Order of Operations: Subtle Distinctions

Some questions we have been asked about the order of operations go beyond the what and why, pondering the relationship of the conventions both to theoretical matters (properties of operations) and to practical matters (evaluating and simplifying expressions). We will see here an important distinction between meaning and processes.

PEMDAS vs properties, I: Meaning vs. Value

Our first question is from 2002:

Order of Operations vs. Associative Property

I volunteer to teach a weekly extra-credit math course in my daughter's 5th grade class. Today I taught a lesson on the correct "Order of Operations" in mathematical expressions (i.e.: parenthesis, exponents, multiplication/division from left to right, and lastly addition/subtraction from left to right).  

That very day, her regular math teacher taught a lesson on the associative property, in which operations in expressions only consisting of addition or multiplication can basically be performed in any order, with the same result. In fact, her homework assignment included some mental math problems in which she was to rearrange the order of operations in multiplication problems in order to more easily solve the equation in her head. For example, 4 x (6 x 25) might be rearranged to (4 x 25) x 6, which is perfectly okay under the associative property.

My question is: does this make the associative property an exception to the order of operations, a corollary, or something totally unrelated?  

Thanks for whatever help you can give me and thanks for the invaluable help your site gives me every week as I prepare my lesson plan (and try to keep one step ahead of the students!).

Jeff has a solid understanding of the order of operations, though he has used the word “equation” for “expression”, and the example involves both the associative and commutative properties. His question is a nice example of the sort of issue a teacher might never think of, but that can arise in students’ minds, usually without Jeff’s clarity. I was the first to answer:

The ideas of Order of Operations, and properties of those operations such as commutativity and associativity, are distinct concepts. The former deals with the mere syntax of how we write algebraic expressions; the latter deals with the semantics - what the operations mean. It's as if I taught you that you should say "I ate a pizza" rather than "Me eat pizza" (teaching grammar), and the next day told you that you could change that sentence to "The pizza was consumed by me" (restating the same fact using different words and word order). These are not contradictory, but two different levels of language understanding.

My comparison to aspects of language is not quite exact, but the basic idea is there: The order of operations determines what a particular expression means, while the properties determine what other expressions will always give the same results, and therefore how we can modify an expression to make it simpler, without messing anything up. An expression has one meaning, but many possible restatements.

So when we write

    4 * (6 * 25)

that means that we multiply 6 times 25, then multiply 4 by that value. It's just a set of instructions for calculating. But when we say that

    4 * (6 * 25) = (4 * 6) * 25

that says that if we make two different calculations, the results will be the same. This is an equation, asserting that this fact is true. And if we say that FOR ALL a, b, and c,

    a * (b * c) = (a * b) * c

this is a theorem, presenting a fact that we can then make use of whenever we want. It says that because of the ways numbers work, when we see this particular arrangement of operations, we can safely rearrange the order in which we do things, and will still get the same answer. It doesn't mean that these expressions don't mean different things (in terms of the operations they say to do), only that their values will always be the same when we carry out those operations.

An expression explicitly says to do a particular calculation in a particular order; but the expression as a result has a particular value, which can also be obtained in other ways. The heart of algebra is the ability to modify expressions and equations.

So, combining the two concepts, if we see an expression like

    4*6*25

without parentheses, although we know that the rules of grammar say we should multiply from left to right, first doing 4*6 and then 24*25, the rules of algebra say we can change the order to make the work easier. We can use the commutative property (swapping the order of the 4 and 6) and say that the expression has the same value as

    6*4*25

and then using the associative property we can do this by first multiplying 4*25 to get 100, then multiplying by 6 to get 600.

The Order of Operations rules told us what this expression _meant_; the properties of multiplication told us tricks we could use to do the calculation (or, later, to rearrange an equation in algebra).

We could say that PEMDAS is about anatomy, while the properties are tools for surgery.

Left-to-right as training wheels

Simultaneously, Doctor Roy was writing, from a very different perspective:

The left to right rule really only applies to division and subtraction. This is because they don't satisfy the associative property of equality. To be really precise, it is because we are really "loose" about the definition of subtraction and division. If we define subtraction properly, it is adding by an inverse. For example, (3 - 5) + 7, is really 3 + (-5) + 7.  By writing it this way, we can rewrite to get 3 + ((-5)+ 7), since addition is associative. We get the same result as before. 

The same applies to division. By dividing, we are really multiplying by a multiplicative inverse. For example 5/3 is really 5 * (1/3).  Written this way, we can use the associative property.

These comments are closely related to my thoughts on the left-to-right rule in Order of Operations: Common Misunderstandings. The commutative property makes it unnecessary to worry about order among additions or multiplications; and if we think of subtraction and division as merely “adding the negative” and “multiplying by the reciprocal”, the equivalent of left-to-right evaluation is automatic.

The "left to right" rule in the order of operations really exists because elementary school/middle school students are generally too young to understand the abstractions that would make the explanation make sense. They are also generally too young to understand the concept of an additive inverse and a multiplicative inverse.  So the "left to right" rule allows us to keep subtraction and division as they are without changing the basic rules of math that students are taught (i.e. that subtraction and division are really addition and multiplication with different names).  Even now, whenever I see something like (3-5)-2, I have no problems associating 3 + [(-5) - 2] to get the same answer, or even switching the order to get (-5) - 2 + 3. It's all really the same, even for multiplication/division.

In effect, with maturity you just see \(3\ -\ 5\ -\ 2\) as \(3 + -5 + -2\), and have no need to think about order. But students new to arithmetic are taught addition, subtraction, multiplication, and division as four separate operations, which need such rules.

So, really, by introducing the associative property (and all the other field axioms), we are introducing concepts like additive identity (i.e. negative numbers) and multiplicative identity (multiplying by 1/3 instead of dividing by 3). This is the point when the somewhat artificial "left to right" rule can be abandoned for a more formal explanation.

I suppose I have given a rather long-winded response, but the question deserved a very full answer. Simply put, you can think of the associative property of multiplication and addition to be an exception to the order of operations, if it makes arithmetic easier. But in a more real sense, the order of operations going from left to right was already more an artificial structure that isn't properly needed except when we want to use the very familiar constructs of division and subtraction.

From this perspective, this aspect of the order of operations is a temporary way to make sense of what later becomes a natural implication of properties, which make everything fit neatly together. This also relates to what we have seen about the reasons for the conventions.

PEMDAS vs properties, II: Complementary, not contradictory

We got a similar question in 2005 about the distributive property:

Order of Operations vs. Distributive Property

Doesn't the distributive law contradict the rules of order of operations?  We should do parentheses before multiplication or division, but the distributive law tells us it's ok to solve x(a + b) by a*x + b*x, in spite of the parentheses.  Is this just an exceptional case?

Do properties tell us not to follow the order of operations? I answered this one:

The roles of the order of operations and the distributive property are complementary, rather than contradictory.  That is, each applies to a different aspect of algebra.

The order of operations tells us what an expression MEANS: if we follow the rules, we will correctly evaluate the expression.

Properties like the distributive property tell us how we can REWRITE an expression without changing its value.

This is essentially my idea above.

So if we are faced with an expression like

  a(b + c)

which (taken at face value) MEANS that we add b and c, then multiply the result by a, we know that we will get the same value if we rearrange it and INSTEAD evaluate

  ab + ac

So properties allow us to safely manipulate an expression to make it easier to evaluate, or to solve a problem, knowing that it still has the same value.  That doesn't change its meaning, only how we actually calculate it.

We could say that the order of operations gives us a starting point, and properties tell us where we can go from there.

But it’s a good insight that what the distributive property does is to allow us to change “add, then multiply” (as forced by parentheses) to “multiply, then add”, or vice versa. What it doesn’t do is to allow us to do the addition in \(ab + ac\) before the multiplications.

PEMDAS cooperates with properties

Here is a somewhat related question from 2006:

Associative Property of Multiplication and Grouping Symbols

Will the associative property of multiplication delete grouping symbols? 

Ex: (a x b) x c 
    (b x c) x a
     a x b x c

My algebra teacher says that each algebraic "tool" has certain things it can do.  While you may be able to find a simple answer he always wants to know which tools you used to teach us to synthesize.  I want to make sure I know what each property can do and what it can't.

I would think that it could delete grouping symbols and not affect the answer because 2 x (3 x 4) = 24, (2 x 3) x 4 = 24, 2 x 3 x 4 = 24.

The question, as I read it, is whether the associative property is the justification for not needing to write all the parentheses. Erika is basically right, but there are some subtleties in the interaction between the property and PEMDAS. I answered:

Note that the associative property says

  (ab)c = a(bc)

which does not EXPLICITLY drop the parentheses, just moves them around.

But also note that when we write it without parentheses, as abc, the order of operations tells us to work left to right, so

  abc = (ab)c

by definition.  We multiply ab first, then multiply that by c.

That is, in itself the associative property is not about dropping parentheses, but only about moving them; but the order of operations interacts with this to do more:

So to drop the parentheses in (ab)c, we can just use the order of operations; those parentheses are not really needed!  But to drop the parentheses in a(bc), we have to first apply the associative property to change it to (ab)c, and THEN use the order of operations.

The order of  operations, again, provides the starting point, while the property is the tool for change.

Now notice Erika’s examples:

• (a × b) × c
• (b × c) × a
• a × b × c

There’s more going on here:

The examples you give don't all have the same order for a, b, and c, so you would have to use the commutative property too, for the middle one to be equal to the others.  The first two have their parentheses on the left, so they don't require associativity.

That is, \((ab)c\) and \((bc)a\), by the order of operations alone, can be written respectively as \(abc\) and \(bca\); the commutative property lets us write the latter as \(abc\).

The net effect of associativity and the order of operations is that we can ignore parentheses where only multiplication (or only addition) is involved; we don't normally pay much attention to which we are actually doing, but it's good to notice the details once in a while!

Note in particular that it is the order of operations alone (that is, the mere fact that an expression has a meaning when written without parentheses) that allows us to not write parentheses; it is the associative property that allows us to ignore parentheses in what we read, and to drop them when we rewrite.

How to distinguish subtraction from negation

While we’re talking about subtleties of meaning, let’s look at this rather different question from 2003, which I have found more students have than you might expect:

Negative Number or Subtraction?

When I see a '-' in front of a number, e.g., in 

  2(5-6-3)-8 

how can I tell whether it's supposed to indicate a negative number, or a subtraction?

This sort of question arises most often when students are deep into the order of operations, being given all sorts of expressions to stretch their intuitions. They may be doing a series of problems like Andrea’s, or like \((5)(-3)-2\), and see it as the product of \(5\), \(-3\), and \(-2\). Then they will ask me, why isn’t it that? I don’t think I’ve ever seen this discussed in a textbook – because authors don’t tend to think this way.

I answered:

If it's between two numbers, it tells you to subtract; if what's before it is not a number (or something equal to a number, like a parenthesized expression), then all you can do is take the negative of what comes after. In other words, a "-" is only to be interpreted as a negative when there is no way to take it as a subtraction.

If you can subtract, do it; if you can’t, then take it as a negation. Subtraction has priority.

For example,

  2-3      says to subtract 3 from 2, NOT to multiply 2 by -3.

  2(-3)    says to take the negative of 3 and multiply by 2,
           because you have to evaluate the expression -3
           inside the parentheses first.

  (1+1)-3  says to add 1 and 1, then subtract 3 from the result.

  2-(1+2)  says to add 1 and 2, then subtract that number from 2.

  -(2+3)   says to add 2 and 3, then take the negative of the result.

  2--3     says to take the negative of 3, then subtract that from 2.

           (Some people would require you to write this as
           2-(-3), but there's only one way to interpret it
           anyway, since the second "-" does not follow a number.)

Next time, we’ll go through a long question that couldn’t fit here, dealing with a similar issue of knowing what an expression actually says vs. what you can do in evaluating it. After that, we’ll close out this series on the order of operations with another subtle way in which students sometimes think that a property (the distributive) conflicts with the order of operations.