Order of Operations

PEMDAS: the agreed order that makes every expression unambiguous.

The idea

The order of operations is the convention that fixes how every arithmetic expression is read. Without it, $5 + 2 \cdot 6$ is ambiguous: working left to right gives $42$, and multiplying first gives $17$. The convention selects one reading. Scientific calculators and mathematical software generally follow it; a basic calculator may instead work in the order you enter the operations.

It ranks the operations, and we carry out the higher-ranked ones first:

1. Parentheses 2. Exponents 3. Multiplication and Division 4. Addition and Subtraction

The list is remembered as PEMDAS. Multiplication and division share one rank, as do addition and subtraction, and within a rank we work left to right. Parentheses are not an operation but an override: they mark a piece to evaluate first. So the convention reads $5 + 2 \cdot 6$ as $17$, and writing $(5 + 2) \cdot 6$ is how to ask for $42$.

The ranking is not arbitrary. Repeated addition is multiplication, and repeated multiplication is exponentiation, so each rank builds on the one below it, and the operations that bind hardest are carried out first.

Ways to work on it

• Walkthrough. PEMDAS through four contrasting expressions.
• Practice. Random small expressions testing precedence and parentheses.
• Hardest. A nested expression with parentheses, exponent, and product.

Not sure where to start? Take the ten-question placement test.