Order of Operations: Common Misunderstandings

Last time I started a series looking at the Order of Operations from various perspectives. This time I want to consider several kinds of misunderstandings we often see.

Multiplication before division?

Here is a question from 2005 from a teacher, “WRW”:

Confusion over Interpretation of PEMDAS

In telling students to "do multiplication and division IN THE ORDER THEY APPEAR," it seems they want to always do multiplication first.  I think they follow the PEMDAS rule BY THE LETTER, so they want to multiply before dividing.

When doing multiplication first, 8 / 2 * 4 = 8 / 8 = 1 

When doing multiplication and division from left to right, 8 / 2 * 4 = 4 * 4 = 16

I agreed with him:

If you think that students have a tendency to misinterpret the rule, you're probably right; but I think the reason is that PEMDAS is a poorly stated version of the rule, and it is easy to misunderstand it as meaning you do Multiplication, then Division, then Addition, then Subtraction.  That's not what the rule is supposed to mean, but many students don't get past the letters and see the meaning!

It's really wiser to think of subtraction as addition of the opposite, and division as multiplication by the reciprocal, and just leave D and S out of PEMDAS entirely, rather than try to fit them into the rules. But we make the rules for people who aren't ready to see things in a mathematically mature way!  (I myself prefer to avoid PEMDAS altogether, and teach the "rules" in a more natural way that leads into this mature perspective.)

Where some people memorize the rule as “Please Excuse My Dear Aunt Sally”, if I use a mnemonic at all, I make it PEMA: “Please Excuse My Attitude”. It’s just Exponent-stuff, Multiplication-stuff, and Addition-stuff, with Parentheses acting as traffic cop, telling you when to do something other than what the signs say. I’ll include my own way of introducing the concept in a later post on why we need the rules.

But, continuing with an example:

Translating these ideas into the case of multiplication and division, when we write

  8 / 2 * 4

we really mean

  8 * 1/2 * 4

which we can do in any order, since multiplication is commutative; clearly, however you do it, it comes out to 16, not 1.  The problem here is that people tend to see this as if it said

    8
  -----
  2 * 4

which means something different.

Note particularly that if we did the multiplication first in my example, then instead of $$8\times\frac{1}{2}\times 4 = 16,$$ we would be doing $$8\times\frac{1}{2\times 4} = 1.$$ Interpreting it correctly, the only number we divide by is the 2.

Also, seeing it this way allows me to rearrange the expression at will (since multiplication, unlike division, is associative and commutative). If I had, for example, $$7 \div 3\times 15,$$ I could think of it as $$7 \times \frac{1}{3}\times 15 = 7 \times 15 \times \frac{1}{3} =7 \times \left(15 \div 3\right) = 7\times 5 = 35$$ In effect, I’m dragging the division sign around with the number following it!

“WRW” answered,

Thanks so much!  I really like the idea of thinking of division as multiplying by the reciprocal and turning the whole multiplication/division portion into just multiplication.  I'll try that out with my students and see if it helps.  Thanks again!

I didn’t mention there the fact that students outside of America are taught mnemonics like BODMAS, and students there sometimes think Division has to be done before Multiplication. Get them together, and you might have quite an argument!

Addition before subtraction?

Here’s a question from another teacher, Monica, the next year:

Incorrect Application of PEMDAS and Order of Operations

I was working with students on the order of operations today and explained that multiplication and division are done from left to right, as are addition and subtraction.  Apparently, they believe they were taught in the past to do all addition and then all subtraction.  I tried to show examples of why that wouldn't work, but they simply did the problem their way, obtained a different answer and asked why it was incorrect.

Are there any examples or explanations that would clearly explain why they must be done from left to right?

The same reasoning I gave for multiplication applies here, as I explained:

It's impossible to show that they MUST be done from left to right; that is nothing more than a convention we all agree on.  Your class has shown that it makes a difference which order you use; that proves that we MUST make some choice that we can all follow.  What that choice is, is not so definite.  But it makes a lot of sense to go left to right, for the following reason.

You can’t prove that a particular grammar is “correct”, as if nature forced us to use it; every language has a different grammar, and each is correct for its own speakers. What makes a grammar correct is only that it is the same grammar used by other speakers of the language. So you’d have to prove that addition and subtraction are done left to right by showing that all the books do it that way.

But we can see why it was a good idea. It’s the same thing I said about multiplication and division:

We define subtraction this way:

  a - b = a + -b

This allows us to think of any subtraction as an addition; we essentially just attach the negative sign to the number following it, rather than taking it as a different operation.  The subtraction requires no extra rules, just the rules we already have for addition.

If we do this, then

  2 - 3 + 4 = 2 + -3 + 4 = 3

That is the same result we get if we do the operations from left to right (and it doesn't depend on whether we do the ADDITIONS from left to right, since addition is commutative!).  If we did the addition first, we would get

  2 - 3 + 4 = 2 - (3 + 4) = 2 + -(3 + 4) = 2 + -7 = -5

Note that this time, the negative sign ended up applying to ALL the following numbers, rather than just to the one after it.

So doing additions first would mean we are really subtracting everything after a subtraction sign. One benefit of replacing subtraction with addition of a negative, as in the multiplication case above, is to be able to move things around. For example, a common trick to evaluate a string of additions and subtractions more easily is to move all the subtractions to the end: $$5 + 3\ -\ 6 + 2\ -\ 8 + 1 = 5 + 3 + 2 + 1\ -\ 6\ -\ 8 = (5 + 3 + 2 + 1)\ -\ (6 + 8) = 11 – 14 = -3.$$ Without left-to-right operations, we couldn’t do this; we would have to look at the whole expression before rewriting any one subtraction as addition of the negative.

So doing additions and subtractions from left to right makes it easier to transform an expression into one involving only addition; and since addition is commutative and associative, it is MUCH nicer to work with!

The rule, therefore, arises from the wish to make expressions easier to handle.  Without it, a lot of algebra would turn out to be a lot harder.  So your students should thank whoever first made this choice!

I closed by referring to the MD question above:

Now, your student's misunderstanding of the rule very likely comes from the use of PEMDAS or something equivalent, which is meant to be only a summary of the rules.  It sounds as if A comes before S, but that twists the intended meaning of the mnemonic.  See this page for another thought:

  Confusion over Interpretation of PEMDAS
    http://mathforum.org/library/drmath/view/66614.html 

That says essentially the same thing I just said, but about multiplication and division, which is an even bigger problem. (Did you know that in other countries they use BODMAS instead of PEMDAS, so students often think division should be done first?)

For another interesting take on left-to-right operations, see

Left Associativity

Where do negatives fit in?

One of the most common difficulties in evaluating expressions is the mixing of negation with exponents. We have had many questions on this; in fact, I could have included this in the series Frequently Questioned Answers. I’ve chosen to use this question from 2002, whose answer covers most of the ideas we bring up (and refers to several other answers):

Negative Squared, or Squared Negative?

After reading your answer in 

   Exponents and Negative numbers
   http://mathforum.org/library/drmath/view/55709.html 

it seems to me that you're ignoring an important fact:  -3 isn't just -1*3, but a number in its own right, i.e., the number 3 units to the left of zero.  If that's the case, then shouldn't -3^2 have the value -3*-3, or 9?  

If -3 was intended to mean -1*3, then shouldn't it be written that way and not implied?

Thank you for your time.

The answer he referred to is an early one from 1997 where Doctor Ken stated that \(-3^2 = -9\), because it means \(-(3^2)\), not \((-3)^2\). Now, if the convention is that negation is done after exponentiation, then that’s all we need to say. But Tom is arguing from the fact that \(-3\) is itself a number, so it has to be kept together. Does that require us to do the negation first? He has a strong argument (and a common one).

I took the question:

I do recognize that it is possible to disagree on -3^2. Dr. Rick's answer to a similar question,

   Squaring Negative Numbers
   http://mathforum.org/library/drmath/view/55713.html 

mentions this disagreement. 
Like you, he notes that if you think of -3 as a single number, it makes sense for the negation to bind more tightly to the 3 than any operation. That reasoning makes some sense, though I think other arguments are stronger. But I do agree that since there _is_ some reason to read it either way, it is prudent always to include parentheses one way or the other, to clarify your intent, i.e., to write either -(3^2) or (-3)^2.

Minimally, we can say that it is wisest to avoid this form, either because it is easy to read wrongly, or because all the books teach it wrong (depending on your opinion)! In fact, I find that the books I most respect never show such a form (making it hard to find examples to point to!), while others, on the contrary, emphasize it because students always get it wrong unless it is drilled into them.

Occasionally people will try to argue the point based on the behaviors of particular calculators or spreadsheet programs. However, these are really irrelevant, since they all define their own input formats, and programmers (of which I am one) are notorious for choosing what's easiest for them, rather than what is most appropriate for the user. 

I've noted in several answers in our archives that some calculators, and Excel, use non-standard orders of operation without apology. But calculators in particular just don't use standard algebraic notation in the first place.

I’ll be including a link to one of these discussions at the bottom. But the main point is simply that calculators have to follow a convention that suits the way you enter expressions on them, which is different from print. (As calculators have come to display expressions more like type, however, they have been forced to follow conventions more closely.)

We also get questions from people who claim they learned long ago something different from what their children or grandchildren are learning (either the whole PEMDAS business, or some part of it like this one):

There also seems to be a generational difference, with older people (including some teachers) claiming that they were taught to interpret -3^2 as (-3)^2.  

I suspect that what has changed is not the rules governing "order of operation" (operation precedence), but that schools are introducing the issue earlier, before students get into algebra proper. That means that they start by looking at expressions for which it is less clear why the rules make sense. I think you will rarely find examples of "-3^2" in practice, because there is no need for mathematicians to write it. You will find "-x^2" frequently.

Conventions of algebra apply primarily where we have variables; in arithmetic you don’t normally write long expressions with numbers, but just evaluate as you go. Basic four-function calculators were designed for such use, and don’t follow the order of operations. And the conventions make more sense on their home turf (with variables than with numbers):

If you approach the idea starting with numerical expressions like -3^2, you are thinking of -3 as a number and assuming that the expression says to square it. If you approach it first using variables, having first discovered that "-" in a negative number is actually an operator, then it is easier to see why -x^2 should be taken as the negative of the square. So I'll start with the latter, and then it becomes natural to treat numbers the same way we treat variables.

The point here is that in arithmetic, we see the negative sign as part of the number (“the number I’m attached to is negative”); in algebra, we see it primarily as an operator (“negate the thing that follows”). And the convention arises in the latter context. It isn’t primarily meant for use with numbers, but with variables.

Negation as multiplication or addition

Now, in an expression like -x, clearly "-" is a (unary) operator, which takes a value "x" and converts it to its opposite, or negative. The expression "-x" is not just a single symbol, but a statement that something is to be done to a value. As soon as we start combining symbols like this, as in -x^2 or -x*y, we have to decide what order to use in evaluating them.

The trouble is that the "order of operations" rules as commonly taught (PEMDAS) don't mention negatives. So if we are going to go by the rules, we have to figure out how a negative relates to them. Well, there are two ways to express a negative in terms of binary operations.

There is no N in PEMDAS, or even in many fuller explanations of the convention. To see where it fits, we need to think about how its meaning relates to the other operations. How do mathematicians think of negation?

One is as multiplication by -1:

   -x = -1 * x

Treating it this way, clearly

    -x^2 = -1 * x^2 = -(x^2)

That is, since -x means a product, we have to do the exponentiation first.

So if we think of negation as a kind of multiplication, it belongs right in there with MD. And, if you think about it, you’ll realize it doesn’t matter whether we think of it as being done first, last, or left-to-right: If you negate a factor before multiplying or dividing, you get the same answer as if you negate after multiplying or dividing:

$$(-x)\cdot y = -(x\cdot y)$$

The other way to talk about negation is as the additive inverse, subtracting x from 0:

    -x = 0 - x

(This is why the "-" sign is used for both negation and subtraction.) Using this view, we see that

    -x^2 = 0 - x^2 = -(x^2)

In particular, we would like to be able to replace subtraction with negation wherever we find it, and not mess things up: \(x\ -\ y^2 = x + -y^2\), which would not be true if the latter meant \(x + (-y)^2\). This is why the subtraction idea is applicable even when we are not actually subtracting from 0.

So both views of negation produce the same interpretation, which does exponents first, and it is logical to put negation here in the order of precedence.

So if PEMDAS really means PE(MD)(AS), with operations in parentheses being done together, we can extend it as PE(NMD)(AS), where negation is definitely done after exponents and before addition.

But the fact is that there is no authority decreeing these rules; just as in the grammar of English, we get the "rules" by observing how the language is actually used, not by deducing them from some first principles. The order of operations is just the grammar of algebra. So the real question is, how do mathematicians really interpret negatives and exponents combined in an expression?

If you look in books, you will rarely find "-3^2" written out, but you will often find polynomials with negative coefficients. And you will find that

    -x^2 + 3x - 2

is read as the negative of the square of x, plus three times x, minus 2.

So, even though there is not a lot of evidence of usage with numbers, usage in polynomials (with variables) is clearly on the side of negation-after-exponentiation, and we want to be consistent.

I have come to believe that the order of operations is what it is largely so that polynomials can be written efficiently. If "-x^2" meant the square of -x, then we would have to write this as

    -(x^2) + 3x - 2

to make it mean what we intend. Since powers are the core of a polynomial, we ensure that powers are evaluated first, followed by products and negatives (the two ways to write a coefficient) and then sums (adding the terms).

Since we can easily see that this is how -x^2 is universally interpreted, it makes sense to treat -3^2 the same way.

Addition comes last so that a polynomial is a sum of terms; negation goes with multiplication in order to have the same base in every term, without having to use parentheses to avoid accidentally changing a base to \(-x\).

For some other discussions of this issue that haven’t already been mentioned, see

Precedence of Unary Operators

Negative Numbers Combined with Exponentials

Negative vs. Subtraction in Order of Operations

And for a long discussion with a programmer, see

Order of Operations and Negation in Excel

That is the discussion I referred to above, about how calculators and programs have different needs, as well as whether -3 should be thought of as a unitary entity.