What Does PEMDAS Mean in Math and Why Does Everyone Argue About It?

You’ve probably seen the Facebook posts. Some viral math problem like $6 \div 2(1 + 2)$ starts a civil war in the comments section. One camp screams the answer is 1, the other is dying on the hill that it’s 9. Both sides think they’re geniuses. Both sides are usually half-wrong about how math actually works. At the center of this chaos is a six-letter acronym that most of us learned in fifth grade and then immediately started misusing. So, what does PEMDAS mean in math, and why is it causing so much digital blood to be spilled?

Math isn’t just about numbers. It’s a language. Just like you need grammar to make sure "Let's eat, Grandma" doesn't turn into a felony, math needs a hierarchy to ensure we all read the same "sentence" the same way. PEMDAS is that hierarchy. It’s the Order of Operations. Without it, a simple grocery receipt could be interpreted a dozen different ways.

The Acronym That Rules Your Calculator

PEMDAS stands for Parentheses, Exponents, Multiplication, Division, Addition, and Subtraction. Most people remember it through the mnemonic "Please Excuse My Dear Aunt Sally." It sounds sweet, right? Like Aunt Sally just accidentally bumped into someone at a wedding. But in reality, PEMDAS is a rigid set of rules designed to eliminate ambiguity.

Here is the breakdown of what each letter actually signals in an equation:

Parentheses are the "do this first" signal. They are the loudest voice in the room. If there are numbers shoved inside brackets or parentheses, you handle them before you even look at the rest of the line.

Exponents come next. These are your little "raised" numbers, like squares ($x^2$) or cubes ($x^3$). They represent repeated multiplication, and they take priority over almost everything else.

Then things get tricky. Multiplication and Division are the middle children. People often think Multiplication comes first because the 'M' is before the 'D'. That’s a lie. Honestly, it’s the biggest mistake people make. They are actually on the same level of priority. You handle them as they appear, moving from left to right.

Addition and Subtraction are the final step. Like their older siblings, they share a rank. You don’t do all the plus signs and then the minus signs. You just read the equation like a book—left to right—and solve them as you hit them.

The Left-to-Right Trap

Let’s talk about why people get those viral math problems wrong. It’s almost always because they treat PEMDAS as a strict six-step ladder rather than a four-layer cake.

Think about this: $10 - 3 + 2$.

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If you strictly follow the letters P-E-M-D-A-S, you see "A" (Addition) before "S" (Subtraction). You might add $3 + 2$ first to get 5, then do $10 - 5 = 5$.

Wrong.

Because Addition and Subtraction are equal, you go left to right. $10 - 3$ is 7. Then $7 + 2$ is 9. The difference between 5 and 9 is huge when you're talking about engineering a bridge or, more realistically, splitting a pizza bill.

The same applies to Multiplication and Division. In the problem $12 \div 3 \times 2$, if you multiply first, you get 2. If you divide first (the correct way, since it’s on the left), you get 8. This isn't just "math being weird." It’s a convention. It’s a standard we all agreed on so that we don’t accidentally blow things up.

Why Do We Even Need an Order?

Imagine you’re building a shed. The instructions say: "Paint the wood and nail it to the frame." If you nail it first, then paint, you’re fine. If you paint the wood while it’s in a pile and then nail it, you might have sticky hands, but the shed still looks okay.

Math doesn't allow that kind of flexibility.

In the 16th and 17th centuries, as algebra started becoming more complex, mathematicians realized they needed a system. René Descartes and others helped popularize the use of superscripts for exponents. As symbols like $\div$ and $\times$ became standard, the potential for confusion skyrocketed. By the early 1900s, textbooks began to codify these rules into what we now know as the Order of Operations.

It’s basically a treaty. It's an international agreement that says "we will all process information in this specific sequence."

Variations Around the World

Interestingly, PEMDAS is a very American thing. If you grew up in the UK, Canada, or Australia, you probably learned BODMAS or BEDMAS.

• BODMAS: Brackets, Orders, Division, Multiplication, Addition, Subtraction.
• BEDMAS: Brackets, Exponents, Division, Multiplication, Addition, Subtraction.

"Brackets" is just a different word for parentheses. "Orders" is another word for exponents. Even though the letters are swapped—D before M in BODMAS versus M before D in PEMDAS—the rule remains the same: they are equal in priority. It’s just a regional dialect of the same mathematical language.

The Secret "G" You Weren't Taught

Some modern teachers are moving away from PEMDAS and using GEMA.

Why? Because parentheses aren't the only way to group things. You have square brackets $[ ]$, curly braces ${ }$, and even the bar in a fraction (called a vinculum). GEMA stands for:

1. Grouping Symbols
2. Exponents
3. Multiplicative Operations (Multiplication and Division)
4. Additive Operations (Addition and Subtraction)

GEMA is actually a lot more accurate. It reminds the student that multiplication and division are essentially the same "type" of math, just as addition and subtraction are. It prevents the "Aunt Sally" trap of thinking you must add before you subtract.

Historical Nuance and the "Implied Multiplication" Debate

Here is where it gets really nerdy. There is a concept called "multiplication by juxtaposition." This is when you have a number right next to a parenthesis, like $2(3)$.

Some older textbooks and some specific scientific communities argue that this kind of "implied" multiplication has a higher priority than regular division. This is exactly why that viral problem $6 \div 2(1 + 2)$ is so controversial.

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If you follow modern PEMDAS strictly:

1. $1 + 2 = 3$ (Parentheses)
2. $6 \div 2 \times 3$
3. $3 \times 3 = 9$ (Left to right)

But if you follow the "implied multiplication" rule that was common in the early 20th century:

1. $1 + 2 = 3$
2. $6 \div 2(3)$
3. $6 \div 6 = 1$ (Treating the $2(3)$ as a single unit)

Most modern calculators (like Texas Instruments or Casio) have been updated to follow the strict PEMDAS/left-to-right rule, but if you dig up a calculator from the 1980s, you might get a different answer. This isn't because math changed; it's because our conventions for interpreting symbols evolved.

Real-World Consequences of Getting it Wrong

Does this actually matter in real life? Usually, no. If you’re at a grocery store, you aren't calculating $5 + 2^3 \times (4 - 1)$.

But in computer science, it’s everything.

Excel, Python, and C++ all rely on the Order of Operations. If a software engineer writes a line of code for a banking app that calculates interest, and they mess up the grouping, the bank could lose millions—or worse, take it from you.

Spreadsheets are a classic trap. If you type =10+5/5 into Excel, it will give you 11. It divides 5 by 5 first, then adds 10. If you wanted the average of 10 and 5, you'd need to type =(10+5)/5 to get 3. A single set of parentheses is the difference between a functional budget and a total mess.

How to Master the Order of Operations

If you want to stop being confused by what does PEMDAS mean in math, stop looking at it as a list. Start looking at it as a hierarchy of power.

The "powerful" operations happen first. Exponents are "stronger" than multiplication. Multiplication is "stronger" than addition.

When you see a complex problem, take a breath.

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1. Look for groups. Solve the inside of the parentheses first.
2. Look for "elevated" numbers (exponents) and flatten them.
3. Scan from left to right for any multiplication or division. Knock them out as you see them.
4. Scan from left to right for addition or subtraction.

If you do it in that specific order, you’ll get the right answer every single time.

Actionable Steps for Success

Understanding the theory is great, but applying it is where the rubber meets the road. To truly get comfortable with the order of operations, follow these practical steps:

• Rewrite the equation after every step. Don’t try to do the whole thing in your head. If you solve the parentheses, write out the new, simplified equation on the line below. This prevents "mental drift."
• Circle your "equal priority" blocks. When you see a string of multiplication and division, draw a circle around that whole section and solve it left to right before moving to the addition.
• Test your calculator. Type a classic "trick" problem like $6 \div 2 \times 3$ into your phone. If it gives you 9, it’s using standard Order of Operations. If it gives you 1, you’re using an older or non-standard logic system.
• Use Parentheses in your own writing. If you’re sending a budget or a math problem to someone else, don't rely on them knowing PEMDAS. Add extra parentheses to make your intent crystal clear. Instead of 10 + 5 / 2, write 10 + (5 / 2).

Math is a tool, not a torture device. Once you realize PEMDAS is just a set of "traffic laws" for numbers, the confusion disappears. You stop guessing and start calculating. Next time you see a viral math problem on your feed, you can just smile, know the real answer, and keep scrolling.