Order of Operations

Why we need an agreed order at all

The symbols cannot settle it, so we agree on one order and everybody uses it. An agreement like that is called a convention. Nobody proved it, and a different choice would have worked just as well, as long as everyone made the same one. That is what it buys: a written line means the same thing to its writer and its reader.

The tiers, and why they are ranked this way

The agreed order sorts operations into tiers and works through the tiers from the top down. From first to last:

1. Grouping. Anything inside parentheses $(\;)$, brackets $[\;]$, or braces $\{\;\}$ is evaluated first, working from the innermost group outward.
2. Exponents. Powers are evaluated next.
3. Multiplication and division. These two share one tier and are evaluated together, left to right.
4. Addition and subtraction. These also share one tier, evaluated together, left to right.

Why this ranking and not some other? Nothing in arithmetic forces it. But one choice saves far more writing than the rest, and you can see which by comparing the two readings of $5 + 3 \times 4$.

You met in an earlier lesson that multiplication is a short way of writing repeated addition, so $3 \times 4$ is a short way of writing $4 + 4 + 4$. Take the multiplication first, and the line counts three $4$s with a lone $5$ beside them:

Take the addition first, as $(5 + 3) \times 4 = 32$, and the line counts eight $4$s with nothing standing beside them. The $5$ has stopped being a quantity of its own and has become part of how many copies there are. The two readings describe different collections, so the symbols alone cannot pick between them. Only the agreement can.

It picks the first one, and the reason is practical. The shape we write most often is a few products added together, the way a shopping bill is. Ranking multiplication above addition lets every one of those products be written bare. We get to write $5 + 3 \times 4$ and never have to write $5 + (3 \times 4)$. The other ranking would work perfectly well if everyone used it, but then almost every bill would need parentheses.

Exponents sit above multiplication for the same practical reason. A power is a short way of writing repeated multiplication. In $2^3$ the small raised $3$ counts how many copies of $2$ are multiplied together:

Read $3^2$ the same way. It is $3 \times 3 = 9$, not $3 \times 2$. Ranking powers above multiplication lets a power be written bare too, so $5 \times 2^3$ means $5 \times 8 = 40$.

Grouping sits above all of them for a different reason. A pair of parentheses is the one instruction you write by hand. It says “treat what is inside me as one quantity,” and that overrides every ranking below it.

Why same-tier operations go left to right

Multiplication and division sit on the same tier, and so do addition and subtraction. When two operations from one tier appear together, the rule is to work strictly left to right. Without a rule of that kind, some chains of subtractions would give different values depending on how a reader grouped them.

The same reasoning settles division. The expression $16 \div 4 \div 2$ means $(16 \div 4) \div 2 = 4 \div 2 = 2$, not $16 \div (4 \div 2) = 16 \div 2 = 8$. Read same-tier operations strictly from left to right, and you will never have to guess.

What a grouping symbol actually tells you

Grouping symbols are the one place where you, the writer, get to override the ranking of the tiers. A pair of parentheses says “treat everything inside me as one quantity, and settle it before any operation outside is allowed to use it.”

A grouping symbol does not mean “do everything inside first, then ignore the order of operations for the rest of the expression.” Once a group is settled down to a single number, the operations outside it still obey the same tiers as always. Look at $4 + 2 \times (5 - 1)$. The parentheses tell you to settle $5 - 1 = 4$ first, and that group then becomes the single number $4$. But from that moment the rest of the line is back to ordinary rules, so the multiplication still happens before the addition.

When you do want an operation like addition to happen before a multiplication, parentheses are exactly the tool. Writing $(4 + 2) \times 5$ forces the sum first and gives $6 \times 5 = 30$, a different number from the un-grouped $4 + 2 \times 5 = 14$. Grouping is how you bend the order on purpose.

Nested groups: innermost first

When a group sits inside another group, settle the innermost one first and work outward. Brackets and braces behave exactly like parentheses; the different shapes are only there to make each matching pair easy to spot at a glance.

The fraction bar is an invisible grouping symbol

A horizontal fraction bar quietly groups everything above it and everything below it, as though the numerator and the denominator each wore their own parentheses. You evaluate the entire top and the entire bottom first, then divide. This is why writing a quotient as a stacked fraction is so much safer than squeezing it onto one line.

Written on one line, that fraction is $(6 + 4) \div (2 + 3)$, with both pairs of parentheses put in by hand. It is not $6 + 4 \div 2 + 3$, which comes to $6 + 2 + 3 = 11$ under the tiers. Dropping the parentheses changes which numbers are divided, and that is what changes the value from $2$ to $11$. Whenever a stacked fraction has to go onto one line, write those parentheses in.

A larger expression, start to finish

The reliable habit is to scan the whole expression, find the highest tier present, do one operation from it, rewrite the line, and repeat. Writing one step at a time makes each step easier to check.

Common mistakes

You can now read any grouped calculation and get the one value it names. The last lesson of the chapter asks a different question about that same expression. When a number multiplies a group, the calculation can be rewritten in a second form entirely, and the value stays put.