Operations & Integers

Why everyone needs the same reading order

You grab a $$6$ sandwich and two $$5$ drinks, so the total is $6 + 2 \times 5$ dollars. Read that strictly left to right and you get $8 \times 5 = 40$ — a forty-dollar snack run. Compute the drinks first and you get $6 + 10 = 16$, which is what the cashier will actually charge. Both readings look reasonable, and that’s exactly the problem: a math expression — numbers joined by $+$, $-$, $\times$, $\div$, and powers like $3^2$ — is useless if two people can read it two ways. So math fixed one official reading order, the same way a language fixes its grammar. Every expression means exactly one number; this module is about how to find it.

The order isn’t arbitrary

Here’s the logic hiding under the rule. $2 \times 5$ is packed addition — shorthand for $5 + 5$. And $3^2$ is packed multiplication — shorthand for $3 \times 3$. The convention that makes expressions readable is simply: unpack the densest shorthand first. Powers unpack before $\times$ and $\div$; $\times$ and $\div$ unpack before $+$ and $-$; and parentheses outrank everything, because they’re the writer saying “treat this as one thing.”

Look at the snack bill again with those eyes: in $6 + 2 \times 5$, the $2 \times 5$ is one bundled number waiting to be opened. The $+$ can’t grab the $2$, because the $2$ is already spoken for. That’s why “multiply before add” gives the sensible $$16$ — it’s not a decree to memorize so much as a way of seeing which numbers are already glued together.

Walk one through before the rule

Try this one — every kind of move shows up once:

Written as a ladder, that order is:

Most people remember it as PEMDAS (“Please Excuse My Dear Aunt Sally”). The four letters are the easy part. The part worth drilling is that M ties with D and A ties with S — four operations, but only two priority levels between them.

Before you click each round, say the move out loud. When $12 \div 3 \times 2$ comes up, decide: $\div$ or $\times$? (It’s a tie — so the leftmost one fires.) When $5 + 2^3$ appears, notice the exponent wins even though the $+$ sits first in the line. And in $10 - 4 + 5$, the $-$ goes first for the same tie reason.

Why the classic traps feel right

“Multiply always comes before divide.” The mnemonic itself plants this one — PEMDAS spells M before D, so it looks ranked. But dividing by $3$ is the same move as multiplying by $\tfrac{1}{3}$; they’re one operation in two costumes, and neither can outrank the other. Watch what happens if you believe the ranking: $18 \div 3 \times 2$ becomes $18 \div 6 = 3$. The real reading is left to right — $(18 \div 3) \times 2 = 12$. Same digits, answers four times apart. The identical logic covers $+$ and $-$: subtracting $4$ is adding $-4$, so in $10 - 4 + 5$ you take the $-$ first (getting $11$) rather than regrouping the tail into $10 - 9 = 1$.

”$-3^2$ is $9$.” We say “negative three squared,” and the ear hears $-3$ as one number. But on paper, an exponent grabs only the symbol it’s touching — here, just the $3$. The minus sign means “the opposite of” and applies afterwards: $-3^2 = -(3^2) = -9$. To square the whole negative number you must glue it together with parentheses: $(-3)^2 = (-3)(-3) = 9$.

Watch one worked all the way through

The expression below mixes everything. Before you read each line, predict the next move — parentheses first, then the exponent, then… which of the $\times$ and $\div$ fires first here, and why?

Each step applies exactly one operation — the highest-priority one currently available — and boxes in what just changed. If your prediction and the boxed move ever disagree, that line is telling you exactly which rung of the ladder to revisit.

The one thing to remember

An expression is one number wearing layers of shorthand, and you unwrap the densest layer first: parentheses, then powers, then $\times/\div$, then $+/-$ — and inside a tied layer, you just read left to right like a sentence. When a minus sign is involved, parentheses decide whether it’s part of the number or applied after.

The one rule, in strict order

Worked example

For $20 - 3 \times 2^2 + (6 - 1)$:

Classic traps

Quick reference