as “4 2^3*6/1+5-“, but if you wish you can stack the operands first and add the operators later, so that the above can be also validly rendered, say, thus: “5 1 1 6 3 4 2^*//+-“. Which, admittedly, looks confusing, but comes useful in certain cases.
The rule a^b^c = a^(b^c) makes sense to me because (a^b)^c = a^(bc).
This type of questions are the examples shown i Algebra classes to show the importance of verifying the associativity and distributivity of operations before we move away from thinking of them as binary operations 🙂
Just to clear up any debate at all, 6/2(1+2)=9?
Yep! 6/2(1+2)=9.
Thanks! One more quick question,
It seems that the actual question of 6/2(1+2) is a bad question, i.e. not clearly explained, and it should be written as (6/2)*(1+2) or if they mean it the other way it should be 6/(2*(1+2)). Is this right? Or is 6/2(1+2) actually the right way to ask the question?
then should we treat 6/2Y as 6Y/2???
if we let Y=1+2
then 6/2(1+2)= 6/2Y or 6Y/2????
It doesn’t look right, but that is absolutely correct. If you want to be especially careful you can write:
6/2Y = (6)*(1/2)*(Y)
and
6Y/2 = (6)*(Y)*(1/2)
Writing equations in text throws everybody.
Using parentheses does make things more obvious, but they are not necessary, so long as everyone is using the standard convention.
for such a type of calculations use the famous rule BODMAS (bracket order divison multiplication addition subtraction).
-PRASHANT
So, division occurs first before multiplication?
Is this a different problem: 6 / (2(1+2)) when compared to 6/ 2(1+2).
I was taught to see and solve the original problem as if it were written: 6/ (2(1+2)).
I would add 1+2, then multiply the sum by 2 and finally divide into 6, giving an answer of ‘1’.
Division and multiplication are treated like they’re the same thing (really, there are the same thing).
So the first thing you wrote there is the same as 6/2/(1+2) while the second is the same as 6/2*(1+2).
The answer to 6/2(2+1) would have been 9 had the formula been written as follows:
6/2 x (2+1). But it wasn’t.
The answer, as written, is 1.
There is a very important difference between the way in which the two formulas were written. The parentheses themselves! The parentheses denote a relationship to the numbers or formulas within the parentheses to to any number or variable immediately multiplied or divided outside of the parentheses. Together, they form a discrete unit or independent variable. Hence,
6/2(2+1)= 6/[the entire unit determined by the relationship of the 2 to the formula within the parentheses]. In this case, the relationship is multiplication. The 2 is multiplied to the subunit of (2+1). Together, the 2(3) form a discrete unit whose independent relationship to the 6 is by division. Therefore, the correct way to view this problem is 6/[discrete unit] or 6/2(3) or 6/6=1.
Had there been no parentheses at all, the answer could have indeed been 9.
6/2 x (2+1), or [discrete unit of 6/2] whose relationship to the second discrete unit of (2+1) is by multiplication.
6/2=3 multiplied by (2+1) or 3×3=9.
Parentheses are important!
Using the distributive property
6/2(1+2) would be like 6/2(x+y)
6 / 2 * (1+2)
6 / (2*1 + 2*2) would be like 6/(2x + 2y)
6 / (2 + 4)
6 / 6
=1
If I follow PEMDAS in every step correctly:
Step 1: Solve everything inside the (), therefore 6/2(2+1) becomes 6/2(3)
so then I would tick that P to show that I have started with the parentheses.
But as I move onto step 2, I realise that the parentheses are still there :
6/2(3) [note that the brackets have not moved {unsolved}]. then I look at PEMDAS again & ask myself “are there anymore parentheses?”
Therefore
Step 2: Solve (deal) with the parenthese first which will leave me with 6/6. I then look at PEMDAS again & aske myself “are there any parentheses?” & answer “No”
I think the rest of the steps are clear.
@Jack Nichols:
6/2(1+2) = 6/2(x+y) true.
a(b+c)= (ab+ac)
6/2 = a
1 = b
2 = c
Using the distributive property. = (6/2*x+6/2*y) = (3*x+3*y) = (3*1+3*2) = 9
Remi – Former math teacher here – You can’t assign 6/2 to a. If they meant 6/2 to be together they would have written the problem as (6/2)(1 + 2). Order of operations PEMDAS, and I wont get redundant with the solution, but the answer is 1. You must resolve that parentheses! Anyone with the answer of 1 you get full credit! 🙂
use PEDMAS
6/2(1+2) = 1
Solution:
(1+2) =3
6/2(3)
6*1/2(3) = 1
@Sue
Order of operations PEMDAS?
Excell (Math from OpenOffice), and even a Casio Scientific calculator, even the Win calculator in scientific mode, and i bet all smart calculators that will ever fall into my hands, do not know the order of operation cause they all answer 9
For all of you thinking is 1 … have it your way 🙂 … for me is 9.
Case closed.
At first, all I see this equation doing is exposing a flaw in PEMDAS if followed with conventional wisdom. After distribution, the equation reads 6/2(3). Thus, we are at an apparent standstill since neither division nor multiplication comes first. Ayesha’s explanation clears things up considerably by making the equation purely multiplication with 1/2.
Aw, crap. At first Ayesha’s solution looked right, but now not so much. The answer to her equation is 6*1.5=9.
To avoid the “paradox” of 6/2(3), why couldn’t the variable distributive method be used to eliminate the parenthesis all together? 6/2(1+2)=6/2+4=3+4=7
First, my apologies, I did not realize I had responses to my post until today!
@Roger – not so much a flaw of PEMDAS, but mathematicians changing the function of the parentheses! Math is very exact and the mere fact we are having this discussion shows that there is a problem with parentheses. If you follow the distributive rule along with PEMDAS, the answer is 1 but it you ignore the distributive rule it’s 9. Their answer to this dilemma? “It’s not that kind of problem”. WHAT? Not acceptable, it should be the same answer whether you are in grade school or high school!
@Remi – I think this is a discrepancy of the parentheses. Mathematicians need to be clear of its use in a problem. (See my comment to Roger).
When you deal with parentheses, you must remove them. Anytime you have parentheses you need to apply PEMDAS. SO…
6/ 2(1 +2) = 6 / 2(3) the parentheses are not automatically removed at this point and replaced with multiplication. To remove the parentheses, you must multiply by whatever is attached to it and only that. Which is the 2, not 6/2, just the 2. If they meant to multiply by 6/2 they would have written it (6/2)(1 +2).
6/ 2(3) = 6/6=1
If you want to go the distribution route, then you understand that
a(b + c) = ab+bc
You need to understand, only that 2 is distributed, not the 6/2 so…
6 / 2(1 + 2) = 6 / (2*1 + 2*2) = 6 / (2 + 4) = 6 / (6)
since that (6) is not being multiplied by anything (or you can assume it’s multiplied by 1) the parentheses can be removed
6/6 = 1
@Roger – PS – if you apply PEMDAS strictly, and not apply that Left to right stuff (which I always thought was questionable), it is 1
Todd F.’s response is almost identical to what I was going to post. If the author of this equation 6÷2(1+2) wanted to make it clear, doesn’t his/her keyboard have a “x” or a “•”?? If that were the case we would all get 9. But noooooo, he/she had to express the second part of the equation as if it were one figure by which the 6 is divided. I too learned that division and multiplication are equal, so should be done left-to-right–but there has to be a multiplication SYMBOL in place for that!
Here is a thought two actually. First IF you do distributive property you have to treat the 6/2 as 3 and then distribute it out so 3(1+2)= 3+6 =9.
Also just check your work to make sure it is right by plugging in x for any number on the left side of the = sign. For example 6/2(1+2)=9 so 6/x(1+2)=9. Start solving for x. 6/x*3=9 divide by 3 on the left to get rid of it and do the same thing on the left you get 6/x=9/3 which is 6/x=3 and and 6 divided by what number = 3 that’s right 2. Would that work if the answer was one no.
If you go back and check your work with 9 and 1 as different solutions you find that the answer cannot be 9.
The answer is 1:
6 / 2 ( 1 + 2 ) = 1
Think about the order of operations: PEMDAS
Parentheses Exponents Multiplication Division Addition and Subtraction
6 / 2 ( 1 + 2 ) = x
6 / 2 (3) = x
6 / 6 = x
1 = x
Now, if you put this equation just as it is into a calculator (copy and paste it into Google) it will give you the answer 9. However, that is incorrect.
Time to check our work. This is what guaranteed me that 1 was the only correct answer.
Solve the equation again, but this time put the variable in a different spot; then solve it twice: once with 9 and once with 1.
Original equations:
6 / 2 ( 1 + 2 ) = 9 _____________________6 / 2 ( 1 + 2 ) = 1
Substitute variable:
x / 2 ( 1 + 2 ) = 9 ______________________x / 2 ( 1 + 2 ) = 1
Solve equations for x:
x = 9 [ 2 ( 1 + 2 ) ] _____________________x = 1 [ 2 ( 1 + 2 ) ]
x = 9 [ 2 ( 3 ) ] _________________________x = 1 [ 2 ( 3 ) ]
x = 9 [ 6 ] _____________________________ x = 1 [ 6 ]
x = 54 ________________________________x = 6
Since we have the original equation, we already know that the answer should be 6. Since the answer is not x=6 in the first equation (with a solution of 9), 9 cannot be a correct solution. In the second equation (with a solution of 1) the answer is x=6, therefore confirming that 1 is the correct solution to the original equation.
You can use this method again by substituting a variable into all the different spots in the two equations, but the results will remain the same: Incorrect answers with 9 as the solution and correct answers as 1 as the solution. Therefore:
6 / 2 ( 1 + 2 ) = 1
You’ve got it wrong. PEMDAS says:
6 / 2 ( 1 + 2 ) = 1
You didn’t do the multiplication first. You did the division first.
Division comes first, because it creates a numerator (before the division sign), and a denominator (after the division sign). So everything after the division sign gets calculated first.
2(1+2) = 6.
So the numerator is 6, and the denominator is 6.
6/6 = 1
If there was ever a time for the glorious clarification that only a word problem can provide, this is it!
It doesn’t seem to matter. Everyone is so strict on this left to right rule without realizing that math is still a developing concept. If this left to right rule is definitely not a one size fits all. I don’t think we should take the rules of literature and brutally apply them to math.
I need to know the answer. Me and my wife are at each others heads right now. I say 1 and she says 9
6/2(1+2)= 6/2(3)=6/6=1 PEMDAS
The following article explains the view that the expression cannot be evaluated without clarification/context.
http://cnewstead.co.uk/maths/divtimesplus.php
I’m with Sue the math teacher on this one, but it would be nice if the site’s Mathematician and Physicist came back and explained why she is wrong (according to them).
Someone else floated the idea of substituting a variable in but they substituted one for the whole expression, which leaves the parentheses problem intact. What about if x=2, so that 6/x(1+x) ?
I was taught that, absent other brackets or symbols, the number before the parenthesis is “linked” to the parenthetical expression, and must be treated as distributive. I would thus transform this as 6 / (x + x^2). But the Mathematician is saying I should rewrite as 6 * (1/x) * (1+x) which would be equivalent to (6 + 6x) * (1/x) or (6/x) + (6x/x).
Similarly, if we see an expression such as
7xy/4y everyone I know would read that as ” seven-ex-why divided by four-why.”The lack of multiplication symbols links the variables to their constant. Would we really reinterpret that as equal to 7*x*y*(1/4)*y ?
Ultimately, does the parentheses mean simply “multiply,” or does the absence of a symbol between items create a linkage beyond that?
The answer is pretty simple.
First, PEMDAS can’t be taken literally. Its more PE(MD)(AS), because multiplication and division have the same precedence, as well substraction and addition. PEMDAS is one of many mnemonics, others include PEDMAS (source: http://en.wikipedia.org/wiki/Order_of_operations#Mnemonics ) So, if you come with multiplication and division in one equation you simply doing it when they appear.
So, having:
6/2(1+2)
we actually have:
6/2*(1+2)
6/2*3
Because of PEMDAS we know, that MD have same prcedence, so we simply doing division first and multiplication later, and the answer is:
6/2 = 3
3*3 = 9.
To make totally sure that Sue will get it there is simple example. You can change the division to multiplication:
i.e. 4 / 2 = 4 * (1/2) = 2
hence:
6/2*3 = 6 * (1/2) * 3 = 6 * 0.5 * 3 = 9.
ps. English isn’t my first language, sorry for any mistakes.
In PEMDAS the first step is to deal with the equations INSIDE THE PARENTHESIS, not the equations next to it. This is math people. The only thing that makes sense in the world. It is very definitive and the should be no questions in interpretation. When you see 2(3) it is the same as 2*3. When you see 6/2(1+2) it is the same as 6/2*3. The parenthesis are gone and all that is left is to multiply and divide the equation starting from left to right. There is only one right answer and it is 9. Ask any scientific calculator and it will tell you the same. Also if the majority of the world and mathematical/scientific community agrees that the answer is 9 then please don’t confuse more people by trying to convince them otherwise.
6/2(2+1)=9
6/(2(2+1))=1
Note the extra parenthesis.
There is only right or wrong in math. No grey areas, no interpretation, no guessing.
Number are not just “linked” together. They are added, subtracted, multiplied, divided, rooted, or exponentialized (I might of made this word up, but you get it). They are not linked together though. 2*3 is the same as 2(3). Multiplication.
Oliver: “There is only right or wrong in math. No grey areas, no interpretation, no guessing.” The fact that there is so much debate about this would say otherwise.
I think I am going with the linked article that says the statement is ambiguous. I have been looking at problems online and such, and division is shown clearly with a fractional representation; numerator and denominator. No ambiguity. I have the Pi Cubed ap for my IPhone and it doesn’t even have a division symbol. You have to enter the equation with a decided top and bottom. That would clear it all up.
everyone who gets this wrong removes the parenthesis incorrectly.
6/2(1+2)
is not the same as
6/2*(1+2)
look up distributive property.
Used correctly PEDMAS works you get the wrong answer if you repove the () and insert * or x after the 2.
If you solve this without adding the above mention symbols and factor the 2 into the parenthesis to get rid of the parenthesis (the first step in PEDMAS) you get the correct answer.
I agree that parenthesis “kick ass,” but i disagree with
2. Then, exponents get dealt with (including roots). Hence a^b*c = (a^b)*c.
but this
Hence a^b(c) = a^bc
6/2(1+2)—- it’s like a fraction 6/2( which is fraction) and then (1+2) but if you put it 6:2(1+2) then it’s a whole different story
or you can solve it like this
6
———————————————-
2(1+2)
=1
Those of you who say 2(3) should be treated differently than 2*3 or 2*(3) because of the distributive property, that is, you feel 2(3) should be handled as part of the P step in PEMDAS, how would you solve this: 2(1 + 2)^2 ?
If we follow your thinking, then
2(3)^2 (or (2 + 4)^2, depending on when you apply your distributive property) =
6^2 =
36
While the rest of us would say
2 * 3^2 =
2 * 9 =
18.
If you ignore the exponent in favor of applying the distributive property, you’re incorrectly applying the exponent to the leading multiplicative term (2) as well. The exponent ONLY applies to the quantity inside the parenthetical term it qualifies. In other words, my proposed problem is equivalent to 2(1+2)(1+2) = 18, not (2+4)(2+4)=36.
As I see it, the problem with invoking the Distributive property to explain your position is that the distributive property is inherently a multiplicative property and therefore should not be applied until the MD stage of the PEMDAS convention. Therefore, the original problem stands as 6/2(3) = 3(3) = 9
Q.E.D 🙂