Why have an order of operations, anyway?

What we refer to as “order of operations” is a set of rules that mathematicians generally agree upon as a convention (which just means “the way that things are usually done”) so there is no confusion about what anyone means when they write a certain numerical expression. For example, if there were no such thing as the order of operations, we could make one expression mean many different things.

For example, consider the numerical expression 6 · 2 + 4 ÷ 2 – 1

If we use our normal order of operations (first all division and multiplication from left to right, then all addition and subtraction from left to right), we get this:

6 · 2 + 4 ÷ 2 – 1
12 + 2 – 1
14 – 1
13

If, on the other hand, we decided that addition and subtraction should come first, and then division and multiplication, here is what we would get:

6 · 2 + 4 ÷ 2 – 1
6 · 6 ÷ 1
36 ÷ 1
36

Or we might decide that we should evaluate all operations as they come, from left to right, with no precedence given to anything. Then we would get:

6 · 2 + 4 ÷ 2 – 1
12 + 4 ÷ 2 – 1
16 ÷ 2 – 1
8 – 1
7

As you can see, it’s important that we agree on what we mean when we write a numerical expression; otherwise, no matter how well we understand math, we won’t be able to talk about it with anyone else because we’ll all be getting different answers!

"Please excuse my dear Aunt Sally."

You probably know our order of operations as PEMDAS:

Parentheses
Exponents
Multiplication
Division
Addition
Subtraction

This acronym is slightly misleading because you might think that multiplication should always be done before division, and addition should always be done before subtraction, but we could just as well write PEDMAS, or PEDMSA, or PEMDSA.

Order of operations is a four-step process:

1. Simplify any expressions inside parentheses.
2. Simplify any number written with an exponent.
3. Do all multiplications and divisions from left to right.
4. Do all additions and subtractions from left to right.

Why OUR order of operations?

Just because our order of operations is a convention doesn't mean it's completely arbitrary.

A good argument for our version of order of operations is the distributive property¹. We say that multiplication distributes over addition. Since multiplication takes precedence over addition in our order of operations, we write it this way:

a · (b + c) = a · b + a · c

However, if we decided on an order of operations in which we did addition first, then multiplication, the distributive property would look like this:

a · b + c = (a · b) + (a · c)

The distribution is much less clear this way, because we’re no longer distributing anything over an expression in parentheses. Exponents distribute over multiplication in a similar way to multiplication distributing over addition:

(a · b)^c = a^c · b^c

Again, if we decided that we should do multiplication before exponents, the distribution would look like this:

a · b^c = (a^c) · (b^c)

You can’t see the distribution as clearly with this order of operations as you can with ours. The distributive property is probably one of the reasons mathematicians decided on the order of operations we use.

1. http://mathforum.org/library/drmath/view/58237.html

Examples

Here are some examples of how to use our order of operations.

4(3 – 1)²

First, I simplify the expression in parentheses:
4(2)²

Then, I evaluate my exponent:
4(4)

Finally, I do my multiplication:
16

8 + 2(6 – 2) ÷ 4

First, I simplify the expression in parentheses:
8 + 2(4) ÷ 4

Then, I do all my multiplications and divisions in order from left to right:
8 + 8 ÷ 4
8 + 2

Finally, I do my addition:
10

6 + 36 ÷ 6 + 36 · 2

First, I do all my multiplications and divisions in order from left to right:
6 + 6 + 36 · 2
6 + 6 + 72

Then, I do my additions:
12 + 72
84

Further reading

Check out these articles at the Ask Dr. Math website:

History of the Order of Operations

More on Order of Operations