### Some misunderstandings about order of operations

Order of operations is a set of rules that we use to evaluate mathematical expressions. The rules are as follows:

1. Perform all operations inside parentheses.
2. Apply exponents.
3. Perform all multiplications and divisions, working from left to right.
4. Perform all additions and subtractions, working from left to right.

I like to introduce the topic by showing my students two ways to calculate $3 + 4 \times 5$ and asking them which one is correct. I also show them that different calculators will give different answers to the same problem, depending on the type of logic that they use. A four-function calculator gives the answer 35, but a scientific calculator gives 23. In fact, Windows calculator gives both answers, depending on the mode (standard or scientific).

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

$50 - (3 + 5 \times 2)$

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 $4(-3)$ or $\displaystyle \left( \frac34 \right)^2$. In these cases, the expressions inside the parentheses have already been simplified, but the parentheses still serve a useful purpose.

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,

$\dfrac{a+b}{c+d} = (a+b)/(c+d).$

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:

$13 + 7 - 3^2 = 20 - 3^2 = 20 - 9 = 11.$

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 $1 + 2 - 3 \times 4$.

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

$4x + 3 = 20$

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.