Some misunderstandings about order of operations
by David Radcliffe
In this article, I will describe some misunderstandings about order of operations, and suggest a better way of thinking about the topic. Please add your ideas in the comments.
Order of operations is a set of rules that we use to evaluate mathematical expressions. The rules are as follows:
- Perform all operations inside parentheses.
- Apply exponents.
- Perform all multiplications and divisions, working from left to right.
- Perform all additions and subtractions, working from left to right.
I like to introduce the topic by showing my students two ways to calculate 
Many students, and some teachers, have misunderstandings about these rules. There are some fine points of the rules that many books omit, and these omissions lead to confusion. Here are some of these misunderstandings.
1. Order of operations applies within parentheses.
Some students know that they should simplify within parentheses first, but they do not realize that the order of operations also applies while working inside parentheses. For example, in the expression
a student might be unaware that the multiplication must be performed first.
2. Simplifying inside parentheses does not mean eliminating parentheses.
Some teachers describe the first step as getting rid of parentheses. This causes confusion when students are asked to simplify expressions such as 
3. Grouping is sometimes indicated by means other than parentheses.
Parentheses are often used to show grouping, but we also use brackets [ ] or braces { }. A century ago, it was also common to use a vinculum, which is a vertical bar written above a part of an expression. The vinculum is rarely used today, but it persists in the notation for square roots.
Grouping is sometimes implied. When an expression is written as a vertical fraction, then it is implied that the terms in the numerator are grouped together, as are the terms in the denominator. For example,
Expressions in the exponent also have implied parentheses. These parentheses must be entered explicitly when using a calculator.
4. Multiplication and division have equal priority, and must be performed in order from left to right. The same is true for addition and subtraction.
This point is discussed in an excellent blog post by David Ginsburg, so I will not discuss it in this post. Please read his post!
5. Order of operations is an oversimplification.
If we reflect on how we actually simplify expressions, we will realize that we do not always strictly follow the order of operations. Sometimes we perform calculations in parallel, and we may even perform an operation of lower priority first. For example:
This is a violation of the rules of operations, since we performed the addition first. And yet, we know that this violation is harmless; the answer is still correct. Is there a good way to formulate the rules for order of operations to allow for this flexibility?
6. Mathematical expressions are recursive.
Recursion is a difficult topic, but it is crucial for understanding mathematical expressions. Our notation forces us to write mathematical expressions in a linear order, but the true structure is better reflected by a tree. For example, here is the tree structure for 
Expression tree for 1 + 2 − 3 × 4.
It is important that students learn to see this recursive structure. The structure can be shown in other ways, such as circling 1+2 and 3×4.
7. The last operation is the most important.
In beginning algebra, every expression is either a single number, a sum, a difference, a product, a quotient, or a power. The type of the expression is determined by the last operation to be performed. This operation is at the root node of the expression tree.
We use the last operation when we are solving a linear equation. To solve the equation
we first subtract 3 from both sides, because we must undo the last operation, which was adding 3.
The last operation is the key to applying the rules for derivatives in calculus. If the last operation is addition then we use the sum rule, and if the last operation is multiplication then we use the product rule.
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Great post – and very helpful for a new maths teacher to consider the misconceptions held by students :-) Thank you! Your observation about recursion – or the packaged/tree-like structure of expressions is right on the money. I suspect most of us have intuitively grokked this pass-the-parcel game, but well worth being explicit to students about this. Maybe even build an activity presenting the expression and ask students to draw the tree.
I think the tree structure for arithmetical expression should be taught as soon as operations are introduced. There is, on the face of it, nothing hard about it and it relieves the necessity memorize strange rules and work through ambiguous cases. There is evidence that many students do not know how to parse arithmetical expressions correctly.
Parenthetically, if you will pardon the pun, the tree structure is equivalent to a composition of function machines approach in which the (not necessarily commutative) operations are function machines with multiple inputs (2 for addition, multiplication and subtraction)
Your tree is kinda neat, but I would guess that some students would be confused that your multiplication and addition here is on the same level. We, as mathematicians and teachers, know what’s going on, but they might get thrown.
Anyone using this tree might consider modifying it a bit. But I love the tree idea, for sure.
“1. Order of operations applies within parentheses.”…
“It’s often best to treat it as a new separate problem, inside the parens, so you start all over. Generally, work from the inside out”
4. Multiplication and division have equal priority, and must be performed in order from left to right. The same is true for addition and subtraction.
We JUST showed them Commutativity and Associativity of both addition and multiplication. So order matters? Or doesn’t? –><– Best to train them to see all divisions as multiplications (Not 5/2 but 5* (½) ) . Likewise all subtraction should be viewed as adding the negative. (Not 5 – 3 but 5 + (-3) ). Then commutativity and associativity give us LOTS of options.
5. Order of operations is an oversimplification.
Great!. SOO glad some are teaching this. Otherwise, students see us subconsciously violating these rules. 1) Teach it as operator rank, not operator precedence. The operands play the role of 'enlisted', and the operators play the role of officers. 2) Show, with the algebraic calculators, hitting these buttons [3] [+] [7] [-] [3] [x²][=]. Note carefully what the display says at each stage, esp. when you hit the [-]. The mindless robot is already processing that phrase, and has already added 3 and 7, before seeing the rest of the problem. “How does it DO that?”
Reality: We obey operator rank in a local sense, disregarding what is happening elsewhere. Can also talk about unary and binary operators.
If the Adding of negatives is the rule as in 5-3=5+(-3), then wouldn’t the stated above equation 13 + 7 – 3^2 = 20 – 3^2 = 20 – 9 = 11. be 13+7+(-3)^2 or 13+7+9 =29 instead?
One thing I don’t understand is why the order of operations is left to right, rather than right to left (for say division / mutliplication).
When we talk about function composition and matrix operations, it is right to left. Numbers are nothing more than reduced forms of vectors, we can think of them as 1d objects. They expand to vectors, matrices, and finally tensors.
So we are talking about the lowest dimensional mathematical construct which does have some unique properties. But why make the exception of going left to right rather than right to left which is the convention for all higher ordered mathematical objects?
You seem to have missed that the distributive property means that implied multiplication must be done before regular multiplication and division.
That is not true. Distribution is a necessity of Algebra because order of operations cannot be performed in order if there are variables that are unknown numbers. Distribution never has to be performed in order of operations if there are only numbers.
So for you if you start with ‘6/2a’, when you replace ‘a’ by 3 you write 6/2*3 = 3*3 = 9 and not 6/(2*3) = 6/6 = 1 (here writing 2a = (2*3) is similar that saying that the implied multiplication must be done before regular multiplication/division)
Also, where do coefficients fit in the order of operations?
Coefficients are numbers that are being multiplied to a variable (a variable is an unknown number that needs to be solved or a number that can replace the variable)
Example: 2y-3
This means that if a number replaces y then 2y is “2 multiplied to y.”
If 4 replaces y, then the example turns to 2×4-3.
Here by order of operations, 2×4 would be first, which equals to 8, then the next operation would be 8-3. Result: 5
As a side note, the 2 is a coefficient of the linear term 2y and -3 is the coefficient of the constant term (in Algebra, terms are separated by addition or subtraction). “But -3 has no variable, how can it be a coefficient?” you may ask. -3 technically has a variable of y to the zero power. The variable (or any number) to the zero power is equal to 1 and -3 multiplied to 1 is still -3. And if you are wondering why something to the zero power is equal to 1, that is a completely different topic. Search it up on the internet!
so… which is actually correct? 35 or 23?
23 would have to be correct as the order of operations prescribes multiplications and divisions before additions and subtractions. The four function calculator simply evaluates 3+4 once it is entered, then multiplies the sum it calculates by 7. However the scientific calculator gets the entire expression in ram and applies the order of operations finding that 4×5=20 then adding the 20 which replaces 4×5 to the previous addend of 3, to get 23
I taught myself that if I change division to multiplying by the reciprocal and change subtraction to adding the opposite, the result will be correct. And after scanning the internet, I am happy to find others that do the same. But isn’t this the same as performing division before multiplication and subtraction before addition? Answer: yes it is. So why isn’t this taught instead of PEMDAS? Here is what should be taught:
1. Operations within Grouping (parenthesis, brackets, fraction bar, square root… etc.)
2. Exponents
3. Division
4. Multiplication
5. Subtraction
6. Addition
Also, I failed to mention that the engineer who got 1 as the answer kept saying the distributive property can be used with numbers/letters or all numbers. He kept asking the engineer who got 9 as the answer if he understood that this could be done. “Engineer Answer 9” never answered or refuted what “Engineer Answer 1” said, so I was not sure of “Engineer Answer 9s” take on whether or not the distributive property should be used with all numbers as it is used with unknown variables/known numbers.
Thank you,
Shandra Hart
This message actually preceded my shorter message. I deleted some links that I think may have kept the following message from being accepted. Not sure.
Dear Paul Couvrey,
I am an elementary teacher. I majored in English. I am certified to teach regular ed K-5, the arts K-5, and English to speakers of other languages K-12. So…no math degree. I want to understand the issue with the following because so many mathematicians and high school math teachers are saying one of the 3 items I’ve posted below. Please help!
I did not have Facebook when this problem was posted back in 2011.
6/2(1+2)
Note: sometimes displayed with an obelus. Other problem on Facebook: 48/2(9+3)
Some people argued that…
1. The answer is 9: (6/2)*(1+2)=?, (6/2)*(3)=?, (3)*(3)=9. These people followed PEMDAS based on MD having left to right precedence depending on which comes first. I had never heard of PEMDAS until this problem and another like it in July 2017. I graduated high school in 1999. These people said that once the 1+2 was solved inside the parenthesis, that the 2 before the parenthesis is not to be multiplied by the 3. That the student must now start from the beginning (6/2)*(3)=?
2a. The answer is 1: 6/[2(1+2)]=?, *Some simplified inside the parenthesis first and multiplied by the 2 before the parenthesis. In doing so, these people followed PEMDAS literally and also saw 2(1+2) as a unit. 6/[2(3)]=?, 6/[2*3]=?, 6/6=1, I at first, solved it that way until I started to see all of the arguments. People calling others idiots, etc. I was amazed by the anger, yet it was funny to me no matter how wrong, correct, or confused I was by my own participation in trying to solve this problem.
2b. The answer is 1: 6/[2(1+2)]=?, *Some used the distributive property and in doing so…followed PEMDAS literally giving multiplication precedence over division. So, in doing so, they saw 2(1+2) as a unit.
6/[2(3)]=?, 6/[(2*1)+(2*2)]=?, 6/[2+4]=?, 6/6=1
3. That the problem is ambiguous and that mathematicians do not write problems in this way. That they would have added parenthesis to the original problem to avoid the two different interpretations. That if a teacher wanted students to have an answer of 9, then they should add parentheses to show (6/2)*(1+2). That if a teacher wanted students to have an answer of 1, then they should add parenthesis and brackets to show 6/[2(1+2)].
On a YouTube video (can’t recall which), I followed the comments section argument between 2 engineers. They argued over the course of a year. They were insulting each other, but I was still able to see how each dealt with the 2 before the parenthesis.
*The engineer who saw 9 as the answer saw 2 by itself and said that once the inside of the parentheses were added, then the person solving the problem must start at the beginning of the problem and work left to right based on PEMDAS.
*The engineer who saw 1 as the answer saw 2(1+2) as a unit and said that the distributive property must be used. [(2*1)+(2*2)], [2+4], 6. I will say that when I solved the problem, that I first solved for 1+2 then multiplied by the 2 before the parenthesis. [2(3)], 6.
Also, I read through the comments section of the article and found a comment and a reply.
September 20, 2016 at 11:44 pm
“You seem to have missed that the distributive property means that implied multiplication must be done before regular multiplication and division.”–Allen Helmer (I am not sure if Allen was replying to you or someone else in the comments)
“That is not true. Distribution is a necessity of Algebra because order of operations cannot be performed in order if there are variables that are unknown numbers. Distribution never has to be performed in order of operations if there are only numbers.”–Paul Couvrey (This is the reply about the distributive property and whether or not one must use the distributive property if the problem is all numbers)
Because the entire problem is made of numbers, does that mean that the distributive property was not necessary in the way the engineer who arrived at 1 did? Since the problem uses all numbers? Also, I did not use the distributive property, but I still multiplied the 3 by the 2 before the parenthesis.
Anyway, I hope you understand my writing. Please let me know any response to what I have written.
Thank you,
Shandra Hart
7.20.17
Is the hierarchy of operations affected by implied multiplication? What I mean is, if you have the equation: a ÷ b(c), would you still divide “a” by “b” first?….or multiply b & c first?
Your have a misunderstanding of order of operations. It is not a set of steps to perform. Order of operations is a rule of math which describes the relative order of priorities of various operators. It does not tell you to do this first or that first. It does not tell you to perform this from left to right or this before that. It does not tell you to perform anything in any specified order. It just outlines the order of the priorities.
Now from this rule many teaching methods have been devised. Among them PEMDAS and BODMAS. Those teaching methods are a series of step performed in order which, when performed correctly, will ensure that the mathematician will always be abiding by the order of operations.
Although they both mean exactly the same thing I’ll take PEMDAS as an example.
PEMDAS states to do the parentheses first. Order of operations simply states that parentheses have the highest priority. That doesn’t mean they need to be done first.
PEMDAS states to do exponents next. Order of operations simply states that exponents have a higher priority than multiplication & division and addition & subtraction. That does not mean they need to be done first.
PEMDAS states to do multiplication and division next from left to right. Order of operations simply states that multiplication and division have a higher priority than addition or subtraction and are of equal priority. That does not mean they need to be done before addition or subtraction and due to the associative and commutative laws of multiplication it also does not mean you need to evaluate it from left to right.
PEMDAS states to do addition and subtraction last from left to right. Order of operations simply states that addition and subtraction have the lowest priority and are of equal priority. That does not mean due to the commutative law of addition you need to evaluate it from left to right.
The problem I have come across many times when not abiding by the strict steps of PEMDAS but still correctly abiding by the priorities of order of operations is that many people who fail to understand the basic laws of mathematics then claim I am doing it wrong as I MUST do multiplication and division before addition and subtraction and I MUST do multiplication and division from left to right and I MUST do addition and subtraction from left to right. This patent nonsense is caused by people not understanding that PEMDAS is not order of operations and that order of operations is NOT a series of steps to follow.
Hello Fido:
Although I may have selected a different word here and there, this explanation about the differences between what is taught when students make the transition from arithmetic and mathematics — i.e., PEMDAS or BODMAS — is about as good as I’ve seen.
The nonsensical expressions I’ve seen placed on the internet are obviously designed to start an argument between those currently being taught “programmer’s math” and those that have been taught for eons to follow the old school conventions of mathematics. It’s really unfortunate that many have adopted the simplistic philosophy that “easier is better,” so lets come up with a cute mnemonic device and teach everyone that it’s “the law” … but I suppose that the essence of technology has this flawed logic at its very foundation.
Thank you for attempting to set things right.
,
in many situations there exist mistake of order that are observed by prof dr mircea orasanu and prof drd horia orasanu and so followed in many situations ,and thus appear in calculus of series ,or are present in calculus of CONSTRAINTS OF OPTIMIZATIONS with important aspects ,therefore these imposed other relations
Do you still follow PEMDAS even inside parentheses?
yes
The PEMDAS mnemonic isn’t failsafe, so be careful… but, generally speaking, one should always follow the proper Order of Operations — even inside grouping symbols (like parentheses).
I’m still confused about when to completely drop the parentheses. Some problems we keep bringing the parentheses down with the answer….for example 5-2+(4+1)sqared
=5-2+(5)squared……when do I finally get rid of the parentheses?….maybe this isn’t a good example, but I hope you understand what I am saying.
Hello Barbie ~
Although I sincerely hope that you’re no longer confused about this, I wanted to take a crack at your question (just in case). The convention of using grouping symbols — like parentheses — is to establish a priority of the various mathematical operations used to explain how to solve a given problem.
If by “squared,” you mean to say that (4+1) is to be raised to the second power, you would first find the sum of 4 + 1, which is 5 and, next, find the product of 5 squared, which is 25. From there, it’s simple arithmetic: 5 – 2 + 25, which brings one to the ultimate solution: 28.
I’ve seen many discussions on the internet in which students will not budge from the PEMDAS and/or the BODMAS mnemonic(s) they’re being taught in schools. If you wish to excel in any type of science-related career field, you will — literally — have to think outside of the box… or the grouping symbols, in this case.
Many educational practitioners are teaching from established curricula established by God knows who? In my opinion, this doesn’t usually mean that the original source of the mathematics being taught came from a science practitioner. Please bear that in mind.
Thank you for that answer. My apologies I’m so late, I’m just seeing it. So, It took me a few weeks of constantly and slowly going over the problems until it finally clicked. I also noticed that the answer is hardly ever a huge number. I was intimidated for no reason. and yes, thinking outside of the box was definitely a trick I used…I based it on Numerology, then, wala…I understood BASIC algebra…not the harder stuff just yet lol.