# The Order of Operations (PEMDAS/BODMAS) that you learned in elementary school... IS WRONG!!!

Well, okay, it's not incorrect, but it's flawed and by a mathematician's standards: morally wrong.

I'm sure at one point you boringly learned the order of operations. These are the set of rules that tell you whether you should do multiplication before division or addition before subtraction to get the correct answer on your math problem.
1) Parentheses (brackets)
2) Exponents x^x
3) Multiplication 2*2
4) Division 2/2
5) Addition 2+2
6) Subtraction 2-2
7) Get the right answer :)

Except, you don't always get the right answer.

For example: 8-2+1. Is it 5 because 8-3=5? Or is it 7 because 6+1=7?

Is 6/3/3 equal to 2/3 or 6/1?

The issue here is that focusing on the order of operations can lead to ambiguity and obscures the real beauty of mathematics.

A mathematician will tell you that 8-2+1 is actually 8+(-2)+1, which is unambiguously equal to 7 even though the standard order of operations (PEMDAS) taught in the US tell you to add first and gives you 5. But in reality, if you want 5 to be your answer, then you need some parentheses like so: 8-(2+1).

But why is the ambiguity even possible? It's because fundamentally, all these operations are different procedures that start with two numbers and in some way combine them to make a single number. Each operation takes two number, no more, as an input and gives you an output. If you want to be entirely unambiguous then you would have to put parentheses around everything.
It would take something like 1+2+3+4*5-18/3 and make it look like ((1+2)+(3+((4*5)-(18/3)))).
Then, there would be no need to know any order of operations. You would just evaluate the innermost parentheses first and always get the same answer.

But this isn't the only way. You can actually change the order of these operations by changing where the parentheses are like in (1+(2+3)) and ((1+2)+3). But the only way to do this is if you know what the underlying mathematical is.

For example, if you want to multiply the results of 3+4 by 5 or just (3+4)*5 you can either do the addition first and get 7 and multiply that by 5 and get 35. OR you can multiply first as long as you multiply the 5 by both the 3 and the 4 which would yield 15+20. The latter example is distribution and in both cases you get 35.

This even works for exponents like ((3*2)^2) which simplifies to 6^2 and ultimately 36. But you can also square both of them before you multiply which becomes 3^2 * 2^2 or 9*4 and finally 36.

So, the TRUE Order of Operations is this:

1) Parentheses first
2) Learn Math (basically what multiplication, division, exponentiation, and the rest are really doing)
3) Do whatever you want.

All this doesn't mean that we don't have a conventional order of operation in mathematics, but deciding to do multiplication before addition helps us get rid of LOTS of redundant parentheses. Also, learning things like the Associative Property(ies) of Multiplication/Addition helps get rid of a lot more. The parentheses are still there, but they're just implied.

The order of operations learned in school is very different. It's just a set of mechanical instructions that dictate just one of the MANY ways you can use algebra. It locks you in a single path in the beautiful landscape of mathematics. Like a computer, it gives you the right answer but cannot actually give you any insight on what it is that you're actually doing.

So, the order of operations isn't technically wrong, since it generally gives you the right answer, but it is morally wrong because it turns you into a robot.

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Text and Transcript are property of MinutePhysics Productions.

### Comments

The way I learned it, you do either multiplication OR division, then addition OR subtraction, working from left to right. This avoids ambiguity, and as long as one is sharp on the arithmetic, the same answer will be obtained, every time.
The problem with "PEMDAS" as you've shown it isn't that it is wrong, but that it is incomplete and needs to be complemented with additional policies.  I also find parenthesized unary expressions -(....) throw off those following the common towing of the PEMDAS line too.
I don't know if I was taught this, or if I just figured it out, but the way I do algebra, there are no minus operations, only negative terms. So I would approach 8-2+1 as 8 + (-2) +1, so that there is no "order" with regard to addition and subtraction, because there is no subtraction: only the addition of negative terms.
Greg C. Since the "P" of PEMDAS is parentheses, I do not understand your comment "expressions like -(...) throw off those" following the order of operations expressed by PEMDAS. Insert any value inside the parentheses, and then reverse them, as the "P" rule dictates. Lets say (...) = (1 - 5). Then x -(...) = x -(1 - 5) = x + (-1) + (-(-5)) or x + (-1) + 5 or x + 4.

I actually read this post hoping to understand what BODMAS stands for. I get that Brackets replace Parentheses, and since the division IS multiplication, just as subtraction IS addition, there can be no distinction in the order of operations, provided we recall that subtraction of a positive number is equivalent to addition of a negative number, but what does the "O" stand for, that obviously must be equivalent to an exponent?
I find it troubling that you tutor math, and have no clue how to correctly apply PEMDAS. PEMDAS is NOT the rule. It is an acronym to help us remember the order of operations. If you do not understand the actual order of operations, then the acronym is no help at all. First of all, PEMDAS is 4 steps, NOT 6.

P- Parentheses (perform all operations INSIDE parentheses)

E- Exponents

MD- Multiply and Divide as they appear left to right.

AS- Add and Subtract as they appear left to right.

Your first example, 8-2+1, is completely ridiculous! Applying PEMDAS correctly will give you the correct answer 7. Step 4, AS, says add and subtract AS THEY APPEAR LEFT TO RIGHT!

Apply order of operations correctly, and you will arrive at the correct answer!

Wow, I'm impressed, and not in a good way. This whole article is imbecile, at best.
If you can't even get 8-2+1 correct you really can't blame the order of operations for your weak math skills. (even if you add first, WHICH ISN'T what the order of operations states, -2+1=-1, this is like 1st grade math).
Secondly, you obviously haven't got a clue what you are talking about as you completely confuse the order of operations (the basis of all modern math and its rules) with the ACRONYMS created to help remember it (but way to often misunderstood as you so clearly prove here). The order of operations have 4 steps, not 6 as division is just the multiplication of fractions and subtraction the addition of negative numbers. If you actually bothered to learn it and follow it there are no ambiguities, those you imply comes only from ignorance.
You refer to distribution (something that really only applies to algebra when working with variables) and yet doesn't understand it. Distribution DOES NOT break the order of operations or give a different answer than NOt distributing, if that would happen you've made a mistake!
You can still be creative and find paths using associative and commutative properties, neither of those however ever break the order of operations, AS LONG AS YOU HAVE A CLUE WHAT YOU ARE DOING.

LOL, no, Order of Operations is NOT wrong.  Your Understanding of Order of Operations is wrong.  PEMDAS, BODMAS, BIDMAS, Punkt Vor Strich, and my own creation, GENMS all describe the exact same Rules of Math that are over 500 years old, Always apply, and are based on Logic and the History of Math.

• Grouping symbols
• Exponents and roots
• Negation and inversion
• Multiplication and fractions
• Summation of positive and negative numbers

Or Simply Simplify the Terms in the Expression then Sum.  If it's not a plus sign it's a Term.

You can't even get Basic Addition and Subtraction right.  The rule is that the Negative Sign Sticks To The Number On Its Right!  Ever heard of the Commutative Property of math?  It enables us to Move the Terms around in an Expression and always arrive at the same answer.  Therefore 8-2+1=8 + -2 + 1=-2 + 8 + 1=8 + 1 + -2, etc.!

Please, Everyone, Do NOT pay an idiot like this guy to tutor you or your kid!
This would be a good read if the person who wrote it actually understood how the order of operations actually worked. It's...
1. Parenthesis
2. Exponents
3. Multiplication and division (L to R)
4. Addition and subtraction (L to R)

When it comes to 3 and 4 they are don in order left to right. So 8-2+1 would never be 5 when done correctly. Addition doesn't come before subtraction, it's which ever is first left to right after any multiplication or division, if any, is done.
Wow!  Math nerds sure are a snarley lot!  Poor Brice just pointed out a common misunderstanding and got a heap of insults for his efforts.  Lighten up,
The way I learned it, is:

1. Parentheses
2. Exponents
3. Multiplication and division (Left to Right)
4. Addition and subtraction (Left to Right)

However, this still runs into issues. Here is a BETTER example of where order of operations runs afoul.

1+1+1+1/2

On a graphing calculator, you would get 3.5, because the calculator is trained to do order of operations. But to the typical person trying to figure this out, or to a conventional calculator, you get two. Guess what? The dumb calculator is right! You are adding 4, then dividing it by half, effectively averaging it. I can even show you four blocks and taking away half of them.

What is actually done here? 3+(1/2). This is wrong! Order of operations is wrong! And if math actually matters (like, say, you need to determine the correct amount of force needed to stop a train), you lose.

1. Parentheses
2. Exponents
3. All other math from left to right.

If you think a process should come first (for example 3 + (3/2) rather than 3 + 3 / 2), wrap it in parentheses.

3+3/2 is 6/2. 3+(3/2) is 3 + fraction 3/2. These are two different processes, and a formula that decides to confuse them is wrong. Please pass this on!

i really can't believe you people or your teachers actually got through school. i hope you didn't use math on your job. look, of course you do your p's and e's first if you have any then you remember how you learned to read, from left to right. good, because that's exactly how you do math. with no written instructions if there's simply a string of digits and operational symbols ploped down on a line on a piece of paper in front of you work them from left to right taking them one symbol at a time except down the rd. in bldg. #7 and they may take some liberties with shortcuts, skipping "p"'s ect. but this isn't for you, but if your teacher shows you a red apple and says it is blue and the red apple shows up on the test you probably should say it's blue as long as you're aware of what you're doing. my dear aunt sally without instruction , throw it in the garbage where it belongs.
It is somewhat disturbing that you tutor math, yet don't seem to understand how to implement BEDMAS.  (B) (E)(MA)(AS) THOSE ARE the 4 steps. Left to right.
So how does one figure out the order of operations if it read
2X + 6 -1(6÷3)=6× - 2

I could come up with a harder one but I don't have the time. Just curious, tbh. I feel cheated. I used PEMDAS for the SAT, ACT and ASVAB. Not to mention, in life. It must be a small percentage of problems where PEMDAS can't be used.
I think that the PEMDAS acronym is useful when we start to work on equations. The problems arise if the teacher makes the mistake of telling the pupils that this means multiplication comes before addition and addition comes before
subtraction. The word is easily pronounceable which makes it a useful tool. It is just up to the instructor to explain to the students that multiplication and division are a pair and addition and subtraction are a pair. Perhaps the proper way to write the acronym is PE(MD)(AS). Any good instructor should be able to explain that the paired operations are done from left to write. If you didn't have that explained to you then your instructor did a poor job.

"from left to RIGHT." Never post anything until you reread it. I really do know the difference between right and write.

PEMDAS, if taken literally, is completely wrong!!!
Consider:    30 / 5 * 3     and    13 - 7 + 1
If taken literally, PEMDAS says to do complete ALL multiplication before doing any division,
so  30 / 5 * 3  is  30 / 15  =  2    (not 6/3)
and  13 - 7 + 1   is   13 - 8  =  5   (not 6+1).

Please don't argue with ME about the above application of "PEMDAS".
That is exactly what PEMDAS says -- unless you add parentheses to disambiguate "PEMDAS"
by indicating that (MD) is also (DM) and (AS) is also (SA).

If you want it to mean something else, then write it as  "PE(MD)(AS)" or PE[DM][SA];
then, you must figure out how to pronounce those brackets
(and explain why addition and subtraction share the same level in a "precedence table").

A further problem (with PEMDAS and BODMAS) is that neither subtraction nor division is associative,
which makes  "A o B o C"  ambiguous whenever the "o" operator stands for "+" or "*".

Now, I suppose the Left-to-right association rule is OK as a tie-breaker
(and that's what I'm used to assuming in Fortran, C, Java, Perl, etc.)
but it really should be Right-To-Left for exponatiation!

That's why I NEVER use "PEMDAS".

Instead, I teach the different strength levels of arithmetic operators:
0.  Incrementation (i.e. the "NEXT" operator)
1.  Addition (repeated incrementing)
2.  Multiplication (repeated addition)
3.  Exponentiation (repeated multiplication)
(Sometimes. I'll mention tetration, but say not to worry about it!)

When a student understands these strength levels, it is much easier to teach the other rules of exponents by describing "reduction in strength".

Of course, each of these operations has an inverse operator, too (decrement, subtract, divide, root),
and that notion also helps with exponent rules (as well as precalculus!)

The order of operations used throughout mathematics, science, technology and many computer programming languages is expressed here:

exponents and roots
multiplication and division
addition and subtraction

This means that if a mathematical expression is preceded by one binary operator and followed by another, the operator higher on the list should be applied first.

The commutative and associative laws of addition and multiplication allow adding terms in any order, and multiplying factors in any order—but mixed operations must obey the standard order of operations.

It is helpful to treat division as multiplication by the reciprocal (multiplicative inverse) and subtraction as addition of the opposite (additive inverse). Thus 3 ÷ 4 = 3 × ¼; in other words the quotient of 3 and 4 equals the product of 3 and ¼. Also 3 − 4 = 3 + (−4); in other words the difference of 3 and 4 equals the sum of 3 and −4. Thus, 1 − 3 + 7 can be thought of as the sum of 1, −3, and 7, and add in any order: (1 − 3) + 7 = −2 + 7 = 5 and in reverse order (7 − 3) + 1 = 4 + 1 = 5, always keeping the negative sign with the 3.

The root symbol √ requires a symbol of grouping around the radicand. The usual symbol of grouping is a bar (called vinculum) over the radicand. Other functions use parentheses around the input to avoid ambiguity. The parentheses are sometimes omitted if the input is a monomial. Thus, sin 3x = sin(3x), but sin x + y = sin(x) + y, because x + y is not a monomial. Some calculators and programming languages require parentheses around function inputs, some do not.

Symbols of grouping can be used to override the usual order of operations. Grouped symbols can be treated as a single expression. Symbols of grouping can be removed using the associative and distributive laws, also they can be removed if the expression inside the symbol of grouping is sufficiently simplified so no ambiguity results from their removal.

This is an excerpt from Wikipedia.

In short, Division is the inverse operation to multiplication therefore treated at the same level and the same goes for addition and subtraction.

PEMDAS is just an acronym, not to be taken literally. PEMDAS is not the rule itself.

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Bruce-Alan M.
"If you want it to mean something else, then write it as "PE(MD)(AS)" or PE[DM][SA];
then, you must figure out how to pronounce those brackets
(and explain why addition and subtraction share the same level in a "precedence table")."

No, you cannot pronounce the brackets but IAME doesn't exactly roll off the tongue either

You don't want anyone to argue YOUR point because you are closed minded and in your own head KNOW you are right.
Brice, I am sorry to say but your answer post shows a lack of sophistication in your understanding of maths and that you do not understand what order of operations is or what a nmemonic is.<br>
You are making the erroneous assumption that the mnemonic is the rule.<br>
It is not the rule - it is only a memory aid to remember the rules.<br>
Unfortunately mtoo many maths teachers do not make this point clear and students walk away thinking the nmemonic is a rule to follow.<br>
<br>
The mnemonics PEMDAS, PEDMAS, BODMAS, BOMDAS are all equally good mnemonics to remember the rues by.<br>
Other nmemonics are PEMDSA, PEDMSA, BODMSA, and BOMDSA although these are not "good" mnemonics since they are not prounouncable and so not so memorable.<br> \$25p/h

Brice B.

Tutor for High School Level Mathematics and English

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