PEMDAS: What It Actually Stands For (And Where It Trips People Up)

Please Excuse My Dear Aunt Sally has carried millions of students through algebra — and quietly set up just as many of them to get the right steps and the wrong answer.

Every year or two, a math problem goes viral for no reason other than the comments underneath it. Something like 6 ÷ 2(1 + 2) gets posted, and within hours it's split into two camps who are both completely confident and completely certain the other side is wrong. One side gets 1. The other gets 9. Both sides did "PEMDAS." That's the part that should worry you a little, because it means knowing the acronym isn't the same as knowing the rule.

What Each Letter Actually Stands For

PEMDAS stands for Parentheses, Exponents, Multiplication, Division, Addition, Subtraction — the order in which a multi-step expression should be evaluated. Take 2 × (3 + 4)² − 5. Parentheses go first: 3 + 4 = 7. Then the exponent: 7² = 49. Then multiplication: 2 × 49 = 98. Then subtraction: 98 − 5 = 93. Skip the order, and you'd get a different number every time depending on which operation you happened to do first.

That much is genuinely simple, and it's why the acronym has stuck around since well before anyone's grandparents were in school. The letters map cleanly onto six words, the sentence is silly enough to remember, and the underlying idea — some operations have to happen before others — is one every student eventually needs. The trouble isn't the concept. It's what the acronym leaves out.

The Rule PEMDAS Doesn't Say Out Loud

Look at the acronym again: Multiplication comes before Division. Addition comes before Subtraction. Read literally, that looks like an instruction — do all the multiplying first, then all the dividing, then all the adding, then all the subtracting. It isn't. Multiplication and division share equal priority and are worked left to right in the order they appear. The same goes for addition and subtraction. PEMDAS puts M before D and A before S purely because "PEMDAS" is easier to say than an acronym with no fixed order for four of its six letters — not because one operation genuinely outranks the other.

This isn't a minor footnote. A study reviewing decades of order-of-operations research points to earlier work that tested 381 prospective teachers in the United States on multi-step expressions and found the vast majority made at least one order-of-operations error, with roughly a quarter of the errors matching exactly what you'd expect from reading the acronym as six sequential steps rather than four ranked ones. These weren't students still learning the rule. They were adults about to teach it.

A separate analysis in the Journal of Mathematics Education at Teachers College found the same pattern showing up specifically where multiplication sits next to division, or addition sits next to subtraction, in an expression — exactly the spots where the acronym's fixed letter order and the actual left-to-right rule pull in different directions. The researchers' proposed fix wasn't a better acronym; it was teaching students to physically reorganize an expression before solving it, so the ambiguous cases never come up in the first place.

PEMDAS has four priority levels, not six. The letter order inside tiers 3 and 4 is mnemonic convenience, not a ranking.

The free Mental Math Speed Test is a good way to see how automatic your arithmetic has become — and multi-step problems are exactly where the automatic version of a skill and the correct version of a skill can quietly diverge:

Why the Viral Problems Never Actually Resolve

Back to 6 ÷ 2(1 + 2). Solve the parentheses first: 6 ÷ 2(3). From there, the disagreement isn't about PEMDAS at all — it's about what "2(3)" means. Read it as 6 ÷ 2 × 3, working strictly left to right, and you get 9. Read the juxtaposition — a number sitting directly against parentheses with no operator — as a single fused term, and you'd divide 6 by the whole "2(3)" block first, getting 1. Both readings are internally consistent. What's actually happening is that the expression was written in a genuinely ambiguous shorthand, and different textbooks, calculators, and countries don't all resolve that shorthand the same way. The acronym isn't failing here; the notation is doing something PEMDAS was never designed to arbitrate. Well-written math avoids the ambiguity entirely by adding a second set of parentheses or an explicit multiplication sign — which is exactly why textbook problems rarely produce this fight and social media problems always do.

How to Use It Without Getting Tripped Up

Three habits fix nearly every PEMDAS mistake. First, treat M/D and A/S as two single steps rather than four — when you hit a run of multiplications and divisions, work it left to right in one pass instead of hunting for "the multiplication" first. Second, remember that "parentheses" is really shorthand for grouping symbols in general — brackets, braces, and the line in a fraction all behave the same way, even though none of them are literally parentheses. Third, when an expression looks deliberately ambiguous, as the viral ones usually are, that's a sign the expression is badly written, not a sign you've misunderstood the rule.

Outside the US, the same rule travels under different acronyms — BODMAS and BEDMAS are the same four-tier priority with regional letter swaps, which is worth knowing if you ever compare notes with someone who learned math in the UK or Canada. The rule underneath is identical either way.

Where PEMDAS Fits Among Memory Tricks

PEMDAS belongs to a much larger family of first-letter mnemonics — short devices that compress an ordered list into something pronounceable. ROY G BIV does it for the colors of the rainbow, HOMES does it for the Great Lakes, and SOH CAH TOA does it for the three basic trig ratios. They all trade off the same way PEMDAS does: fast to memorize, easy to forget the fine print underneath. If you want the fuller picture of why these tricks work — and how to build your own for something PEMDAS-shaped in your own life — the mnemonics overview and full memory techniques hub cover the underlying principles.

The order-of-operations habit is really a small piece of a bigger skill — holding a sequence of rules in mind and applying them correctly under a little pressure. That skill is trainable the same way any other cognitive skill is, and it's worth checking where yours currently stands. Cognitive Train's brain tests page has assessments that measure the reasoning and working-memory pieces PEMDAS actually leans on, and the site's full set of free brain training tools is built for exactly this kind of practice — a little, regularly, on the skills that quietly underlie a hundred other things.