Number · Integers, powers & roots
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Negatives and the order of operations
Why the order of operations exists, why −3² and (−3)² are different numbers, and how the sign rules for negatives follow from what multiplication means.
Number · Integers, powers & roots
Negatives and the order of operations
Why the order of operations exists, why −3² and (−3)² are different numbers, and how the sign rules for negatives follow from what multiplication means.
Why it works
The order of operations is a convention with a reason. could mean two things; maths picks one so every reader gets the same answer. Powers bind tightest, then multiplication/division, then addition/subtraction — because is shorthand for repeated addition and for repeated multiplication, the shorthands get unpacked first:Left-to-right ( first) gives 28 — a different, wrong number. Brackets exist to override the convention: .
and are different numbers. The square binds tighter than the minus, so means — the minus waits outside. Only a bracket puts the minus inside the square: . This one distinction is worth a mark on almost every calculator paper.
Sign rules aren't magic. : multiplying by means "take copies" — reverse direction, twice the size — so reversing a negative lands positive: . Same-signs multiply or divide to positive, opposite signs to negative: , .
Adding and subtracting negatives: think of a number line. (walk right), (walk left). Subtracting a negative removes a debt — a gain: . The "two minuses make a plus" chant only applies to those touching minus signs — it does NOT turn into ; that's a walk left from , landing .
Substitution with negatives: bracket everything. With : , and . Writing the bracket before substituting is the habit that stops every sign error at once.