PEMDAS: Mastering the Sequence of Mathematical Operations

In nearly every U.S. middle school, students are taught to remember the simple phrase: "Please excuse my dear Aunt Sally." But why are we apologizing for her? Did she commit a fashion faux pas like wearing white after Labor Day?

The mystery may never be solved. But truthfully, "Please Excuse My Dear Aunt Sally," or PEMDAS, is simply a mnemonic. It's a memory aid used by educators to help us recall information through a memorable rhyme, phrase, or acronym. Let’s dive into how to use this tool for solving equations.

What Is PEMDAS?

PEMDAS is an acronym and mnemonic that represents a set of rules used to clarify the order in which operations should be performed to correctly evaluate mathematical expressions. PEMDAS stands for:

Sometimes, the mnemonic "BEDMAS" is used, where "B" stands for "brackets," and serves the same purpose as "parentheses." The mnemonics essentially convey the same order of operations to reach the correct answer, but they use slightly different terminology based on regional preferences. For example, BEDMAS is more commonly used in Canada, while PEMDAS is prevalent in the U.S.

(Note that multiplication and division are of equal precedence in the order of operations, so the flipped order in BEDMAS doesn't change anything.)

History Behind the Order of Operations

The order of operations — as we understand it in the U.S. today — likely took shape in the late 18th century. By the 20th century, the system became more widely accepted, thanks in part to the boom of the U.S. textbook industry.

In an email, math and science historian Judith Grabiner states that concepts like the order of operations are better viewed as "conventions, like red means stop and green means go, not universal mathematical facts."

"However, once a convention is set," she explains, "the traffic light analogy applies: Everyone must follow the same rules, and those rules must be 100 percent clear and unambiguous."

Math and ambiguity are not a good match.

Why Is PEMDAS Important?

PEMDAS ensures uniformity in mathematical outcomes. Essentially, when different individuals solve the same problem, they follow the same steps and arrive at identical results. Ignoring the proper order of operations could lead to incorrect answers.

Disregarding or altering this sequence can yield varying results, which can be particularly problematic in fields like science, engineering, and finance, where accuracy in calculations is paramount.

Using PEMDAS to Solve Math Problems

Imagine it's finals week, and you are tasked with solving the following equation:

No need to stress. This is where our dear Aunt Sally comes in. Each word in the phrase, "Please excuse my dear Aunt Sally," corresponds to a math term (starting with the same letter), guiding us on which operation to prioritize.

Start with Parentheses

Before tackling the equation, PEMDAS tells us to ask one simple question: "Are there any parentheses?" If the answer is "yes," the first step is to simplify whatever is inside those parentheses.

In this case, we see "2 x 3" inside the parentheses. So, we begin by multiplying 2 and 3, which gives us 6. Now, the equation becomes this:

Alright, now it's time to bring in the exponents! In written form, exponents appear as a small number raised to the upper right of a larger number. Take a look at the 5²? That tiny "2" is the exponent.

In this case, the small number two tells us to multiply 5 by itself. And 5 times 5 equals 25, resulting in this:

Now that we've handled the parentheses and exponent(s), let's move on to the next operations: multiplication and division.

Multiply and Divide

It's important to note that multiplication doesn't always come before division. At least, not always.

Imagine you're working with a different equation that has both multiplication and division symbols. In this case, your task is to handle both operations in order from left to right.

This concept is best demonstrated with an example. If the equation is 8 ÷ 4 x 3, you would first divide 8 by 4, which gives 2. Then — and only then — you would multiply that 2 by 3. Let's now return to our original math problem:

Whoever came up with this equation kept it straightforward; there's no division to deal with, and just one multiplication symbol. Praise be to the exam gods.

Now, let's move on and multiply 6 by 4, which gives us 24.

Time to Add and Subtract

Just like multiplication and division, addition and subtraction are part of the same step. Once again, we perform these two operations in order, from left to right. So, we need to subtract 24 from 9.

By performing this operation, we end up with a negative result of -15.

The number 25, however, is positive. So, the equation currently involves a negative 15 added to a positive 25. Adding these together gives a positive 10.

And that’s the solution to our puzzle.

Exercise caution when working with double parentheses.

Before we say goodbye, there are a few more things you should keep in mind. You may eventually come across a complicated equation filled with various operations nestled between parentheses. It could look something like this:

Don't stress! When dealing with math problems that involve multiple operations, the PEMDAS rule ensures you get the right answer consistently. Simply follow the PEMDAS steps within the parentheses before tackling the rest of the equation.

In this case, you'll start by calculating the exponent (2³), then handle the subtraction inside the parentheses, and lastly, proceed to the multiplication in the following set of parentheses. It's that simple! (By the way, the solution to this equation is 2 1/3, or 2.33 if you prefer decimals.)

Beyond PEMDAS

Here are some additional PEMDAS-like rules and techniques for handling arithmetic expressions: