What Is PEMDAS?
Learn what the acronym PEMDAS stands for and find out how it can help you remember the order of operations and answer arithmetic and algebra problems correctly.
Jason Marshall, PhD
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What Is PEMDAS?
Today we’re going to talk about an infamous acronym in math that you may know (or at least have once known) as “PEMDAS.” What is PEMDAS you say? Well, I’m glad you asked: It’s the widely agreed upon standard “order of operations” that we use when solving arithmetic and algebra problems. Okay, but what does “order of operations” mean? Again, I’m glad you asked because that’s exactly what we’re going to find out today. So let’s stop asking questions and let’s start answering them!
What is the Order of Operations?
The order of operations in arithmetic and algebra is used to figure out which part of a problem to solve first. It’s easiest to see what this means with an example, so let’s take a look at the problem 2+3×4. Hmm, it looks like there are two ways to do this problem. The first is to do the addition first to get (2+3)x4=5×4=20, and the second is to do the multiplications first to get 2+ (3×4)=2+12=14.
Those two answers—20 and 14—are obviously different! Which means that one of them must be right and the other must be wrong…right? That’s correct. So how do you know which is which? That’s where the order of operations—which you can remember using the acronym PEMDAS—comes into play.
PEMDAS Step 1: Parenthesis
Here’s how PEMDAS works. The first step in solving an arithmetic problem is to first do all of the parts that are inside parenthesis. If a problem doesn’t have parenthesis, you can skip this step. But for a problem like (2+3)x4, the first thing you need to do is solve the problem 2+ 3 since that part of the problem is inside parenthesis. Which, since 2+3=5, means that (2+3) x4 simplifies to 5×4=20.
When you’re reading and writing math, “parenthesis” can look like “( )”, “[ ]”, or even “{ }”. The second style, “[ ]”, are often called “brackets,” and the last style, “{ }”, are often called “braces.” Sometimes you’ll see more than one style used together when several sets of parenthesis are needed in the same problem. For example, you can make a problem like 2 / ((4 + 3) x 6) easier to read by instead writing it 2 / [(4 + 3) x 6]. Using different styles of parenthesis in this problem simply makes it easier to see how many parenthesis you have nested together. In situations like this where you have multiple sets of nested parenthesis, the plan of action is always to work from the inside out.
PEMDAS Step 2: Exponentiation
The second step in solving arithmetic problems is exponentiation. In other words, raising numbers to exponent powers. As with parenthesis, if a problem doesn’t have any exponents, you can skip this step. But in a problem like 2^2 + 3 there is an exponent, and since there are no parenthesis the first thing you need to do is square the number 2. Which means that 2^2 + 3 simplifies to 4 + 3 = 7. As another example, in the problem (3 x 4)^2 + 3, you first need to do the part in parentheses (since “P” comes before “E” in PEMDAS), so the problem becomes (3 x 4)^2 + 3 = 12^2 + 3 = 144 + 3 = 147.
PEMDAS Step 3: Multiplication & Division
The third step in solving arithmetic problems is to do any multiplication or division from left to right across the problem. For example, in 2+3×4, since there are no parenthesis to worry about and there is no exponentiation to do, the first thing you need to do is solve the multiplication problem 3×4 (since “M” comes before “A” in PEMDAS). Which means that 2+3×4 simplifies to 2+12=14. Notice how similar this problem is to the problem (2+3)x4 that we looked at before. But importantly, the answers to the two problems are different since the parenthesis take precedence over multiplication.
PEMDAS Step 4: Addition & Subtraction
The fourth and final step is to do any remaining addition or subtraction from left to right across the problem. For example, in 2+3×4–6, we first need to do the multiplication to get 2 +12–6, and then we need to work from left to right doing the addition and subtraction problems as we encounter them. So doing the addition problem first (since it’s the left-most problem) we find that 2+12–6 is equal to 14–6, and then doing the subtraction gives us 14–6=8.
One thing to notice about PEMDAS is that the ordering of the pair of letters MD and the pair of letters AS don’t actually matter since multiplication has the same precedence as division and addition has the same precedence as subtraction. In other words, instead of MD in PEMDAS we could write DM or instead of AS in PEMDAS we could write SA. While that would turn PEMDAS into either PEDMAS, PEDMSA, or PEMDSA, it wouldn’t really matter since everything would still work the same. The quick and dirty tip to remember is that in PEMDAS you do not do all multiplication, then all division, then all addition, then all subtraction. Instead, you work from left to right across the problem doing all the multiplication and division as you encounter it and then afterwards from left to right again doing all the addition and subtraction as you encounter it.
How to Remember PEMDAS? And Why PEMDAS?
So those are all the rules about PEMDAS that you need to know. But how can you possibly remember all of them? While we’re at it, why do we use PEMDAS and not some other ordering like PEASMD (where the precedence of addition and subtraction are switched with the precedence of multiplication and division)? And how do you use PEMDAS in tricker situations? Unfortunately we’re all out of time for today. So we’ll answer these questions next time.
Wrap Up
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Until next time, this is Jason Marshall with The Math Dude’s Quick and Dirty Tips to Make Math Easier. Thanks for reading, math fans!
Math image, AJC1 at Flickr, CC BY-SA 2.0
About the Author
Jason Marshall is the author of The Math Dude’s Quick and Dirty Guide to Algebra. He provides clear explanations of math terms and principles, and his simple tricks for solving basic algebra problems will have even the most math-phobic person looking forward to working out whatever math problem comes their way.